Revisiting the Effectiveness of LLM Pruning for Test-Time Scaling

Anshuman Chhabra; Ocean Monjur; Shahriar Kabir Nahin

REVIEW 3 major objections 1 minor 1 cited by

Unstructured pruning can improve test-time scaling performance in reasoning LLMs, sometimes outperforming the original full models.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review

2026-07-01 09:14 UTC pith:MFCSPPSD

load-bearing objection Unstructured pruning can match or beat dense models on TTS according to the experiments, but the gains appear tied to hand-tuned layer-wise sparsity allocations that the paper does not separate from the pruning effect. the 3 major comments →

arxiv 2604.25098 v2 pith:MFCSPPSD submitted 2026-04-28 cs.AI cs.CLcs.LG

Revisiting the Effectiveness of LLM Pruning for Test-Time Scaling

Ocean Monjur , Shahriar Kabir Nahin , Anshuman Chhabra This is my paper

classification cs.AI cs.CLcs.LG

keywords LLM pruningtest-time scalingunstructured pruningstructured pruningreasoning benchmarksmodel compressionTTS performance

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether unstructured pruning harms test-time compute scaling on reasoning tasks in the same way structured pruning does. Experiments on the s1.1-7B and Qwen3-8B models across four benchmarks show that removing specific weights instead of whole layers often boosts TTS results over structured approaches and can exceed the unpruned baseline. This challenges the view that pruning must reduce reasoning effectiveness under increased test-time compute. A sympathetic reader would care because it indicates selective compression might preserve or enhance scaling behavior rather than trade it away.

Core claim

The paper claims that unstructured pruning augments TTS performance compared to structured pruning, and at times can even outperform the unpruned full-weight LLMs on reasoning benchmarks. It further shows that different layer-wise sparsity allocation strategies affect these outcomes, challenging the conventional notion that pruning always reduces TTS performance.

What carries the argument

Unstructured pruning with layer-wise sparsity allocation strategies that selectively remove individual weights while keeping full layer structure intact.

Load-bearing premise

The unstructured pruning methods rely on carefully chosen layer-wise sparsity allocation strategies.

What would settle it

Reproducing the experiments on the same models and benchmarks and finding that unstructured-pruned versions underperform or match but do not exceed the full models on TTS metrics would falsify the central claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

Unstructured pruning can produce more efficient reasoning models that retain or gain TTS effectiveness.
Layer-wise sparsity allocation choices determine whether pruning helps or hurts TTS performance.
Pruning at the weight level does not inherently degrade reasoning under test-time compute scaling.
This opens the possibility that selective parameter removal can be compatible with or supportive of scaling laws for reasoning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the claim holds, certain weights may interfere with optimal reasoning scaling and their removal could be net beneficial.
The result suggests testing whether similar gains appear when unstructured pruning is combined with other compression techniques such as quantization.
It raises the question of whether the effect generalizes to larger models or non-reasoning tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Unstructured pruning can match or beat dense models on TTS according to the experiments, but the gains appear tied to hand-tuned layer-wise sparsity allocations that the paper does not separate from the pruning effect.

read the letter

The main point is that this paper finds unstructured pruning sometimes matches or exceeds the unpruned baseline on test-time scaling for reasoning tasks on s1.1-7B and Qwen3-8B, while structured pruning hurts performance. They run this across four benchmarks and also test different layer-wise sparsity strategies.

The experiments provide concrete comparisons that challenge the prior view based on structured methods, and that is the useful part. Running the same setups on two models gives a bit of breadth.

The soft spot is exactly what the stress-test note flags: the abstract calls the layer-wise sparsity allocation an important parametric choice and the results rely on carefully chosen versions of it. If those allocations involve extra search or tuning per model and benchmark, then the headline claim that pruning itself retains or improves TTS effectiveness does not fully hold up. The comparison to the dense model and to structured pruning is not apples-to-apples without showing the result survives standard or untuned unstructured pruning. The abstract also gives no error bars, statistical tests, or controls, so the strength of the "consistently show" and "outperform" statements is hard to judge.

This is for researchers working on compression for reasoning models. A reader in that area would get some empirical pointers from the comparisons, but the tuning issue needs addressing before the central claim lands cleanly.

I would send it to peer review so the methods section and any ablations on the sparsity choices can be checked directly.

