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<!DOCTYPE html><!--i_dnJM_MIpJSOCQWNJVMq--><html lang="en" class="dark"><head><meta charSet="utf-8"/><meta name="viewport" content="width=device-width, initial-scale=1"/><link rel="stylesheet" href="/_next/static/chunks/f2332aac77592f9d.css" data-precedence="next"/><link rel="preload" as="script" fetchPriority="low" href="/_next/static/chunks/e83606e8fa9cc796.js"/><script src="/_next/static/chunks/36d6595f0156cd7e.js" async=""></script><script src="/_next/static/chunks/040e9cea20a8d9c7.js" async=""></script><script src="/_next/static/chunks/d4dffb5a0973d49c.js" async=""></script><script src="/_next/static/chunks/turbopack-967f27a9fd33556d.js" async=""></script><script src="/_next/static/chunks/59d0ad1b64f8544e.js" async=""></script><script src="/_next/static/chunks/4d80e004cf4896dd.js" async=""></script><script src="/_next/static/chunks/350ee4303b732916.js" async=""></script><script src="/_next/static/chunks/8de849ca74fc071f.js" async=""></script><script src="/_next/static/chunks/e62b91212ee7f8ff.js" async=""></script><script src="/_next/static/chunks/f19fe44237e54646.js" async=""></script><script src="/_next/static/chunks/cb0a883bafeb6805.js" async=""></script><meta name="theme-color" media="(prefers-color-scheme: dark)" content="#0A0A0A"/><meta name="theme-color" media="(prefers-color-scheme: light)" content="#fff"/><title>QwQ: Reflect Deeply on the Boundaries of the Unknown — Zen LM Blog | Zen LM</title><meta name="description" content="GITHUB HUGGING FACE MODELSCOPE DEMO DISCORD"/><meta property="og:title" content="Zen LM - Open Foundation Models"/><meta property="og:description" content="Zen AI model family by Hanzo AI. 14 frontier models from 4B to 1T+ parameters for code, reasoning, vision, and multimodal tasks."/><meta property="og:url" content="https://zenlm.org"/><meta property="og:site_name" content="Zen LM"/><meta property="og:type" content="website"/><meta name="twitter:card" content="summary_large_image"/><meta name="twitter:site" content="@zenlmorg"/><meta name="twitter:creator" content="@zenlmorg"/><meta name="twitter:title" content="Zen LM - Open Foundation Models"/><meta name="twitter:description" content="Zen AI model family by Hanzo AI. 14 frontier models from 4B to 1T+ parameters for code, reasoning, vision, and multimodal tasks."/><script src="/_next/static/chunks/a6dad97d9634a72d.js" noModule=""></script><script defer src="/zen-decorate.js"></script></head><body class="antialiased"><div hidden=""><!--$--><!--/$--></div><script>((a,b,c,d,e,f,g,h)=>{let i=document.documentElement,j=["light","dark"];function k(b){var c;(Array.isArray(a)?a:[a]).forEach(a=>{let c="class"===a,d=c&&f?e.map(a=>f[a]||a):e;c?(i.classList.remove(...d),i.classList.add(f&&f[b]?f[b]:b)):i.setAttribute(a,b)}),c=b,h&&j.includes(c)&&(i.style.colorScheme=c)}if(d)k(d);else try{let a=localStorage.getItem(b)||c,d=g&&"system"===a?window.matchMedia("(prefers-color-scheme: dark)").matches?"dark":"light":a;k(d)}catch(a){}})("class","theme","system",null,["light","dark"],null,true,true)</script><!--$--><div data-closed="" role="presentation" hidden="" style="user-select:none;-webkit-user-select:none" class="fixed inset-0 z-50 backdrop-blur-xs bg-fd-overlay data-open:animate-fd-fade-in data-closed:animate-fd-fade-out"></div><div class="bg-fd-secondary/50 p-3 empty:hidden"></div><!--/$--><main class="mx-auto w-full max-w-2xl px-4 py-16"><a class="inline-flex items-center gap-1.5 text-sm text-fd-muted-foreground hover:text-fd-foreground mb-10 transition-colors" href="/blog">← Back to Blog</a><div class="mb-8"><time class="text-xs font-mono text-fd-muted-foreground uppercase tracking-wider">November 27, 2024</time><h1 class="text-3xl font-bold mt-2 mb-3">QwQ: Reflect Deeply on the Boundaries of the Unknown</h1><p class="text-fd-muted-foreground text-lg mb-4">GITHUB HUGGING FACE MODELSCOPE DEMO DISCORD</p><div class="flex items-center gap-3 pt-4 border-t border-fd-border"><span class="text-sm text-fd-muted-foreground">By <!-- -->Zen LM Team</span><div class="flex gap-1.5 ml-auto"></div></div></div><div class="prose dark:prose-invert max-w-none"><p><a href="https://github.com/QwenLM/zen" rel="noreferrer noopener" target="_blank">GITHUB</a> <a href="https://huggingface.co/Qwen" rel="noreferrer noopener" target="_blank">HUGGING FACE</a> <a href="https://modelscope.cn/organization/qwen" rel="noreferrer noopener" target="_blank">MODELSCOPE</a> <a href="https://huggingface.co/spaces/Qwen/QwQ-32B-preview" rel="noreferrer noopener" target="_blank">DEMO</a> <a href="https://discord.gg/yPEP2vHTu4" rel="noreferrer noopener" target="_blank">DISCORD</a></p>
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<p><em>Note: This is the pronunciation of QwQ: /kwju:/ , similar to the word “quill”.</em></p>
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<p>What does it mean to think, to question, to understand? These are the deep waters that QwQ (Qwen with Questions) wades into. Like an eternal student of wisdom, it approaches every problem - be it mathematics, code, or knowledge of our world - with genuine wonder and doubt. QwQ embodies that ancient philosophical spirit: it knows that it knows nothing, and that’s precisely what drives its curiosity. Before settling on any answer, it turns inward, questioning its own assumptions, exploring different paths of thought, always seeking deeper truth. Yet, like all seekers of wisdom, QwQ has its limitations. This version is but an early step on a longer journey - a student still learning to walk the path of reasoning. Its thoughts sometimes wander, its answers aren’t always complete, and its wisdom is still growing. But isn’t that the beauty of true learning? To be both capable and humble, knowledgeable yet always questioning? We invite you to explore alongside QwQ, embracing both its insights and its imperfections as part of the endless quest for understanding.</p>
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<h1 class="flex scroll-m-28 flex-row items-center gap-2" id="limitations"><a data-card="" href="#limitations" class="peer">Limitations</a><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-link size-3.5 shrink-0 text-fd-muted-foreground opacity-0 transition-opacity peer-hover:opacity-100" aria-hidden="true"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></h1>
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<p><strong>QwQ-32B-Preview</strong> is an experimental research model developed by the Qwen Team, focused on advancing AI reasoning capabilities. As a preview release, it demonstrates promising analytical abilities while having several important limitations:</p>
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<ol>
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<li><strong>Language Mixing and Code-Switching</strong> : The model may mix languages or switch between them unexpectedly, affecting response clarity.</li>
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<li><strong>Recursive Reasoning Loops</strong> : The model may enter circular reasoning patterns, leading to lengthy responses without a conclusive answer.</li>
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<li><strong>Safety and Ethical Considerations</strong> : The model requires enhanced safety measures to ensure reliable and secure performance, and users should exercise caution when deploying it.</li>
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<li><strong>Performance and Benchmark Limitations</strong> : The model excels in math and coding but has room for improvement in other areas, such as common sense reasoning and nuanced language understanding.</li>
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</ol>
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<h1 class="flex scroll-m-28 flex-row items-center gap-2" id="performance"><a data-card="" href="#performance" class="peer">Performance</a><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-link size-3.5 shrink-0 text-fd-muted-foreground opacity-0 transition-opacity peer-hover:opacity-100" aria-hidden="true"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></h1>
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<p>Through deep exploration and countless trials, we discovered something profound: when given time to ponder, to question, and to reflect, the model’s understanding of mathematics and programming blossoms like a flower opening to the sun. Just as a student grows wiser by carefully examining their work and learning from mistakes, our model achieves deeper insight through patient, thoughtful analysis. This process of careful reflection and self-questioning leads to remarkable breakthroughs in solving complex problems. Our journey of discovery revealed the model’s exceptional ability to tackle some of the most challenging problems in mathematics and programming, including:</p>
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<ul>
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<li>GPQA: A Graduate-Level Google-Proof Q&A Benchmark, a challenging benchmark for evaluating scientific problem-solving abilities through grade school level questions.</li>
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<li>AIME: American Invitation Mathematics Evaluation, which tests mathematical problem solving with arithmetic, algebra, counting, geometry, number theory, and probability and other secondary school math topics.</li>
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<li>MATH-500: The 500 test cases of the MATH benchmark, a comprehensive dataset testing mathematical problem-solving.</li>
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<li>LiveCodeBench: A challenging benchmark for evaluating code generation and problem solving abilities in real-world programming scenarios.</li>
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</ul>
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<p>Specifically, QwQ demonstrates remarkable performance across these benchmarks, achieving impressive scores of 65.2% on GPQA, showcasing its graduate-level scientific reasoning capabilities; 50.0% on AIME, highlighting its strong mathematical problem-solving skills; 90.6% on MATH-500, demonstrating exceptional mathematical comprehension across diverse topics; and 50.0% on LiveCodeBench, validating its robust programming abilities in real-world scenarios. These results underscore QwQ’s significant advancement in analytical and problem-solving capabilities, particularly in technical domains requiring deep reasoning.</p>