Referee Report

3 major / 1 minor

Summary. The paper claims that unstructured pruning of reasoning LLMs (s1.1-7B and Qwen3-8B) can augment test-time scaling (TTS) performance on four reasoning benchmarks relative to structured pruning and, in some cases, even exceed the unpruned dense baseline. It further examines the role of layer-wise sparsity allocation strategies as a key hyperparameter for unstructured methods and concludes that carefully undertaken pruning can retain or improve TTS effectiveness, challenging prior assumptions about pruning degrading reasoning under test-time compute scaling.

Significance. If the empirical findings are robust, the result would be significant for efficient LLM deployment: it suggests that unstructured pruning need not trade off against TTS gains and could enable smaller models that match or surpass dense counterparts on reasoning tasks. The work provides new empirical comparisons across models and benchmarks with no reduction to author-defined fitted quantities.

major comments (3)

[Abstract / Experiments] Abstract and experimental results: the headline claim that unstructured pruning 'at times can even outperform the unpruned full-weight LLMs' is presented without reported error bars, statistical significance tests, or controls for random seed variability, making it impossible to assess whether observed gains exceed noise (soundness concern noted in review).
[Abstract] Abstract: the central result explicitly depends on 'carefully chosen layer-wise sparsity allocation strategies' described as 'an important parametric choice.' No ablation is described that isolates whether the TTS gains survive under non-tuned (e.g., uniform or random) per-layer densities; if the allocations function as an additional hyperparameter search compensating for pruning damage, the comparison to structured pruning and the dense baseline is not controlled.
[Experiments] Experimental setup: the manuscript supplies no details on exact pruning implementations (e.g., which unstructured method, magnitude vs. other criteria), exact sparsity schedules, or how TTS is measured (number of samples, temperature, stopping criteria), preventing reproduction or assessment of confounds.

minor comments (1)

[Abstract] Abstract: 'lays the foundation' should be 'lay the foundation' for subject-verb agreement.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback highlighting areas where additional rigor and detail would strengthen the manuscript. We address each major comment below and commit to revisions where the current version is lacking.

read point-by-point responses

Referee: [Abstract / Experiments] Abstract and experimental results: the headline claim that unstructured pruning 'at times can even outperform the unpruned full-weight LLMs' is presented without reported error bars, statistical significance tests, or controls for random seed variability, making it impossible to assess whether observed gains exceed noise (soundness concern noted in review).

Authors: We agree this is a valid soundness concern. The revised manuscript will report all key results with error bars computed over multiple random seeds and will include statistical significance tests (e.g., paired t-tests) to establish whether the observed gains over the dense baseline exceed variability. revision: yes
Referee: [Abstract] Abstract: the central result explicitly depends on 'carefully chosen layer-wise sparsity allocation strategies' described as 'an important parametric choice.' No ablation is described that isolates whether the TTS gains survive under non-tuned (e.g., uniform or random) per-layer densities; if the allocations function as an additional hyperparameter search compensating for pruning damage, the comparison to structured pruning and the dense baseline is not controlled.

Authors: The manuscript already positions layer-wise allocation as a key hyperparameter and studies several strategies. To directly address whether gains persist without tuned allocations, the revision will add an explicit ablation comparing the reported tuned allocations against uniform and random per-layer sparsity on the same models and benchmarks. revision: yes
Referee: [Experiments] Experimental setup: the manuscript supplies no details on exact pruning implementations (e.g., which unstructured method, magnitude vs. other criteria), exact sparsity schedules, or how TTS is measured (number of samples, temperature, stopping criteria), preventing reproduction or assessment of confounds.

Authors: We agree that reproducibility requires these details. The revised manuscript will specify the exact unstructured pruning criterion (magnitude pruning), the per-layer sparsity schedules, and the full TTS protocol (sample count, temperature, decoding strategy, and stopping criteria) in a dedicated experimental setup subsection. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical comparisons with no derivation chain

full rationale

The paper reports experimental results from running unstructured and structured pruning on two LLMs across four reasoning benchmarks, measuring TTS performance. No equations, fitted parameters renamed as predictions, or self-citation load-bearing steps appear in the abstract or described content. Layer-wise sparsity allocation is explicitly called an 'important parametric choice' for the methods, but the headline claims rest on direct benchmark measurements rather than any reduction to those choices by construction. The work is self-contained against external benchmarks and contains no mathematical derivation that could be circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is an empirical study; the abstract mentions no explicit free parameters, mathematical axioms, or newly postulated entities beyond standard LLM pruning techniques.

pith-pipeline@v0.9.1-grok · 5756 in / 960 out tokens · 28780 ms · 2026-07-01T09:14:15.115836+00:00 · methodology

0 comments

read the original abstract

Large Language Models (LLMs) now exhibit remarkable reasoning capabilities through test-time compute scaling (TTS), with impressive performance across math and coding benchmarks. In parallel, research in model compression has developed pruning methods that seek to remove redundant/detrimental parameters without sacrificing task performance. The intersection of these two research advancements lays the foundation for our work. Specific to reasoning LLMs, prior work has shown that structured pruning (methods which remove entire set of layer blocks), significantly degrades TTS reasoning performance. However, in this work, we revisit this assumption and investigate whether unstructured pruning (methods that carefully remove only certain redundant/detrimental weights) exhibits similar limitations. Surprisingly, our extensive experiments across four reasoning benchmarks on two reasoning LLMs: s1.1-7B and Qwen3-8B, consistently show that unstructured pruning augments TTS performance compared to structured pruning, and at times can even outperform the unpruned full-weight LLMs. Furthermore, we also empirically study the impact of different layer-wise sparsity allocation strategies, which are an important parametric choice for instantiating these unstructured methods. These findings challenge the conventional notion that pruning always reduces TTS performance and in fact, suggest that carefully undertaken pruning can retain TTS effectiveness.

Figures

Figures reproduced from arXiv: 2604.25098 by Anshuman Chhabra, Ocean Monjur, Shahriar Kabir Nahin.

**Figure 1.** Figure 1: Overview of structured and unstructured pruning for LLMs and their impact on test-time scaling (TTS) reasoning performance. As identified in prior work, removing entire layer blocks via structured pruning makes LLMs more susceptible to producing incoherent chains of thought, ultimately resulting in incorrect answers. However, as our findings show, this is not the case for unstructured pruning, where TTS pe… view at source ↗

**Figure 2.** Figure 2: Comparing structured (ShortGPT) and unstructured (Magnitude, Wanda) pruning methods on four long-chain reasoning datasets. Unstructured pruning is employed uniformly at both 10% and 20% sparsity rates, while structured pruning removes 1 and 2 layer blocks. It is evident that unstructured pruning retains or surpasses unpruned LLM performance, whereas structured pruning leads to substantial degradation. et a… view at source ↗

**Figure 3.** Figure 3: Comparing different layer-wise sparsity allocation strategies (Uniform, Owl, and LayerIF) with global sparsity rates of 10% and 20%. Performance is averaged across AIME24, GPQA-Diamond, AMC23, and MATH500 benchmarks while varying thinking tokens from 512 to 8192. proach for improving/preserving TTS capabilities. Remarks on RQ1 Findings. Experiments across all four reasoning datasets and both LLMs demonstr… view at source ↗

**Figure 4.** Figure 4: s1.1-7B results, when attention and feed-forward MLP layers are pruned in isolation at different sparsity view at source ↗

**Figure 5.** Figure 5: Qwen3-8B results, when attention and feed-forward MLP layers are pruned in isolation at different view at source ↗

Review history (2 revisions) →

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · cited by 1 Pith paper · 4 internal anchors

[1]

Available: https://arxiv.org/abs/2401.00625

SliceGPT: Compress large language models by deleting rows and columns. InThe Twelfth Inter- national Conference on Learning Representations. Hadi Askari, Shivanshu Gupta, Fei Wang, Anshuman Chhabra, and Muhao Chen. 2025. LayerIF: Estimat- ing layer quality for large language models using influence functions. InThe Thirty-ninth Annual Con- ference on Neura...

work page arXiv 2025
[2]

Training Verifiers to Solve Math Word Problems

Training verifiers to solve math word prob- lems.Preprint, arXiv:2110.14168. Rémi Coulom. 2006. Efficient selectivity and backup operators in monte-carlo tree search. InInternational conference on computers and games, pages 72–83. Springer. Kazuki Egashira, Mark Vero, Robin Staab, Jingxuan He, and Martin Vechev. 2024. Exploiting llm quanti- zation.Advance...