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<h1 class="flex scroll-m-28 flex-row items-center gap-2" id="demo-cases"><a data-card="" href="#demo-cases" class="peer">Demo Cases</a><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-link size-3.5 shrink-0 text-fd-muted-foreground opacity-0 transition-opacity peer-hover:opacity-100" aria-hidden="true"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></h1>
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<p>In the following examples, we invite you to witness the profound depths of QwQ-32B-Preview’s contemplative abilities. Like a seeker of wisdom on an endless journey of discovery, the model demonstrates its capacity for deep introspection - questioning its own assumptions, engaging in thoughtful self-dialogue, and carefully examining each step of its reasoning process. Through these cases, you’ll observe how QwQ embodies the timeless understanding that true knowledge emerges not from hasty conclusions, but from patient inquiry and the courage to challenge one’s own certainties in the eternal pursuit of truth.</p>
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<p>Example: Logical Reasoning Next</p>
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<p>User</p>
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<p>Please add a pair of parentheses to the incorrect equation: 1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479, to make the equation true.</p>
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<p>QwQ-32B-Preview</p>
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<p>Let’s tackle this problem step by step. The equation given is:</p>
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<p>1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479</p>
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<p>But it’s incorrect, and I need to add a pair of parentheses to make it true. Okay, first things first, I should understand what the current value of the left side is without any parentheses.</p>
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<p>Let’s recall the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). So, multiplication comes before addition.</p>
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<p>So, calculating step by step:</p>
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<p>2 * 3 = 6</p>
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<p>4 * 5 = 20</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>Now, plugging these back into the equation:</p>
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<p>1 + 6 + 20 + 42 + 72</p>
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<p>Adding these up:</p>
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<p>1 + 6 = 7</p>
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<p>7 + 20 = 27</p>
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<p>27 + 42 = 69</p>
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<p>69 + 72 = 141</p>
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<p>So, without any parentheses, the left side equals 141, but it should be 479. That’s quite a difference. So, I need to add a pair of parentheses to make it equal to 479.</p>
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<p>I need to think about where to place the parentheses to alter the order of operations to achieve the desired result.</p>
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<p>Let me consider different possibilities:</p>
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<ol>
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<li>Parenthesizing individual terms:</li>
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</ol>
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<p>For example, (1 + 2) * 3 + 4 * 5 + 6 * 7 + 8 * 9</p>
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<p>Calculate inside the parentheses first:</p>
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<p>(1 + 2) = 3</p>
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<p>Then, 3 * 3 = 9</p>
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<p>Now, the equation becomes:</p>
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<p>9 + 4 * 5 + 6 * 7 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>4 * 5 = 20</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 9 + 20 + 42 + 72 = 143</p>
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<p>Still not 479.</p>
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<ol start="2">
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<li>Parenthesizing more terms:</li>
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</ol>
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<p>What if I do (1 + 2 * 3) + 4 * 5 + 6 * 7 + 8 * 9</p>
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<p>First, calculate inside the parentheses:</p>
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<p>1 + 2 * 3</p>
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<p>Here, multiplication comes before addition, so 2 * 3 = 6, then 1 + 6 = 7</p>
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<p>So, the equation becomes:</p>
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<p>7 + 4 * 5 + 6 * 7 + 8 * 9</p>
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<p>Again, following order of operations:</p>
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<p>4 * 5 = 20</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 7 + 20 + 42 + 72 = 141</p>
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<p>Same as before.</p>
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<ol start="3">
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<li>Let’s try (1 + 2) * (3 + 4) * 5 + 6 * 7 + 8 * 9</li>
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</ol>
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<p>First, calculate inside the parentheses:</p>
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<p>(1 + 2) = 3</p>
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<p>(3 + 4) = 7</p>
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<p>So, 3 * 7 * 5 = 105</p>
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<p>Now, the equation becomes:</p>
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<p>105 + 6 * 7 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 105 + 42 + 72 = 219</p>
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<p>Still not 479.</p>
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<ol start="4">
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<li>Maybe trying to group more terms:</li>
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</ol>
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<p>(1 + 2 * 3 + 4) * 5 + 6 * 7 + 8 * 9</p>
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<p>First, calculate inside the parentheses:</p>
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<p>1 + 2 * 3 + 4</p>
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<p>Multiplication first: 2 * 3 = 6</p>
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<p>Then, 1 + 6 + 4 = 11</p>
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<p>So, 11 * 5 = 55</p>
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<p>Now, the equation becomes:</p>
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<p>55 + 6 * 7 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 55 + 42 + 72 = 169</p>
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<p>Getting closer, but still not 479.</p>
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<ol start="5">
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<li>Let’s try a different approach. Maybe grouping the additions together:</li>
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</ol>
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<p>1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)</p>
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<p>First, calculate inside the parentheses:</p>
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<p>2 * 3 = 6</p>
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<p>4 * 5 = 20</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 6 + 20 + 42 + 72 = 140</p>
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<p>Then, 1 + 140 = 141</p>
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<p>Same as before.</p>
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<ol start="6">
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<li>Perhaps grouping two multiplications together:</li>
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</ol>
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<p>1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9</p>
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<p>First, calculate inside the parentheses:</p>
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<p>(3 + 4) = 7</p>
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<p>So, 2 * 7 * 5 = 70</p>
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<p>Now, the equation becomes:</p>
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<p>1 + 70 + 6 * 7 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 1 + 70 + 42 + 72 = 185</p>
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<p>Still not 479.</p>
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<ol start="7">
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<li>Maybe grouping more multiplications:</li>
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</ol>
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<p>1 + 2 * 3 + 4 * (5 + 6) * 7 + 8 * 9</p>
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<p>First, calculate inside the parentheses:</p>
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<p>(5 + 6) = 11</p>
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<p>So, 4 * 11 * 7 = 308</p>
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<p>Now, the equation becomes:</p>
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<p>1 + 2 * 3 + 308 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>2 * 3 = 6</p>
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<p>8 * 9 = 72</p>
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<p>So, 1 + 6 + 308 + 72 = 387</p>
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<p>Closer, but still not 479.</p>
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<ol start="8">
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<li>Let’s try grouping the last terms:</li>