work page internal anchor Pith review Pith/arXiv arXiv 2006
[3]

InAdvances in Neural In- formation Processing Systems, volume 28

Learning both weights and connections for efficient neural network. InAdvances in Neural In- formation Processing Systems, volume 28. Curran Associates, Inc. Babak Hassibi, David G Stork, and Gregory J Wolff

work page
[4]

Scaling Laws for Neural Language Models

Optimal brain surgeon and general network pruning. InIEEE international conference on neural networks, pages 293–299. IEEE. 9 Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. 2021. Measuring mathematical problem solving with the MATH dataset. InThirty- fifth Conference on Neural Informati...

work page internal anchor Pith review Pith/arXiv arXiv 2021
[5]

InThe Twelfth Inter- national Conference on Learning Representations

Let’s verify step by step. InThe Twelfth Inter- national Conference on Learning Representations. Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harri- son Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe

work page
[6]

SpinQuant: LLM quantization with learned rotations

Let’s verify step by step. InThe Twelfth Inter- national Conference on Learning Representations. Chihung Lin and Chang-Fu Kuo. 2025. Roles and potential of large language models in healthcare: A comprehensive review.Biomedical Journal, 48(5):100868. Zechun Liu, Changsheng Zhao, Igor Fedorov, Bilge Soran, Dhruv Choudhary, Raghuraman Krishnamoor- thi, Vikas...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[7]

Kimi k1.5: Scaling Reinforcement Learning with LLMs

A simple and effective pruning approach for large language models. InThe Twelfth International Conference on Learning Representations. Kimi Team, Angang Du, Bofei Gao, Bowei Xing, Changjiu Jiang, Cheng Chen, Cheng Li, Chenjun Xiao, Chenzhuang Du, Chonghua Liao, Chuning Tang, Congcong Wang, Dehao Zhang, Enming Yuan, Enzhe Lu, Fengxiang Tang, Flood Sung, Gu...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[8]

Al- right, let’s start

Then, find r2 where r is the radius of that sphere, and express it as a fraction p/q with p and q coprime, then find p+q . Al- right, let’s start. First, let me recall some formulas. For a rect- angular box with length, width, and height x, y, z, the surface area is2(xy+yz+xz) =

work page
[9]

The problem is about finding the smallest sphere that can contain such a box

The volume is xyz= 23 . The problem is about finding the smallest sphere that can contain such a box. The sphere’s radius will depend on the box’s dimensions. ..........= −1 + 2 √ 6≈ −1 + 4.898≈3.898 , which is positive. Therefore, if x=y= 2 , then z= 23/(22) = 23/4 = 5.75 . Alternatively, if x=y≈3.898 , then z= 23/x 2. Let me compute x2 exactly: x=−1 + 2 √

work page
[10]

Then z= 23/(25−4 √ 6)

There- fore, x2 = (−1+2 √ 6)2 = 1−4 √ 6+24 = 25−4 √ 6≈25−9.798≈15.202 . Then z= 23/(25−4 √ 6). Let me rationalize that denominator: Multiply numerator and denominator by (25 + 4 √ 6): z= 23(25+4 √ 6) (25−4 √ 6)(25+4 √

work page
[11]

Answer: 721 (Correct) 18 Table 12: Qualitative outputs on AMC23 for Unpruned vs

= 23(25+4 √ 6) 625−(4 √ 6)2 . Answer: 721 (Correct) 18 Table 12: Qualitative outputs on AMC23 for Unpruned vs. Wanda-Uniform 20% pruned model. Query Unpruned Wanda-Uniform 20% If A and B are vertices of a polyhedron, define the distance d(A, B) to be the minimum num- ber of edges of the poly- hedron one must tra- verse in order to con- nect A and B. For e...

work page
[12]

Find the probability that d(Q, R)> d(R, S)

Let Q, R, and S be randomly chosen dis- tinct vertices of a regu- lar icosahedron (regular polyhedron made up of 20 equilateral triangles). Find the probability that d(Q, R)> d(R, S) . The final answer can be written in the form m n , where m and n are rela- tively prime positive in- tegers. What is m+n ? Chain of Thought:Okay, so I need to find the maxim...

work page
[13]