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</ol>
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<p>1 + 2 * 3 + 4 * 5 + 6 * (7 + 8) * 9</p>
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<p>First, calculate inside the parentheses:</p>
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<p>(7 + 8) = 15</p>
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<p>So, 6 * 15 * 9 = 810</p>
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<p>Now, the equation becomes:</p>
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<p>1 + 2 * 3 + 4 * 5 + 810</p>
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<p>Following order of operations:</p>
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<p>2 * 3 = 6</p>
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<p>4 * 5 = 20</p>
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<p>So, 1 + 6 + 20 + 810 = 837</p>
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<p>Now that’s way over 479.</p>
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<p>Maybe I need to group terms in a way that subtraction is implied, but since there are no subtractions in the original equation, perhaps I need to consider that.</p>
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<p>Wait, maybe the problem allows for adding parentheses to change the operations indirectly.</p>
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<p>Alternatively, perhaps the parentheses need to group additions in a specific way.</p>
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<p>Let me try grouping the first few terms:</p>
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<p>(1 + 2 * 3 + 4 * 5) + 6 * 7 + 8 * 9</p>
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<p>First, calculate inside the parentheses:</p>
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<p>2 * 3 = 6</p>
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<p>4 * 5 = 20</p>
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<p>So, 1 + 6 + 20 = 27</p>
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<p>Now, the equation becomes:</p>
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<p>27 + 6 * 7 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 27 + 42 + 72 = 141</p>
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<p>Same as before.</p>
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<ol start="9">
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<li>Maybe grouping further:</li>
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</ol>
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<p>(1 + 2 * 3 + 4 * 5 + 6 * 7) + 8 * 9</p>
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<p>First, calculate inside the parentheses:</p>
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<p>2 * 3 = 6</p>
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<p>4 * 5 = 20</p>
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<p>6 * 7 = 42</p>
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<p>So, 1 + 6 + 20 + 42 = 69</p>
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<p>Now, the equation becomes:</p>
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<p>69 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>8 * 9 = 72</p>
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<p>So, 69 + 72 = 141</p>
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<p>Still not 479.</p>
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<ol start="10">
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<li>Let’s try grouping differently:</li>
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</ol>
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<p>1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)</p>
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<p>First, calculate inside the parentheses:</p>
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<p>3 + 4 * 5 + 6 * 7 + 8 * 9</p>
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<p>Following order of operations inside:</p>
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<p>4 * 5 = 20</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 3 + 20 + 42 + 72 = 137</p>
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<p>Now, 2 * 137 = 274</p>
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<p>Then, the equation becomes:</p>
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<p>1 + 274 = 275</p>
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<p>Still not 479.</p>
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<ol start="11">
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<li>Maybe trying to group the additions and multiplications in a different way:</li>
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</ol>
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<p>1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)</p>
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<p>Wait, I did this earlier and got 141.</p>
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<p>Hmm.</p>
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<p>Let me think differently. Maybe I need to group terms to create a larger multiplication.</p>
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<p>For example:</p>
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<p>1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)</p>
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<p>First, calculate inside the parentheses:</p>
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<p>3 + 4 * 5 + 6 * 7 + 8 * 9</p>
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<p>Following order of operations:</p>
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<p>4 * 5 = 20</p>
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<p>6 * 7 = 42</p>
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<p>8 * 9 = 72</p>
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<p>So, 3 + 20 + 42 + 72 = 137</p>
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<p>Now, 2 * 137 = 274</p>
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<p>Then, 1 + 274 = 275</p>
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<p>Still not 479.</p>
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<ol start="12">
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<li>Let’s try another approach:</li>
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</ol>
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<p>(1 + 2) * (3 + 4) * (5 + 6) * (7 + 8) * 9</p>
|
||
<p>Wait, that’s adding multiple sets of parentheses, which is not allowed since I can only add one pair.</p>
|
||
<p>Back to the original problem: only one pair of parentheses can be added.</p>
|
||
<p>Maybe I need to group a subtraction or division, but those operations aren’t present in the original equation.</p>
|
||
<p>Wait, perhaps I need to consider that adding parentheses can create a negative term, but again, there are no subtractions in the original equation.</p>
|
||
<p>Alternatively, maybe the parentheses are meant to group additions in a specific way to create a larger number.</p>
|
||
<p>Let me try grouping more terms together:</p>
|
||
<p>1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)</p>
|
||
<p>First, inside the parentheses:</p>
|
||
<p>3 + 4 * 5 + 6 * 7 + 8 * 9</p>
|
||
<p>Following order of operations:</p>
|
||
<p>4 * 5 = 20</p>
|
||
<p>6 * 7 = 42</p>
|
||
<p>8 * 9 = 72</p>
|
||
<p>So, 3 + 20 + 42 + 72 = 137</p>
|
||
<p>Now, 2 * 137 = 274</p>
|
||
<p>Then, 1 + 274 = 275</p>
|
||
<p>Same as before.</p>
|
||
<ol start="13">
|
||
<li>Maybe trying to group the last few terms:</li>
|
||
</ol>
|
||
<p>1 + 2 * 3 + 4 * 5 + 6 * (7 + 8 * 9)</p>
|
||
<p>First, inside the parentheses:</p>
|
||
<p>7 + 8 * 9</p>
|
||
<p>Following order of operations:</p>
|
||
<p>8 * 9 = 72</p>
|
||
<p>Then, 7 + 72 = 79</p>
|
||
<p>Now, 6 * 79 = 474</p>
|
||
<p>So, the equation becomes:</p>
|
||
<p>1 + 2 * 3 + 4 * 5 + 474</p>
|
||
<p>Following order of operations:</p>
|
||
<p>2 * 3 = 6</p>
|
||
<p>4 * 5 = 20</p>
|
||
<p>So, 1 + 6 + 20 + 474 = 501</p>
|
||
<p>Close to 479, but still not there.</p>
|
||
<ol start="14">
|
||
<li>Maybe adjusting the grouping slightly:</li>
|
||
</ol>
|
||
<p>1 + 2 * 3 + 4 * (5 + 6 * 7 + 8 * 9)</p>
|
||
<p>First, inside the parentheses:</p>
|
||
<p>5 + 6 * 7 + 8 * 9</p>
|
||
<p>Following order of operations:</p>
|
||
<p>6 * 7 = 42</p>
|
||
<p>8 * 9 = 72</p>
|
||
<p>So, 5 + 42 + 72 = 119</p>
|
||
<p>Now, 4 * 119 = 476</p>
|
||
<p>Then, the equation becomes:</p>
|
||
<p>1 + 2 * 3 + 476</p>
|
||
<p>Following order of operations:</p>
|
||
<p>2 * 3 = 6</p>
|
||
<p>So, 1 + 6 + 476 = 483</p>
|
||
<p>Still not 479.</p>
|
||
<ol start="15">
|
||
<li>Let’s try grouping the first part differently:</li>
|
||
</ol>
|
||
<p>(1 + 2 * 3 + 4) * (5 + 6 * 7 + 8 * 9)</p>
|
||
<p>First, calculate inside the first set of parentheses:</p>
|
||
<p>1 + 2 * 3 + 4</p>
|
||
<p>Following order of operations:</p>
|
||
<p>2 * 3 = 6</p>
|
||
<p>Then, 1 + 6 + 4 = 11</p>
|
||
<p>Now, the second set of parentheses:</p>
|
||
<p>5 + 6 * 7 + 8 * 9</p>
|
||
<p>Following order of operations:</p>
|
||
<p>6 * 7 = 42</p>
|
||
<p>8 * 9 = 72</p>
|
||
<p>So, 5 + 42 + 72 = 119</p>
|
||
<p>Now, 11 * 119 = 1309</p>
|
||
<p>Way over 479.</p>
|
||
<ol start="16">
|
||
<li>Maybe trying to group only two terms:</li>
|
||
</ol>
|
||
<p>1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9</p>
|
||
<p>Wait, I did this earlier and got 185.</p>
|
||
<p>Hmm.</p>
|
||
<p>Let me think differently. Maybe I need to group terms to create a division or subtraction, but since those operations aren’t present, perhaps I need to consider that the parentheses can be used to create a fraction or some other operation, but that might be reaching too far.</p>
|
||
<p>Alternatively, maybe the problem allows for adding a pair of parentheses around the entire equation, but that doesn’t make sense.</p>
|
||
<p>Wait, perhaps the parentheses are meant to group additions in a specific way to create a larger number.</p>
|
||
<p>Let me try grouping more terms together for multiplication.</p>
|
||
<p>For example:</p>
|
||
<p>1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9</p>
|
||
<p>First, inside the parentheses:</p>
|
||
<p>3 + 4 * 5 + 6</p>
|
||
<p>Following order of operations:</p>
|
||
<p>4 * 5 = 20</p>
|
||
<p>So, 3 + 20 + 6 = 29</p>
|
||
<p>Now, 2 * 29 * 7 = 406</p>
|
||
<p>Then, the equation becomes:</p>
|
||
<p>1 + 406 + 8 * 9</p>
|
||