Maybe there’s a better approach

Hmm, this seems complicated. Maybe there’s a better approach. Let me think. Al- ternatively, maybe consider substituting y2 from the equation or something else. Alter- natively, think of this as a quadratic equation in y2? Let me check. Let me rearrange the equation: y4 + (2x2 + 2x−1)y 2 + (x4 + 2x3 + 3x1 + 2x+ 1−16) = 0 . So, if I let y2 =t , then:t2 + (...

work page
[14]

6 and 10 are even

Can we make 29 with 6 and 10? 29 is odd. 6 and 10 are even. So no. If we use 15 twice: 44 - 30 = 14. Can we make 14 with 6 and 10? 10 + 6* (something)? 14 -10 = 4. Not divisible by 6. 14 - 6 = 8. Not divisible by 10. So no Answer: 8 (Incorrect) Chain of Thought:Okay, so I need to fig- ure out the largest amount of money that can’t be formed using coins of...

work page
[15]

The positions that are meta to 1 are 3 and

work page
[16]

The positions that are meta to 3 are 1 and

work page
[17]

A benzene ring has a carboxylic acid, a carbaldehyde, and a cyano group all meta to one another

Therefore, the only common position is 5. So if the third group is at 5, then it is meta to both 1 and 3. Therefore, the three groups are at positions Answer: C (Incorrect) Chain of Thought:Okay, so I need to figure out the correct IUPAC name for this com- plex benzene ring molecule. Let me start by breaking down the information given. First, the molecule...

work page
[18]

But the addition is between the product and 1, so how can we group it differently? Wait, for example, maybe insert- ing parentheses between the multiplications and the addition

So, to have the addition happen earlier, we need to have parentheses that include the ad- dition operator. But the addition is between the product and 1, so how can we group it differently? Wait, for example, maybe insert- ing parentheses between the multiplications and the addition. For example, 2·3·(4·(5 + 1)) which is same as before. Or 2·(3·4·(5 + 1))...

work page 2048

This paper was first reviewed by grok-4.3 on July 1, 2026.

[1] [1]

Available: https://arxiv.org/abs/2401.00625

SliceGPT: Compress large language models by deleting rows and columns. InThe Twelfth Inter- national Conference on Learning Representations. Hadi Askari, Shivanshu Gupta, Fei Wang, Anshuman Chhabra, and Muhao Chen. 2025. LayerIF: Estimat- ing layer quality for large language models using influence functions. InThe Thirty-ninth Annual Con- ference on Neura...

work page arXiv 2025

[2] [2]

Training Verifiers to Solve Math Word Problems

Training verifiers to solve math word prob- lems.Preprint, arXiv:2110.14168. Rémi Coulom. 2006. Efficient selectivity and backup operators in monte-carlo tree search. InInternational conference on computers and games, pages 72–83. Springer. Kazuki Egashira, Mark Vero, Robin Staab, Jingxuan He, and Martin Vechev. 2024. Exploiting llm quanti- zation.Advance...

work page internal anchor Pith review Pith/arXiv arXiv 2006

[3] [3]

InAdvances in Neural In- formation Processing Systems, volume 28

Learning both weights and connections for efficient neural network. InAdvances in Neural In- formation Processing Systems, volume 28. Curran Associates, Inc. Babak Hassibi, David G Stork, and Gregory J Wolff

work page

[4] [4]

Scaling Laws for Neural Language Models

Optimal brain surgeon and general network pruning. InIEEE international conference on neural networks, pages 293–299. IEEE. 9 Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. 2021. Measuring mathematical problem solving with the MATH dataset. InThirty- fifth Conference on Neural Informati...

work page internal anchor Pith review Pith/arXiv arXiv 2021

[5] [5]

InThe Twelfth Inter- national Conference on Learning Representations

Let’s verify step by step. InThe Twelfth Inter- national Conference on Learning Representations. Hunter Lightman, Vineet Kosaraju, Yuri Burda, Harri- son Edwards, Bowen Baker, Teddy Lee, Jan Leike, John Schulman, Ilya Sutskever, and Karl Cobbe

work page

[6] [6]

SpinQuant: LLM quantization with learned rotations

Let’s verify step by step. InThe Twelfth Inter- national Conference on Learning Representations. Chihung Lin and Chang-Fu Kuo. 2025. Roles and potential of large language models in healthcare: A comprehensive review.Biomedical Journal, 48(5):100868. Zechun Liu, Changsheng Zhao, Igor Fedorov, Bilge Soran, Dhruv Choudhary, Raghuraman Krishnamoor- thi, Vikas...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[7] [7]