<p>Following order of operations:</p>
|
||
<p>8 * 9 = 72</p>
|
||
<p>So, 1 + 406 + 72 = 479</p>
|
||
<p>Wait a minute, that’s exactly 479!</p>
|
||
<p>Let me double-check:</p>
|
||
<p>First, inside the parentheses: 3 + 4 * 5 + 6</p>
|
||
<p>4 * 5 = 20</p>
|
||
<p>Then, 3 + 20 + 6 = 29</p>
|
||
<p>Then, 2 * 29 * 7 = 2 * 203 = 406</p>
|
||
<p>Then, 1 + 406 = 407</p>
|
||
<p>Plus 8 * 9 = 72</p>
|
||
<p>So, 407 + 72 = 479</p>
|
||
<p>Perfect! So, by adding parentheses around (3 + 4 * 5 + 6), and multiplying that sum by 2 and 7, we get the desired result.</p>
|
||
<p>So, the corrected equation is:</p>
|
||
<p>1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9 = 479</p>
|
||
<p><strong>Final Answer</strong></p>
|
||
<p><code>\boxed\{1 + 2 \times (3 + 4 \times 5 + 6) \times 7 + 8 \times 9 = 479\}</code></p>
|
||
<p>Example: Math Next</p>
|
||
<p>User</p>
|
||
<p>Find the least odd prime factor of $2019^8+1$.</p>
|
||
<p>QwQ-32B-Preview</p>
|
||
<p>So I need to find the least odd prime factor of $ 2019^8 + 1 $. Hmm, that seems pretty big, but I think I can break it down step by step.</p>
|
||
<p>First, I know that finding prime factors of large exponents can be tricky, but there might be some properties or theorems that can help simplify the process. Since the expression is $ 2019^8 + 1 $, maybe I can look for patterns or ways to factor this expression.</p>
|
||
<p>I recall that expressions of the form $ a^n + b^n $ can sometimes be factored, especially when $ n $ is a power of 2. In this case, $ n = 8 $, which is $ 2^3 $, so that might be useful.</p>
|
||
<p>Let me see if I can factor $ 2019^8 + 1 $. I know that $ a^4 + 1 $ can be factored as $ (a^4 + 1) = (a^2 + \sqrt{2}a + 1)(a^2 - \sqrt{2}a + 1) $, but that involves irrational numbers, which isn’t helpful for factoring into integers. Maybe there’s another way.</p>
|
||
<p>Wait, perhaps I can write $ 2019^8 + 1 $ as $ (2019^4)^2 + 1^2 $, which resembles the sum of squares. I know that over the integers, sums of squares don’t factor nicely, but maybe there’s a different approach.</p>
|
||
<p>Another thought: maybe I can consider smaller exponents first and see if there’s a pattern or a recurring factor that appears.</p>
|
||
<p>Let’s try computing $ 2019^2 + 1 $, $ 2019^4 + 1 $, and so on, and see if I can find any common factors or patterns.</p>
|
||
<p>First, calculate $ 2019^2 + 1 $:</p>
|
||
<p>$ 2019^2 = 2019 \times 2019 $. I don’t need the exact value right now; maybe I can just keep it as $ 2019^2 $ for now.</p>
|
||
<p>So, $ 2019^2 + 1 = 2019^2 + 1 $. Doesn’t seem immediately helpful.</p>
|
||
<p>Next, $ 2019^4 + 1 $. Again, it’s similar to the previous expression, and I don’t see an obvious factorization.</p>
|
||
<p>Maybe I should try to see if there are any small prime numbers that divide $ 2019^8 + 1 $. Since I’m looking for the least odd prime factor, I can start testing small primes like 3, 5, 7, 11, and so on, to see if any of them divide $ 2019^8 + 1 $.</p>
|
||
<p>Let’s start with 3. To check if 3 divides $ 2019^8 + 1 $, I can compute $ 2019^8 + 1 $ modulo 3.</p>
|
||
<p>First, find $ 2019 $ modulo 3. Since $ 2019 = 3 \times 673 $, $ 2019 \equiv 0 \pmod{3} $. Therefore, $ 2019^8 \equiv 0^8 \equiv 0 \pmod{3} $, and $ 2019^8 + 1 \equiv 0 + 1 \equiv 1 \pmod{3} $. So, 3 does not divide $ 2019^8 + 1 $.</p>
|
||
<p>Next, check 5. Compute $ 2019 $ modulo 5. $ 2019 \div 5 = 403 $ with a remainder of 4, so $ 2019 \equiv 4 \pmod{5} $. Then, $ 2019^8 \equiv 4^8 \pmod{5} $. Since $ 4 \equiv -1 \pmod{5} $, $ 4^8 \equiv (-1)^8 \equiv 1 \pmod{5} $. Therefore, $ 2019^8 + 1 \equiv 1 + 1 \equiv 2 \pmod{5} $, so 5 does not divide $ 2019^8 + 1 $.</p>
|
||
<p>Now, check 7. Find $ 2019 $ modulo 7. $ 2019 \div 7 = 288 $ with a remainder of 3, so $ 2019 \equiv 3 \pmod{7} $. Then, $ 2019^8 \equiv 3^8 \pmod{7} $. Let’s compute $ 3^8 $ modulo 7.</p>
|
||
<p>First, find the pattern of powers of 3 modulo 7:</p>
|
||
<p>$ 3^1 \equiv 3 \pmod{7} $</p>
|
||
<p>$ 3^2 \equiv 9 \equiv 2 \pmod{7} $</p>
|
||
<p>$ 3^3 \equiv 6 \pmod{7} $</p>
|
||
<p>$ 3^4 \equiv 18 \equiv 4 \pmod{7} $</p>
|
||
<p>$ 3^5 \equiv 12 \equiv 5 \pmod{7} $</p>
|
||
<p>$ 3^6 \equiv 15 \equiv 1 \pmod{7} $</p>
|
||
<p>Since $ 3^6 \equiv 1 \pmod{7} $, then $ 3^8 = 3^{6+2} = 3^6 \times 3^2 \equiv 1 \times 2 \equiv 2 \pmod{7} $. Therefore, $ 2019^8 + 1 \equiv 2 + 1 \equiv 3 \pmod{7} $, so 7 does not divide $ 2019^8 + 1 $.</p>
|
||
<p>Next, check 11. Compute $ 2019 $ modulo 11. $ 2019 \div 11 = 183 $ with a remainder of 6, so $ 2019 \equiv 6 \pmod{11} $. Then, $ 2019^8 \equiv 6^8 \pmod{11} $. Let’s find a pattern for powers of 6 modulo 11:</p>
|
||
<p>$ 6^1 \equiv 6 \pmod{11} $</p>
|
||
<p>$ 6^2 \equiv 36 \equiv 3 \pmod{11} $</p>
|
||
<p>$ 6^3 \equiv 18 \equiv 7 \pmod{11} $</p>
|
||
<p>$ 6^4 \equiv 42 \equiv 9 \pmod{11} $</p>
|
||
<p>$ 6^5 \equiv 54 \equiv 10 \pmod{11} $</p>
|
||
<p>$ 6^6 \equiv 60 \equiv 5 \pmod{11} $</p>
|
||
<p>$ 6^7 \equiv 30 \equiv 8 \pmod{11} $</p>
|
||
<p>$ 6^8 \equiv 48 \equiv 4 \pmod{11} $</p>
|
||
<p>Therefore, $ 2019^8 + 1 \equiv 4 + 1 \equiv 5 \pmod{11} $, so 11 does not divide $ 2019^8 + 1 $.</p>
|
||
<p>Continuing this process seems tedious. Maybe there’s a better way to approach this problem.</p>
|
||
<p>Let me think differently. Perhaps I can consider the expression $ 2019^8 + 1 $ and see if it can be expressed in a form that reveals its prime factors.</p>
|
||
<p>I recall that for expressions like $ a^4 + 4b^4 $, there’s a Sophie Germain identity: $ a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2) $. Maybe I can manipulate $ 2019^8 + 1 $ into a similar form.</p>
|
||
<p>Wait, $ 2019^8 + 1 = (2019^4)^2 + 1^2 $. This resembles the sum of squares, but over integers, sums of squares don’t factor nicely. Maybe I can consider complex numbers or some other algebraic identities, but that might not help in finding integer prime factors.</p>
|
||
<p>Alternatively, perhaps I can consider modulo properties to find possible prime factors. For example, if $ p $ is an odd prime that divides $ 2019^8 + 1 $, then $ 2019^8 \equiv -1 \pmod{p} $, which implies $ 2019^{16} \equiv 1 \pmod{p} $. This means that the order of 2019 modulo $ p $ divides 16, and since $ 2019^8 \equiv -1 \pmod{p} $, the order can’t divide 8. Therefore, the order must be 16.</p>
|
||
<p>By Fermat’s Little Theorem, the order of 2019 modulo $ p $ divides $ p - 1 $. Therefore, $ p - 1 $ must be divisible by 16. So, $ p \equiv 1 \pmod{16} $.</p>
|
||
<p>This narrows down the possible primes $ p $ that could divide $ 2019^8 + 1 $ to those primes congruent to 1 modulo 16.</p>
|
||
<p>Let me list some primes that are 1 modulo 16: 17, 97, 113, 193, 241, 257, etc.</p>
|
||
<p>Now, I can test these primes to see if any of them divide $ 2019^8 + 1 $.</p>
|
||
<p>Start with 17. Compute $ 2019 $ modulo 17.</p>
|
||
<p>First, find $ 2019 \div 17 $. 17 times 118 is 2006, so $ 2019 - 2006 = 13 $. Therefore, $ 2019 \equiv 13 \pmod{17} $.</p>
|
||
<p>Then, $ 2019^8 \equiv 13^8 \pmod{17} $.</p>
|
||
<p>Compute $ 13^2 \equiv 169 \equiv 16 \pmod{17} $</p>
|
||
<p>$ 13^4 = (13^2)^2 \equiv 16^2 \equiv 256 \equiv 1 \pmod{17} $</p>
|
||
<p>$ 13^8 = (13^4)^2 \equiv 1^2 \equiv 1 \pmod{17} $</p>
|
||
<p>Therefore, $ 2019^8 + 1 \equiv 1 + 1 \equiv 2 \pmod{17} $, so 17 does not divide $ 2019^8 + 1 $.</p>
|
||
<p>Next, check 97. Compute $ 2019 $ modulo 97.</p>
|
||
<p>First, find how many times 97 fits into 2019. 97 times 20 is 1940, and 2019 - 1940 = 79. So, $ 2019 \equiv 79 \pmod{97} $.</p>
|
||
<p>Then, $ 2019^8 \equiv 79^8 \pmod{97} $.</p>
|
||
<p>This seems a bit cumbersome to compute directly. Maybe I can find a pattern or use Fermat’s Little Theorem, which states that $ a^{p-1} \equiv 1 \pmod{p} $ if $ p $ is prime and $ p $ does not divide $ a $.</p>
|
||
<p>Here, $ p = 97 $, so $ 79^{96} \equiv 1 \pmod{97} $. But I need $ 79^8 $, which is not directly helpful. Maybe I can compute $ 79^2 $, then $ 79^4 $, and so on.</p>
|
||
<p>First, compute $ 79^2 \pmod{97} $:</p>
|
||
<p>$ 79^2 = 6241 $. Now, divide 6241 by 97.</p>
|
||
<p>97 times 64 is 6208, so 6241 - 6208 = 33. Therefore, $ 79^2 \equiv 33 \pmod{97} $.</p>
|
||
<p>Next, $ 79^4 = (79^2)^2 \equiv 33^2 \pmod{97} $.</p>
|
||
<p>$ 33^2 = 1089 $. 97 times 11 is 1067, so 1089 - 1067 = 22. Therefore, $ 79^4 \equiv 22 \pmod{97} $.</p>
|
||
<p>Then, $ 79^8 = (79^4)^2 \equiv 22^2 \pmod{97} $.</p>
|
||
<p>$ 22^2 = 484 $. 97 times 4 is 388, so 484 - 388 = 96. Therefore, $ 79^8 \equiv 96 \pmod{97} $.</p>
|
||
<p>Thus, $ 2019^8 + 1 \equiv 96 + 1 \equiv 97 \equiv 0 \pmod{97} $.</p>
|
||
<p>So, 97 divides $ 2019^8 + 1 $. Since 97 is the smallest prime congruent to 1 modulo 16 that we’ve checked and it divides $ 2019^8 + 1 $, it must be the least odd prime factor of $ 2019^8 + 1 $.</p>
|
||
<p>Therefore, the least odd prime factor is 97.</p>
|
||
<p><strong>Final Answer</strong></p>
|
||
<p><code>\boxed\{97\}</code></p>
|
||
<h1 class="flex scroll-m-28 flex-row items-center gap-2" id="reflections-on-the-journey-ahead"><a data-card="" href="#reflections-on-the-journey-ahead" class="peer">Reflections on the Journey Ahead</a><svg xmlns="http://www.w3.org/2000/svg" width="24" height="24" viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" class="lucide lucide-link size-3.5 shrink-0 text-fd-muted-foreground opacity-0 transition-opacity peer-hover:opacity-100" aria-hidden="true"><path d="M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71"></path><path d="M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71"></path></svg></h1>
|
||
<p>The path to understanding reasoning in large language models branches into many directions, each offering unique insights into the nature of machine intelligence. Our research spans multiple domains - from process reward models that shape learning patterns, to LLM critique that fosters deeper analysis, to multi-step reasoning that builds complex thought, to reinforcement learning with system feedback that enables growth through real-world interactions. While the destination remains unclear, each step brings us closer to understanding how machines think and reason. We don’t know precisely where this journey leads, but we continue forward with unwavering determination - toward truth, toward intelligence, toward the realm where amazing happens.</p></div></main><!--$--><!--/$--><script src="/_next/static/chunks/e83606e8fa9cc796.js" id="_R_" async=""></script><script>(self.__next_f=self.__next_f||[]).push([0])</script><script>self.__next_f.push([1,"1:\"$Sreact.fragment\"\n2:I[106,[\"/_next/static/chunks/59d0ad1b64f8544e.js\"],\"RootProvider\"]\n3:I[53113,[\"/_next/static/chunks/4d80e004cf4896dd.js\",\"/_next/static/chunks/350ee4303b732916.js\"],\"default\"]\n4:I[73211,[\"/_next/static/chunks/4d80e004cf4896dd.js\",\"/_next/static/chunks/350ee4303b732916.js\"],\"default\"]\n5:I[10086,[\"/_next/static/chunks/59d0ad1b64f8544e.js\",\"/_next/static/chunks/8de849ca74fc071f.js\",\"/_next/static/chunks/e62b91212ee7f8ff.js\",\"/_next/static/chunks/f19fe44237e54646.js\",\"/_next/static/chunks/cb0a883bafeb6805.js\"],\"\"]\n7:I[89923,[\"/_next/static/chunks/4d80e004cf4896dd.js\",\"/_next/static/chunks/350ee4303b732916.js\"],\"OutletBoundary\"]\n8:\"$Sreact.suspense\"\na:I[89923,[\"/_next/static/chunks/4d80e004cf4896dd.js\",\"/_next/static/chunks/350ee4303b732916.js\"],\"ViewportBoundary\"]\nc:I[89923,[\"/_next/static/chunks/4d80e004cf4896dd.js\",\"/_next/static/chunks/350ee4303b732916.js\"],\"MetadataBoundary\"]\ne:I[6998,[\"/_next/static/chunks/4d80e004cf4896dd.js\",\"/_next/static/chunks/350ee4303b732916.js\"],\"default\"]\n:HL[\"/_next/static/chunks/f2332aac77592f9d.css\",\"style\"]\n"])</script><script>self.__next_f.push([1,"0:{\"P\":null,\"b\":\"i-dnJM_MIpJSOCQWNJVMq\",\"c\":[\"\",\"blog\",\"qwq-32b-preview\"],\"q\":\"\",\"i\":false,\"f\":[[[\"\",{\"children\":[\"blog\",{\"children\":[[\"slug\",\"qwq-32b-preview\",\"d\"],{\"children\":[\"__PAGE__\",{}]}]}]},\"$undefined\",\"$undefined\",true],[[\"$\",\"$1\",\"c\",{\"children\":[[[\"$\",\"link\",\"0\",{\"rel\":\"stylesheet\",\"href\":\"/_next/static/chunks/f2332aac77592f9d.css\",\"precedence\":\"next\",\"crossOrigin\":\"$undefined\",\"nonce\":\"$undefined\"}],[\"$\",\"script\",\"script-0\",{\"src\":\"/_next/static/chunks/59d0ad1b64f8544e.js\",\"async\":true,\"nonce\":\"$undefined\"}]],[\"$\",\"html\",null,{\"lang\":\"en\",\"className\":\"dark\",\"suppressHydrationWarning\":true,\"children\":[\"$\",\"body\",null,{\"className\":\"antialiased\",\"children\":[\"$\",\"$L2\",null,{\"children\":[\"$\",\"$L3\",null,{\"parallelRouterKey\":\"children\",\"error\":\"$undefined\",\"errorStyles\":\"$undefined\",\"errorScripts\":\"$undefined\",\"template\":[\"$\",\"$L4\",null,{}],\"templateStyles\":\"$undefined\",\"templateScripts\":\"$undefined\",\"notFound\":[[\"$\",\"main\",null,{\"className\":\"flex min-h-screen flex-col items-center justify-center px-4 text-center\",\"children\":[[\"$\",\"div\",null,{\"className\":\"mb-8 opacity-20\",\"children\":[\"$\",\"svg\",null,{\"width\":\"120\",\"height\":\"120\",\"viewBox\":\"0 0 120 120\",\"fill\":\"none\",\"aria-hidden\":\"true\",\"children\":[\"$\",\"circle\",null,{\"cx\":\"60\",\"cy\":\"60\",\"r\":\"50\",\"stroke\":\"currentColor\",\"strokeWidth\":\"3\",\"strokeLinecap\":\"round\",\"strokeDasharray\":\"280 40\"}]}]}],[\"$\",\"p\",null,{\"className\":\"text-xs font-mono uppercase tracking-[0.3em] text-fd-muted-foreground mb-4\",\"children\":\"404\"}],[\"$\",\"h1\",null,{\"className\":\"text-3xl font-semibold mb-3\",\"children\":\"Page not found\"}],[\"$\",\"p\",null,{\"className\":\"text-fd-muted-foreground max-w-sm mb-10\",\"children\":\"This page doesn't exist, or it may have moved. 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gap-2\",\"id\":\"performance\",\"children\":[[\"$\",\"a\",null,{\"data-card\":\"\",\"href\":\"#performance\",\"className\":\"peer\",\"children\":\"Performance\"}],[\"$\",\"svg\",null,{\"ref\":\"$undefined\",\"xmlns\":\"http://www.w3.org/2000/svg\",\"width\":24,\"height\":24,\"viewBox\":\"0 0 24 24\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"strokeWidth\":2,\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"className\":\"lucide lucide-link size-3.5 shrink-0 text-fd-muted-foreground opacity-0 transition-opacity peer-hover:opacity-100\",\"aria-hidden\":true,\"children\":[[\"$\",\"path\",\"1cjeqo\",{\"d\":\"M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71\"}],[\"$\",\"path\",\"19qd67\",{\"d\":\"M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71\"}],\"$undefined\"]}]]}]\n16:[\"$\",\"p\",null,{\"children\":\"Through deep exploration and countless trials, we discovered something profound: when given time to ponder, to question, and to reflect, the model’s understanding of mathematics and programming blossoms like a flower opening to the sun. Just as a student grows wiser by carefully examining their work and learning from mistakes, our model achieves deeper insight through patient, thoughtful analysis. This process of careful reflection and self-questioning leads to remarkable breakthroughs in solving complex problems. Our journey of discovery revealed the model’s exceptional ability to tackle some of the most challenging problems in mathematics and programming, including:\"}]\n17:[\"$\",\"ul\",null,{\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"GPQA: A Graduate-Level Google-Proof Q\u0026A Benchmark, a challenging benchmark for evaluating scientific problem-solving abilities through grade school level questions.\"}],\"\\n\",[\"$\",\"li\",null,{\"children\":\"AIME: American Invitation Mathematics Evaluation, which tests mathematical problem solving with arithmetic, algebra, counting, geometry, number theory, and probability and other secondary school math topics.\"}],\"\\n\",[\"$\",\"li\",null,{\"children\":\"MATH-500: The 500 test cases of the MATH benchmark, a comprehensive dataset testing mathematical problem-solving.\"}],\"\\n\",[\"$\",\"li\",null,{\"children\":\"LiveCodeBench: A challenging benchmark for evaluating code generation and problem solving abilities in real-world programming scenarios.\"}],\"\\n\"]}]\n18:[\"$\",\"p\",null,{\"children\":\"Specifically, QwQ demonstrates remarkable performance across these benchmarks, achieving impressive scores of 65.2% on GPQA, showcasing its graduate-level scientific reasoning capabilities; 50.0% on AIME, highlighting its strong mathematical problem-solving skills; 90.6% on MATH-500, demonstrating exceptional mathematical comprehension across diverse topics; and 50.0% on LiveCodeBench, validating its robust programming abilities in real-world scenarios. These results underscore QwQ’s significant advancement in analytical and problem-solving capabilities, particularly in technical domains requiring deep reasoning.\"}]\n19:[\"$\",\"h1\",null,{\"className\":\"flex scroll-m-28 flex-row items-center gap-2\",\"id\":\"demo-cases\",\"children\":[[\"$\",\"a\",null,{\"data-card\":\"\",\"href\":\"#demo-cases\",\"className\":\"peer\",\"children\":\"Demo Cases\"}],[\"$\",\"svg\","])</script><script>self.__next_f.push([1,"null,{\"ref\":\"$undefined\",\"xmlns\":\"http://www.w3.org/2000/svg\",\"width\":24,\"height\":24,\"viewBox\":\"0 0 24 24\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"strokeWidth\":2,\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"className\":\"lucide lucide-link size-3.5 shrink-0 text-fd-muted-foreground opacity-0 transition-opacity peer-hover:opacity-100\",\"aria-hidden\":true,\"children\":[[\"$\",\"path\",\"1cjeqo\",{\"d\":\"M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71\"}],[\"$\",\"path\",\"19qd67\",{\"d\":\"M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71\"}],\"$undefined\"]}]]}]\n1a:[\"$\",\"p\",null,{\"children\":\"In the following examples, we invite you to witness the profound depths of QwQ-32B-Preview’s contemplative abilities. Like a seeker of wisdom on an endless journey of discovery, the model demonstrates its capacity for deep introspection - questioning its own assumptions, engaging in thoughtful self-dialogue, and carefully examining each step of its reasoning process. Through these cases, you’ll observe how QwQ embodies the timeless understanding that true knowledge emerges not from hasty conclusions, but from patient inquiry and the courage to challenge one’s own certainties in the eternal pursuit of truth.\"}]\n1b:[\"$\",\"p\",null,{\"children\":\"Example: Logical Reasoning Next\"}]\n1c:[\"$\",\"p\",null,{\"children\":\"User\"}]\n1d:[\"$\",\"p\",null,{\"children\":\"Please add a pair of parentheses to the incorrect equation: 1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479, to make the equation true.\"}]\n1e:[\"$\",\"p\",null,{\"children\":\"QwQ-32B-Preview\"}]\n1f:[\"$\",\"p\",null,{\"children\":\"Let’s tackle this problem step by step. The equation given is:\"}]\n20:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4 * 5 + 6 * 7 + 8 * 9 = 479\"}]\n21:[\"$\",\"p\",null,{\"children\":\"But it’s incorrect, and I need to add a pair of parentheses to make it true. Okay, first things first, I should understand what the current value of the left side is without any parentheses.\"}]\n22:[\"$\",\"p\",null,{\"children\":\"Let’s recall the order of operations: PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). So, multiplication comes before addition.\"}]\n23:[\"$\",\"p\",null,{\"children\":\"So, calculating step by step:\"}]\n24:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\n25:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n26:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n27:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n28:[\"$\",\"p\",null,{\"children\":\"Now, plugging these back into the equation:\"}]\n29:[\"$\",\"p\",null,{\"children\":\"1 + 6 + 20 + 42 + 72\"}]\n2a:[\"$\",\"p\",null,{\"children\":\"Adding these up:\"}]\n2b:[\"$\",\"p\",null,{\"children\":\"1 + 6 = 7\"}]\n2c:[\"$\",\"p\",null,{\"children\":\"7 + 20 = 27\"}]\n2d:[\"$\",\"p\",null,{\"children\":\"27 + 42 = 69\"}]\n2e:[\"$\",\"p\",null,{\"children\":\"69 + 72 = 141\"}]\n2f:[\"$\",\"p\",null,{\"children\":\"So, without any parentheses, the left side equals 141, but it should be 479. That’s quite a difference. So, I need to add a pair of parentheses to make it equal to 479.\"}]\n30:[\"$\",\"p\",null,{\"children\":\"I need to think about where to place the parentheses to alter the order of operations to achieve the desired result.\"}]\n31:[\"$\",\"p\",null,{\"children\":\"Let me consider different possibilities:\"}]\n32:[\"$\",\"ol\",null,{\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Parenthesizing individual terms:\"}],\"\\n\"]}]\n33:[\"$\",\"p\",null,{\"children\":\"For example, (1 + 2) * 3 + 4 * 5 + 6 * 7 + 8 * 9\"}]\n34:[\"$\",\"p\",null,{\"children\":\"Calculate inside the parentheses first:\"}]\n35:[\"$\",\"p\",null,{\"children\":\"(1 + 2) = 3\"}]\n36:[\"$\",\"p\",null,{\"children\":\"Then, 3 * 3 = 9\"}]\n37:[\"$\",\"p\",null,{\"children\":\"Now, the equation becomes:\"}]\n38:[\"$\",\"p\",null,{\"children\":\"9 + 4 * 5 + 6 * 7 + 8 * 9\"}]\n39:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n3a:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n3b:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n3c:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n3d:[\"$\",\"p\",null,{\"children\":\"So, 9 + 20 + 42 + 72 = 143\"}]\n3e:[\"$\",\"p\",null,{\"children\":\"Still not 479.\"}]\n3f:[\"$\",\"ol\",null,{\"start\":\"2\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Parenthesizing more terms:\"}],\"\\n\"]}]\n40:[\"$\",\"p\",null,{\"children\":\"What if "])</script><script>self.__next_f.push([1,"I do (1 + 2 * 3) + 4 * 5 + 6 * 7 + 8 * 9\"}]\n41:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n42:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3\"}]\n43:[\"$\",\"p\",null,{\"children\":\"Here, multiplication comes before addition, so 2 * 3 = 6, then 1 + 6 = 7\"}]\n44:[\"$\",\"p\",null,{\"children\":\"So, the equation becomes:\"}]\n45:[\"$\",\"p\",null,{\"children\":\"7 + 4 * 5 + 6 * 7 + 8 * 9\"}]\n46:[\"$\",\"p\",null,{\"children\":\"Again, following order of operations:\"}]\n47:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n48:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n49:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n4a:[\"$\",\"p\",null,{\"children\":\"So, 7 + 20 + 42 + 72 = 141\"}]\n4b:[\"$\",\"p\",null,{\"children\":\"Same as before.\"}]\n4c:[\"$\",\"ol\",null,{\"start\":\"3\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Let’s try (1 + 2) * (3 + 4) * 5 + 6 * 7 + 8 * 9\"}],\"\\n\"]}]\n4d:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n4e:[\"$\",\"p\",null,{\"children\":\"(1 + 2) = 3\"}]\n4f:[\"$\",\"p\",null,{\"children\":\"(3 + 4) = 7\"}]\n50:[\"$\",\"p\",null,{\"children\":\"So, 3 * 7 * 5 = 105\"}]\n51:[\"$\",\"p\",null,{\"children\":\"Now, the equation becomes:\"}]\n52:[\"$\",\"p\",null,{\"children\":\"105 + 6 * 7 + 8 * 9\"}]\n53:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n54:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n55:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n56:[\"$\",\"p\",null,{\"children\":\"So, 105 + 42 + 72 = 219\"}]\n57:[\"$\",\"p\",null,{\"children\":\"Still not 479.\"}]\n58:[\"$\",\"ol\",null,{\"start\":\"4\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Maybe trying to group more terms:\"}],\"\\n\"]}]\n59:[\"$\",\"p\",null,{\"children\":\"(1 + 2 * 3 + 4) * 5 + 6 * 7 + 8 * 9\"}]\n5a:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n5b:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4\"}]\n5c:[\"$\",\"p\",null,{\"children\":\"Multiplication first: 2 * 3 = 6\"}]\n5d:[\"$\",\"p\",null,{\"children\":\"Then, 1 + 6 + 4 = 11\"}]\n5e:[\"$\",\"p\",null,{\"children\":\"So, 11 * 5 = 55\"}]\n5f:[\"$\",\"p\",null,{\"children\":\"Now, the equation becomes:\"}]\n60:[\"$\",\"p\",null,{\"children\":\"55 + 6 * 7 + 8 * 9\"}]\n61:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n62:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n63:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n64:[\"$\",\"p\",null,{\"children\":\"So, 55 + 42 + 72 = 169\"}]\n65:[\"$\",\"p\",null,{\"children\":\"Getting closer, but still not 479.\"}]\n66:[\"$\",\"ol\",null,{\"start\":\"5\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Let’s try a different approach. Maybe grouping the additions together:\"}],\"\\n\"]}]\n67:[\"$\",\"p\",null,{\"children\":\"1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)\"}]\n68:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n69:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\n6a:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n6b:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n6c:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n6d:[\"$\",\"p\",null,{\"children\":\"So, 6 + 20 + 42 + 72 = 140\"}]\n6e:[\"$\",\"p\",null,{\"children\":\"Then, 1 + 140 = 141\"}]\n6f:[\"$\",\"p\",null,{\"children\":\"Same as before.\"}]\n70:[\"$\",\"ol\",null,{\"start\":\"6\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Perhaps grouping two multiplications together:\"}],\"\\n\"]}]\n71:[\"$\",\"p\",null,{\"children\":\"1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9\"}]\n72:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n73:[\"$\",\"p\",null,{\"children\":\"(3 + 4) = 7\"}]\n74:[\"$\",\"p\",null,{\"children\":\"So, 2 * 7 * 5 = 70\"}]\n75:[\"$\",\"p\",null,{\"children\":\"Now, the equation becomes:\"}]\n76:[\"$\",\"p\",null,{\"children\":\"1 + 70 + 6 * 7 + 8 * 9\"}]\n77:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n78:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n79:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n7a:[\"$\",\"p\",null,{\"children\":\"So, 1 + 70 + 42 + 72 = 185\"}]\n7b:[\"$\",\"p\",null,{\"children\":\"Still not 479.\"}]\n7c:[\"$\",\"ol\",null,{\"start\":\"7\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Maybe grouping more multiplications:\"}],\"\\n\"]}]\n7d:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4 * (5 + 6) * 7 + 8 * 9\"}]\n7e:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n7f:[\"$\",\"p\",null,{\"children\":\"(5 + 6) = 11\"}]\n80:[\"$\",\"p\",null,{\"children\":\"So, 4 * 11 * 7 = 308\"}]\n81:[\"$\",\"p\",null,{\"children\":\"Now, the equation bec"])</script><script>self.__next_f.push([1,"omes:\"}]\n82:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 308 + 8 * 9\"}]\n83:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n84:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\n85:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n86:[\"$\",\"p\",null,{\"children\":\"So, 1 + 6 + 308 + 72 = 387\"}]\n87:[\"$\",\"p\",null,{\"children\":\"Closer, but still not 479.\"}]\n88:[\"$\",\"ol\",null,{\"start\":\"8\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Let’s try grouping the last terms:\"}],\"\\n\"]}]\n89:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4 * 5 + 6 * (7 + 8) * 9\"}]\n8a:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n8b:[\"$\",\"p\",null,{\"children\":\"(7 + 8) = 15\"}]\n8c:[\"$\",\"p\",null,{\"children\":\"So, 6 * 15 * 9 = 810\"}]\n8d:[\"$\",\"p\",null,{\"children\":\"Now, the equation becomes:\"}]\n8e:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4 * 5 + 810\"}]\n8f:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n90:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\n91:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n92:[\"$\",\"p\",null,{\"children\":\"So, 1 + 6 + 20 + 810 = 837\"}]\n93:[\"$\",\"p\",null,{\"children\":\"Now that’s way over 479.\"}]\n94:[\"$\",\"p\",null,{\"children\":\"Maybe I need to group terms in a way that subtraction is implied, but since there are no subtractions in the original equation, perhaps I need to consider that.\"}]\n95:[\"$\",\"p\",null,{\"children\":\"Wait, maybe the problem allows for adding parentheses to change the operations indirectly.\"}]\n96:[\"$\",\"p\",null,{\"children\":\"Alternatively, perhaps the parentheses need to group additions in a specific way.\"}]\n97:[\"$\",\"p\",null,{\"children\":\"Let me try grouping the first few terms:\"}]\n98:[\"$\",\"p\",null,{\"children\":\"(1 + 2 * 3 + 4 * 5) + 6 * 7 + 8 * 9\"}]\n99:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\n9a:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\n9b:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n9c:[\"$\",\"p\",null,{\"children\":\"So, 1 + 6 + 20 = 27\"}]\n9d:[\"$\",\"p\",null,{\"children\":\"Now, the equation becomes:\"}]\n9e:[\"$\",\"p\",null,{\"children\":\"27 + 6 * 7 + 8 * 9\"}]\n9f:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\na0:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\na1:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\na2:[\"$\",\"p\",null,{\"children\":\"So, 27 + 42 + 72 = 141\"}]\na3:[\"$\",\"p\",null,{\"children\":\"Same as before.\"}]\na4:[\"$\",\"ol\",null,{\"start\":\"9\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Maybe grouping further:\"}],\"\\n\"]}]\na5:[\"$\",\"p\",null,{\"children\":\"(1 + 2 * 3 + 4 * 5 + 6 * 7) + 8 * 9\"}]\na6:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\na7:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\na8:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\na9:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\naa:[\"$\",\"p\",null,{\"children\":\"So, 1 + 6 + 20 + 42 = 69\"}]\nab:[\"$\",\"p\",null,{\"children\":\"Now, the equation becomes:\"}]\nac:[\"$\",\"p\",null,{\"children\":\"69 + 8 * 9\"}]\nad:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\nae:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\naf:[\"$\",\"p\",null,{\"children\":\"So, 69 + 72 = 141\"}]\nb0:[\"$\",\"p\",null,{\"children\":\"Still not 479.\"}]\nb1:[\"$\",\"ol\",null,{\"start\":\"10\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Let’s try grouping differently:\"}],\"\\n\"]}]\nb2:[\"$\",\"p\",null,{\"children\":\"1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)\"}]\nb3:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\nb4:[\"$\",\"p\",null,{\"children\":\"3 + 4 * 5 + 6 * 7 + 8 * 9\"}]\nb5:[\"$\",\"p\",null,{\"children\":\"Following order of operations inside:\"}]\nb6:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\nb7:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\nb8:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\nb9:[\"$\",\"p\",null,{\"children\":\"So, 3 + 20 + 42 + 72 = 137\"}]\nba:[\"$\",\"p\",null,{\"children\":\"Now, 2 * 137 = 274\"}]\nbb:[\"$\",\"p\",null,{\"children\":\"Then, the equation becomes:\"}]\nbc:[\"$\",\"p\",null,{\"children\":\"1 + 274 = 275\"}]\nbd:[\"$\",\"p\",null,{\"children\":\"Still not 479.\"}]\nbe:[\"$\",\"ol\",null,{\"start\":\"11\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Maybe trying to group the additions and multiplications in a different way:\"}],\"\\n\"]}]\nbf:[\"$\",\"p\",null,{\"children\":\"1 + (2 * 3 + 4 * 5 + 6 * 7 + 8 * 9)\"}]\nc0:[\"$\",\"p\",null,{\"children\":\"Wait, I did this earlier and got "])</script><script>self.__next_f.push([1,"141.\"}]\nc1:[\"$\",\"p\",null,{\"children\":\"Hmm.\"}]\nc2:[\"$\",\"p\",null,{\"children\":\"Let me think differently. Maybe I need to group terms to create a larger multiplication.\"}]\nc3:[\"$\",\"p\",null,{\"children\":\"For example:\"}]\nc4:[\"$\",\"p\",null,{\"children\":\"1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)\"}]\nc5:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the parentheses:\"}]\nc6:[\"$\",\"p\",null,{\"children\":\"3 + 4 * 5 + 6 * 7 + 8 * 9\"}]\nc7:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\nc8:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\nc9:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\nca:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\ncb:[\"$\",\"p\",null,{\"children\":\"So, 3 + 20 + 42 + 72 = 137\"}]\ncc:[\"$\",\"p\",null,{\"children\":\"Now, 2 * 137 = 274\"}]\ncd:[\"$\",\"p\",null,{\"children\":\"Then, 1 + 274 = 275\"}]\nce:[\"$\",\"p\",null,{\"children\":\"Still not 479.\"}]\ncf:[\"$\",\"ol\",null,{\"start\":\"12\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Let’s try another approach:\"}],\"\\n\"]}]\nd0:[\"$\",\"p\",null,{\"children\":\"(1 + 2) * (3 + 4) * (5 + 6) * (7 + 8) * 9\"}]\nd1:[\"$\",\"p\",null,{\"children\":\"Wait, that’s adding multiple sets of parentheses, which is not allowed since I can only add one pair.\"}]\nd2:[\"$\",\"p\",null,{\"children\":\"Back to the original problem: only one pair of parentheses can be added.\"}]\nd3:[\"$\",\"p\",null,{\"children\":\"Maybe I need to group a subtraction or division, but those operations aren’t present in the original equation.\"}]\nd4:[\"$\",\"p\",null,{\"children\":\"Wait, perhaps I need to consider that adding parentheses can create a negative term, but again, there are no subtractions in the original equation.\"}]\nd5:[\"$\",\"p\",null,{\"children\":\"Alternatively, maybe the parentheses are meant to group additions in a specific way to create a larger number.\"}]\nd6:[\"$\",\"p\",null,{\"children\":\"Let me try grouping more terms together:\"}]\nd7:[\"$\",\"p\",null,{\"children\":\"1 + 2 * (3 + 4 * 5 + 6 * 7 + 8 * 9)\"}]\nd8:[\"$\",\"p\",null,{\"children\":\"First, inside the parentheses:\"}]\nd9:[\"$\",\"p\",null,{\"children\":\"3 + 4 * 5 + 6 * 7 + 8 * 9\"}]\nda:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\ndb:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\ndc:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\ndd:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\nde:[\"$\",\"p\",null,{\"children\":\"So, 3 + 20 + 42 + 72 = 137\"}]\ndf:[\"$\",\"p\",null,{\"children\":\"Now, 2 * 137 = 274\"}]\ne0:[\"$\",\"p\",null,{\"children\":\"Then, 1 + 274 = 275\"}]\ne1:[\"$\",\"p\",null,{\"children\":\"Same as before.\"}]\ne2:[\"$\",\"ol\",null,{\"start\":\"13\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Maybe trying to group the last few terms:\"}],\"\\n\"]}]\ne3:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4 * 5 + 6 * (7 + 8 * 9)\"}]\ne4:[\"$\",\"p\",null,{\"children\":\"First, inside the parentheses:\"}]\ne5:[\"$\",\"p\",null,{\"children\":\"7 + 8 * 9\"}]\ne6:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\ne7:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\ne8:[\"$\",\"p\",null,{\"children\":\"Then, 7 + 72 = 79\"}]\ne9:[\"$\",\"p\",null,{\"children\":\"Now, 6 * 79 = 474\"}]\nea:[\"$\",\"p\",null,{\"children\":\"So, the equation becomes:\"}]\neb:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4 * 5 + 474\"}]\nec:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\ned:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\nee:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\nef:[\"$\",\"p\",null,{\"children\":\"So, 1 + 6 + 20 + 474 = 501\"}]\nf0:[\"$\",\"p\",null,{\"children\":\"Close to 479, but still not there.\"}]\nf1:[\"$\",\"ol\",null,{\"start\":\"14\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Maybe adjusting the grouping slightly:\"}],\"\\n\"]}]\nf2:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4 * (5 + 6 * 7 + 8 * 9)\"}]\nf3:[\"$\",\"p\",null,{\"children\":\"First, inside the parentheses:\"}]\nf4:[\"$\",\"p\",null,{\"children\":\"5 + 6 * 7 + 8 * 9\"}]\nf5:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\nf6:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\nf7:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\nf8:[\"$\",\"p\",null,{\"children\":\"So, 5 + 42 + 72 = 119\"}]\nf9:[\"$\",\"p\",null,{\"children\":\"Now, 4 * 119 = 476\"}]\nfa:[\"$\",\"p\",null,{\"children\":\"Then, the equation becomes:\"}]\nfb:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 476\"}]\nfc:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\nfd:[\"$\",\"p\",null,{\"children\":\"2 *"])</script><script>self.__next_f.push([1," 3 = 6\"}]\nfe:[\"$\",\"p\",null,{\"children\":\"So, 1 + 6 + 476 = 483\"}]\nff:[\"$\",\"p\",null,{\"children\":\"Still not 479.\"}]\n100:[\"$\",\"ol\",null,{\"start\":\"15\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Let’s try grouping the first part differently:\"}],\"\\n\"]}]\n101:[\"$\",\"p\",null,{\"children\":\"(1 + 2 * 3 + 4) * (5 + 6 * 7 + 8 * 9)\"}]\n102:[\"$\",\"p\",null,{\"children\":\"First, calculate inside the first set of parentheses:\"}]\n103:[\"$\",\"p\",null,{\"children\":\"1 + 2 * 3 + 4\"}]\n104:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n105:[\"$\",\"p\",null,{\"children\":\"2 * 3 = 6\"}]\n106:[\"$\",\"p\",null,{\"children\":\"Then, 1 + 6 + 4 = 11\"}]\n107:[\"$\",\"p\",null,{\"children\":\"Now, the second set of parentheses:\"}]\n108:[\"$\",\"p\",null,{\"children\":\"5 + 6 * 7 + 8 * 9\"}]\n109:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n10a:[\"$\",\"p\",null,{\"children\":\"6 * 7 = 42\"}]\n10b:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n10c:[\"$\",\"p\",null,{\"children\":\"So, 5 + 42 + 72 = 119\"}]\n10d:[\"$\",\"p\",null,{\"children\":\"Now, 11 * 119 = 1309\"}]\n10e:[\"$\",\"p\",null,{\"children\":\"Way over 479.\"}]\n10f:[\"$\",\"ol\",null,{\"start\":\"16\",\"children\":[\"\\n\",[\"$\",\"li\",null,{\"children\":\"Maybe trying to group only two terms:\"}],\"\\n\"]}]\n110:[\"$\",\"p\",null,{\"children\":\"1 + 2 * (3 + 4) * 5 + 6 * 7 + 8 * 9\"}]\n111:[\"$\",\"p\",null,{\"children\":\"Wait, I did this earlier and got 185.\"}]\n112:[\"$\",\"p\",null,{\"children\":\"Hmm.\"}]\n113:[\"$\",\"p\",null,{\"children\":\"Let me think differently. Maybe I need to group terms to create a division or subtraction, but since those operations aren’t present, perhaps I need to consider that the parentheses can be used to create a fraction or some other operation, but that might be reaching too far.\"}]\n114:[\"$\",\"p\",null,{\"children\":\"Alternatively, maybe the problem allows for adding a pair of parentheses around the entire equation, but that doesn’t make sense.\"}]\n115:[\"$\",\"p\",null,{\"children\":\"Wait, perhaps the parentheses are meant to group additions in a specific way to create a larger number.\"}]\n116:[\"$\",\"p\",null,{\"children\":\"Let me try grouping more terms together for multiplication.\"}]\n117:[\"$\",\"p\",null,{\"children\":\"For example:\"}]\n118:[\"$\",\"p\",null,{\"children\":\"1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9\"}]\n119:[\"$\",\"p\",null,{\"children\":\"First, inside the parentheses:\"}]\n11a:[\"$\",\"p\",null,{\"children\":\"3 + 4 * 5 + 6\"}]\n11b:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n11c:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n11d:[\"$\",\"p\",null,{\"children\":\"So, 3 + 20 + 6 = 29\"}]\n11e:[\"$\",\"p\",null,{\"children\":\"Now, 2 * 29 * 7 = 406\"}]\n11f:[\"$\",\"p\",null,{\"children\":\"Then, the equation becomes:\"}]\n120:[\"$\",\"p\",null,{\"children\":\"1 + 406 + 8 * 9\"}]\n121:[\"$\",\"p\",null,{\"children\":\"Following order of operations:\"}]\n122:[\"$\",\"p\",null,{\"children\":\"8 * 9 = 72\"}]\n123:[\"$\",\"p\",null,{\"children\":\"So, 1 + 406 + 72 = 479\"}]\n124:[\"$\",\"p\",null,{\"children\":\"Wait a minute, that’s exactly 479!\"}]\n125:[\"$\",\"p\",null,{\"children\":\"Let me double-check:\"}]\n126:[\"$\",\"p\",null,{\"children\":\"First, inside the parentheses: 3 + 4 * 5 + 6\"}]\n127:[\"$\",\"p\",null,{\"children\":\"4 * 5 = 20\"}]\n128:[\"$\",\"p\",null,{\"children\":\"Then, 3 + 20 + 6 = 29\"}]\n129:[\"$\",\"p\",null,{\"children\":\"Then, 2 * 29 * 7 = 2 * 203 = 406\"}]\n12a:[\"$\",\"p\",null,{\"children\":\"Then, 1 + 406 = 407\"}]\n12b:[\"$\",\"p\",null,{\"children\":\"Plus 8 * 9 = 72\"}]\n12c:[\"$\",\"p\",null,{\"children\":\"So, 407 + 72 = 479\"}]\n12d:[\"$\",\"p\",null,{\"children\":\"Perfect! So, by adding parentheses around (3 + 4 * 5 + 6), and multiplying that sum by 2 and 7, we get the desired result.\"}]\n12e:[\"$\",\"p\",null,{\"children\":\"So, the corrected equation is:\"}]\n12f:[\"$\",\"p\",null,{\"children\":\"1 + 2 * (3 + 4 * 5 + 6) * 7 + 8 * 9 = 479\"}]\n130:[\"$\",\"p\",null,{\"children\":[\"$\",\"strong\",null,{\"children\":\"Final Answer\"}]}]\n131:[\"$\",\"p\",null,{\"children\":[\"$\",\"code\",null,{\"children\":\"\\\\boxed\\\\{1 + 2 \\\\times (3 + 4 \\\\times 5 + 6) \\\\times 7 + 8 \\\\times 9 = 479\\\\}\"}]}]\n132:[\"$\",\"p\",null,{\"children\":\"Example: Math Next\"}]\n133:[\"$\",\"p\",null,{\"children\":\"User\"}]\n134:[\"$\",\"p\",null,{\"children\":\"Find the least odd prime factor of $2019^8+1$.\"}]\n135:[\"$\",\"p\",null,{\"children\":\"QwQ-32B-Preview\"}]\n136:[\"$\",\"p\",nu"])</script><script>self.__next_f.push([1,"ll,{\"children\":\"So I need to find the least odd prime factor of $ 2019^8 + 1 $. Hmm, that seems pretty big, but I think I can break it down step by step.\"}]\n137:[\"$\",\"p\",null,{\"children\":\"First, I know that finding prime factors of large exponents can be tricky, but there might be some properties or theorems that can help simplify the process. Since the expression is $ 2019^8 + 1 $, maybe I can look for patterns or ways to factor this expression.\"}]\n138:[\"$\",\"p\",null,{\"children\":\"I recall that expressions of the form $ a^n + b^n $ can sometimes be factored, especially when $ n $ is a power of 2. In this case, $ n = 8 $, which is $ 2^3 $, so that might be useful.\"}]\n139:[\"$\",\"p\",null,{\"children\":\"Let me see if I can factor $ 2019^8 + 1 $. I know that $ a^4 + 1 $ can be factored as $ (a^4 + 1) = (a^2 + \\\\sqrt{2}a + 1)(a^2 - \\\\sqrt{2}a + 1) $, but that involves irrational numbers, which isn’t helpful for factoring into integers. Maybe there’s another way.\"}]\n13a:[\"$\",\"p\",null,{\"children\":\"Wait, perhaps I can write $ 2019^8 + 1 $ as $ (2019^4)^2 + 1^2 $, which resembles the sum of squares. I know that over the integers, sums of squares don’t factor nicely, but maybe there’s a different approach.\"}]\n13b:[\"$\",\"p\",null,{\"children\":\"Another thought: maybe I can consider smaller exponents first and see if there’s a pattern or a recurring factor that appears.\"}]\n13c:[\"$\",\"p\",null,{\"children\":\"Let’s try computing $ 2019^2 + 1 $, $ 2019^4 + 1 $, and so on, and see if I can find any common factors or patterns.\"}]\n13d:[\"$\",\"p\",null,{\"children\":\"First, calculate $ 2019^2 + 1 $:\"}]\n13e:[\"$\",\"p\",null,{\"children\":\"$$ 2019^2 = 2019 \\\\times 2019 $. I don’t need the exact value right now; maybe I can just keep it as $ 2019^2 $ for now.\"}]\n13f:[\"$\",\"p\",null,{\"children\":\"So, $ 2019^2 + 1 = 2019^2 + 1 $. Doesn’t seem immediately helpful.\"}]\n140:[\"$\",\"p\",null,{\"children\":\"Next, $ 2019^4 + 1 $. Again, it’s similar to the previous expression, and I don’t see an obvious factorization.\"}]\n141:[\"$\",\"p\",null,{\"children\":\"Maybe I should try to see if there are any small prime numbers that divide $ 2019^8 + 1 $. Since I’m looking for the least odd prime factor, I can start testing small primes like 3, 5, 7, 11, and so on, to see if any of them divide $ 2019^8 + 1 $.\"}]\n142:[\"$\",\"p\",null,{\"children\":\"Let’s start with 3. To check if 3 divides $ 2019^8 + 1 $, I can compute $ 2019^8 + 1 $ modulo 3.\"}]\n143:[\"$\",\"p\",null,{\"children\":\"First, find $ 2019 $ modulo 3. Since $ 2019 = 3 \\\\times 673 $, $ 2019 \\\\equiv 0 \\\\pmod{3} $. Therefore, $ 2019^8 \\\\equiv 0^8 \\\\equiv 0 \\\\pmod{3} $, and $ 2019^8 + 1 \\\\equiv 0 + 1 \\\\equiv 1 \\\\pmod{3} $. So, 3 does not divide $ 2019^8 + 1 $.\"}]\n144:[\"$\",\"p\",null,{\"children\":\"Next, check 5. Compute $ 2019 $ modulo 5. $ 2019 \\\\div 5 = 403 $ with a remainder of 4, so $ 2019 \\\\equiv 4 \\\\pmod{5} $. Then, $ 2019^8 \\\\equiv 4^8 \\\\pmod{5} $. Since $ 4 \\\\equiv -1 \\\\pmod{5} $, $ 4^8 \\\\equiv (-1)^8 \\\\equiv 1 \\\\pmod{5} $. Therefore, $ 2019^8 + 1 \\\\equiv 1 + 1 \\\\equiv 2 \\\\pmod{5} $, so 5 does not divide $ 2019^8 + 1 $.\"}]\n145:[\"$\",\"p\",null,{\"children\":\"Now, check 7. Find $ 2019 $ modulo 7. $ 2019 \\\\div 7 = 288 $ with a remainder of 3, so $ 2019 \\\\equiv 3 \\\\pmod{7} $. Then, $ 2019^8 \\\\equiv 3^8 \\\\pmod{7} $. Let’s compute $ 3^8 $ modulo 7.\"}]\n146:[\"$\",\"p\",null,{\"children\":\"First, find the pattern of powers of 3 modulo 7:\"}]\n147:[\"$\",\"p\",null,{\"children\":\"$$ 3^1 \\\\equiv 3 \\\\pmod{7} $\"}]\n148:[\"$\",\"p\",null,{\"children\":\"$$ 3^2 \\\\equiv 9 \\\\equiv 2 \\\\pmod{7} $\"}]\n149:[\"$\",\"p\",null,{\"children\":\"$$ 3^3 \\\\equiv 6 \\\\pmod{7} $\"}]\n14a:[\"$\",\"p\",null,{\"children\":\"$$ 3^4 \\\\equiv 18 \\\\equiv 4 \\\\pmod{7} $\"}]\n14b:[\"$\",\"p\",null,{\"children\":\"$$ 3^5 \\\\equiv 12 \\\\equiv 5 \\\\pmod{7} $\"}]\n14c:[\"$\",\"p\",null,{\"children\":\"$$ 3^6 \\\\equiv 15 \\\\equiv 1 \\\\pmod{7} $\"}]\n14d:[\"$\",\"p\",null,{\"children\":\"Since $ 3^6 \\\\equiv 1 \\\\pmod{7} $, then $ 3^8 = 3^{6+2} = 3^6 \\\\times 3^2 \\\\equiv 1 \\\\times 2 \\\\equiv 2 \\\\pmod{7} $. Therefore, $ 2019^8 + 1 \\\\equiv 2 + 1 \\\\equiv 3 \\\\pmod{7} $, so 7 does not divide $ 2019^8 + 1 $.\"}]\n14e:[\"$\",\"p\",null,{\"children\":\"Next, chec"])</script><script>self.__next_f.push([1,"k 11. Compute $ 2019 $ modulo 11. $ 2019 \\\\div 11 = 183 $ with a remainder of 6, so $ 2019 \\\\equiv 6 \\\\pmod{11} $. Then, $ 2019^8 \\\\equiv 6^8 \\\\pmod{11} $. Let’s find a pattern for powers of 6 modulo 11:\"}]\n14f:[\"$\",\"p\",null,{\"children\":\"$$ 6^1 \\\\equiv 6 \\\\pmod{11} $\"}]\n150:[\"$\",\"p\",null,{\"children\":\"$$ 6^2 \\\\equiv 36 \\\\equiv 3 \\\\pmod{11} $\"}]\n151:[\"$\",\"p\",null,{\"children\":\"$$ 6^3 \\\\equiv 18 \\\\equiv 7 \\\\pmod{11} $\"}]\n152:[\"$\",\"p\",null,{\"children\":\"$$ 6^4 \\\\equiv 42 \\\\equiv 9 \\\\pmod{11} $\"}]\n153:[\"$\",\"p\",null,{\"children\":\"$$ 6^5 \\\\equiv 54 \\\\equiv 10 \\\\pmod{11} $\"}]\n154:[\"$\",\"p\",null,{\"children\":\"$$ 6^6 \\\\equiv 60 \\\\equiv 5 \\\\pmod{11} $\"}]\n155:[\"$\",\"p\",null,{\"children\":\"$$ 6^7 \\\\equiv 30 \\\\equiv 8 \\\\pmod{11} $\"}]\n156:[\"$\",\"p\",null,{\"children\":\"$$ 6^8 \\\\equiv 48 \\\\equiv 4 \\\\pmod{11} $\"}]\n157:[\"$\",\"p\",null,{\"children\":\"Therefore, $ 2019^8 + 1 \\\\equiv 4 + 1 \\\\equiv 5 \\\\pmod{11} $, so 11 does not divide $ 2019^8 + 1 $.\"}]\n158:[\"$\",\"p\",null,{\"children\":\"Continuing this process seems tedious. Maybe there’s a better way to approach this problem.\"}]\n159:[\"$\",\"p\",null,{\"children\":\"Let me think differently. Perhaps I can consider the expression $ 2019^8 + 1 $ and see if it can be expressed in a form that reveals its prime factors.\"}]\n15a:[\"$\",\"p\",null,{\"children\":\"I recall that for expressions like $ a^4 + 4b^4 $, there’s a Sophie Germain identity: $ a^4 + 4b^4 = (a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2) $. Maybe I can manipulate $ 2019^8 + 1 $ into a similar form.\"}]\n15b:[\"$\",\"p\",null,{\"children\":\"Wait, $ 2019^8 + 1 = (2019^4)^2 + 1^2 $. This resembles the sum of squares, but over integers, sums of squares don’t factor nicely. Maybe I can consider complex numbers or some other algebraic identities, but that might not help in finding integer prime factors.\"}]\n15c:[\"$\",\"p\",null,{\"children\":\"Alternatively, perhaps I can consider modulo properties to find possible prime factors. For example, if $ p $ is an odd prime that divides $ 2019^8 + 1 $, then $ 2019^8 \\\\equiv -1 \\\\pmod{p} $, which implies $ 2019^{16} \\\\equiv 1 \\\\pmod{p} $. This means that the order of 2019 modulo $ p $ divides 16, and since $ 2019^8 \\\\equiv -1 \\\\pmod{p} $, the order can’t divide 8. Therefore, the order must be 16.\"}]\n15d:[\"$\",\"p\",null,{\"children\":\"By Fermat’s Little Theorem, the order of 2019 modulo $ p $ divides $ p - 1 $. Therefore, $ p - 1 $ must be divisible by 16. So, $ p \\\\equiv 1 \\\\pmod{16} $.\"}]\n15e:[\"$\",\"p\",null,{\"children\":\"This narrows down the possible primes $ p $ that could divide $ 2019^8 + 1 $ to those primes congruent to 1 modulo 16.\"}]\n15f:[\"$\",\"p\",null,{\"children\":\"Let me list some primes that are 1 modulo 16: 17, 97, 113, 193, 241, 257, etc.\"}]\n160:[\"$\",\"p\",null,{\"children\":\"Now, I can test these primes to see if any of them divide $ 2019^8 + 1 $.\"}]\n161:[\"$\",\"p\",null,{\"children\":\"Start with 17. Compute $ 2019 $ modulo 17.\"}]\n162:[\"$\",\"p\",null,{\"children\":\"First, find $ 2019 \\\\div 17 $. 17 times 118 is 2006, so $ 2019 - 2006 = 13 $. Therefore, $ 2019 \\\\equiv 13 \\\\pmod{17} $.\"}]\n163:[\"$\",\"p\",null,{\"children\":\"Then, $ 2019^8 \\\\equiv 13^8 \\\\pmod{17} $.\"}]\n164:[\"$\",\"p\",null,{\"children\":\"Compute $ 13^2 \\\\equiv 169 \\\\equiv 16 \\\\pmod{17} $\"}]\n165:[\"$\",\"p\",null,{\"children\":\"$$ 13^4 = (13^2)^2 \\\\equiv 16^2 \\\\equiv 256 \\\\equiv 1 \\\\pmod{17} $\"}]\n166:[\"$\",\"p\",null,{\"children\":\"$$ 13^8 = (13^4)^2 \\\\equiv 1^2 \\\\equiv 1 \\\\pmod{17} $\"}]\n167:[\"$\",\"p\",null,{\"children\":\"Therefore, $ 2019^8 + 1 \\\\equiv 1 + 1 \\\\equiv 2 \\\\pmod{17} $, so 17 does not divide $ 2019^8 + 1 $.\"}]\n168:[\"$\",\"p\",null,{\"children\":\"Next, check 97. Compute $ 2019 $ modulo 97.\"}]\n169:[\"$\",\"p\",null,{\"children\":\"First, find how many times 97 fits into 2019. 97 times 20 is 1940, and 2019 - 1940 = 79. So, $ 2019 \\\\equiv 79 \\\\pmod{97} $.\"}]\n16a:[\"$\",\"p\",null,{\"children\":\"Then, $ 2019^8 \\\\equiv 79^8 \\\\pmod{97} $.\"}]\n16b:[\"$\",\"p\",null,{\"children\":\"This seems a bit cumbersome to compute directly. Maybe I can find a pattern or use Fermat’s Little Theorem, which states that $ a^{p-1} \\\\equiv 1 \\\\pmod{p} $ if $ p $ is prime and $ p $ does not divide $ a $.\"}]\n16c:[\"$\",\"p\",null,{\"children\":\"Here"])</script><script>self.__next_f.push([1,", $ p = 97 $, so $ 79^{96} \\\\equiv 1 \\\\pmod{97} $. But I need $ 79^8 $, which is not directly helpful. Maybe I can compute $ 79^2 $, then $ 79^4 $, and so on.\"}]\n16d:[\"$\",\"p\",null,{\"children\":\"First, compute $ 79^2 \\\\pmod{97} $:\"}]\n16e:[\"$\",\"p\",null,{\"children\":\"$$ 79^2 = 6241 $. Now, divide 6241 by 97.\"}]\n16f:[\"$\",\"p\",null,{\"children\":\"97 times 64 is 6208, so 6241 - 6208 = 33. Therefore, $ 79^2 \\\\equiv 33 \\\\pmod{97} $.\"}]\n170:[\"$\",\"p\",null,{\"children\":\"Next, $ 79^4 = (79^2)^2 \\\\equiv 33^2 \\\\pmod{97} $.\"}]\n171:[\"$\",\"p\",null,{\"children\":\"$$ 33^2 = 1089 $. 97 times 11 is 1067, so 1089 - 1067 = 22. Therefore, $ 79^4 \\\\equiv 22 \\\\pmod{97} $.\"}]\n172:[\"$\",\"p\",null,{\"children\":\"Then, $ 79^8 = (79^4)^2 \\\\equiv 22^2 \\\\pmod{97} $.\"}]\n173:[\"$\",\"p\",null,{\"children\":\"$$ 22^2 = 484 $. 97 times 4 is 388, so 484 - 388 = 96. Therefore, $ 79^8 \\\\equiv 96 \\\\pmod{97} $.\"}]\n174:[\"$\",\"p\",null,{\"children\":\"Thus, $ 2019^8 + 1 \\\\equiv 96 + 1 \\\\equiv 97 \\\\equiv 0 \\\\pmod{97} $.\"}]\n175:[\"$\",\"p\",null,{\"children\":\"So, 97 divides $ 2019^8 + 1 $. Since 97 is the smallest prime congruent to 1 modulo 16 that we’ve checked and it divides $ 2019^8 + 1 $, it must be the least odd prime factor of $ 2019^8 + 1 $.\"}]\n176:[\"$\",\"p\",null,{\"children\":\"Therefore, the least odd prime factor is 97.\"}]\n177:[\"$\",\"p\",null,{\"children\":[\"$\",\"strong\",null,{\"children\":\"Final Answer\"}]}]\n178:[\"$\",\"p\",null,{\"children\":[\"$\",\"code\",null,{\"children\":\"\\\\boxed\\\\{97\\\\}\"}]}]\n179:[\"$\",\"h1\",null,{\"className\":\"flex scroll-m-28 flex-row items-center gap-2\",\"id\":\"reflections-on-the-journey-ahead\",\"children\":[[\"$\",\"a\",null,{\"data-card\":\"\",\"href\":\"#reflections-on-the-journey-ahead\",\"className\":\"peer\",\"children\":\"Reflections on the Journey Ahead\"}],[\"$\",\"svg\",null,{\"ref\":\"$undefined\",\"xmlns\":\"http://www.w3.org/2000/svg\",\"width\":24,\"height\":24,\"viewBox\":\"0 0 24 24\",\"fill\":\"none\",\"stroke\":\"currentColor\",\"strokeWidth\":2,\"strokeLinecap\":\"round\",\"strokeLinejoin\":\"round\",\"className\":\"lucide lucide-link size-3.5 shrink-0 text-fd-muted-foreground opacity-0 transition-opacity peer-hover:opacity-100\",\"aria-hidden\":true,\"children\":[[\"$\",\"path\",\"1cjeqo\",{\"d\":\"M10 13a5 5 0 0 0 7.54.54l3-3a5 5 0 0 0-7.07-7.07l-1.72 1.71\"}],[\"$\",\"path\",\"19qd67\",{\"d\":\"M14 11a5 5 0 0 0-7.54-.54l-3 3a5 5 0 0 0 7.07 7.07l1.71-1.71\"}],\"$undefined\"]}]]}]\n17a:[\"$\",\"p\",null,{\"children\":\"The path to understanding reasoning in large language models branches into many directions, each offering unique insights into the nature of machine intelligence. Our research spans multiple domains - from process reward models that shape learning patterns, to LLM critique that fosters deeper analysis, to multi-step reasoning that builds complex thought, to reinforcement learning with system feedback that enables growth through real-world interactions. While the destination remains unclear, each step brings us closer to understanding how machines think and reason. 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