Kimi k1.5: Scaling Reinforcement Learning with LLMs

A simple and effective pruning approach for large language models. InThe Twelfth International Conference on Learning Representations. Kimi Team, Angang Du, Bofei Gao, Bowei Xing, Changjiu Jiang, Cheng Chen, Cheng Li, Chenjun Xiao, Chenzhuang Du, Chonghua Liao, Chuning Tang, Congcong Wang, Dehao Zhang, Enming Yuan, Enzhe Lu, Fengxiang Tang, Flood Sung, Gu...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[8] [8]

Al- right, let’s start

Then, find r2 where r is the radius of that sphere, and express it as a fraction p/q with p and q coprime, then find p+q . Al- right, let’s start. First, let me recall some formulas. For a rect- angular box with length, width, and height x, y, z, the surface area is2(xy+yz+xz) =

work page

[9] [9]

The problem is about finding the smallest sphere that can contain such a box

The volume is xyz= 23 . The problem is about finding the smallest sphere that can contain such a box. The sphere’s radius will depend on the box’s dimensions. ..........= −1 + 2 √ 6≈ −1 + 4.898≈3.898 , which is positive. Therefore, if x=y= 2 , then z= 23/(22) = 23/4 = 5.75 . Alternatively, if x=y≈3.898 , then z= 23/x 2. Let me compute x2 exactly: x=−1 + 2 √

work page

[10] [10]

Then z= 23/(25−4 √ 6)

There- fore, x2 = (−1+2 √ 6)2 = 1−4 √ 6+24 = 25−4 √ 6≈25−9.798≈15.202 . Then z= 23/(25−4 √ 6). Let me rationalize that denominator: Multiply numerator and denominator by (25 + 4 √ 6): z= 23(25+4 √ 6) (25−4 √ 6)(25+4 √

work page

[11] [11]

Answer: 721 (Correct) 18 Table 12: Qualitative outputs on AMC23 for Unpruned vs

= 23(25+4 √ 6) 625−(4 √ 6)2 . Answer: 721 (Correct) 18 Table 12: Qualitative outputs on AMC23 for Unpruned vs. Wanda-Uniform 20% pruned model. Query Unpruned Wanda-Uniform 20% If A and B are vertices of a polyhedron, define the distance d(A, B) to be the minimum num- ber of edges of the poly- hedron one must tra- verse in order to con- nect A and B. For e...

work page

[12] [12]

Find the probability that d(Q, R)> d(R, S)

Let Q, R, and S be randomly chosen dis- tinct vertices of a regu- lar icosahedron (regular polyhedron made up of 20 equilateral triangles). Find the probability that d(Q, R)> d(R, S) . The final answer can be written in the form m n , where m and n are rela- tively prime positive in- tegers. What is m+n ? Chain of Thought:Okay, so I need to find the maxim...

work page

[13] [13]

Maybe there’s a better approach

Hmm, this seems complicated. Maybe there’s a better approach. Let me think. Al- ternatively, maybe consider substituting y2 from the equation or something else. Alter- natively, think of this as a quadratic equation in y2? Let me check. Let me rearrange the equation: y4 + (2x2 + 2x−1)y 2 + (x4 + 2x3 + 3x1 + 2x+ 1−16) = 0 . So, if I let y2 =t , then:t2 + (...

work page

[14] [14]

6 and 10 are even

Can we make 29 with 6 and 10? 29 is odd. 6 and 10 are even. So no. If we use 15 twice: 44 - 30 = 14. Can we make 14 with 6 and 10? 10 + 6* (something)? 14 -10 = 4. Not divisible by 6. 14 - 6 = 8. Not divisible by 10. So no Answer: 8 (Incorrect) Chain of Thought:Okay, so I need to fig- ure out the largest amount of money that can’t be formed using coins of...

work page

[15] [15]

The positions that are meta to 1 are 3 and

work page

[16] [16]

The positions that are meta to 3 are 1 and

work page

[17] [17]

A benzene ring has a carboxylic acid, a carbaldehyde, and a cyano group all meta to one another

Therefore, the only common position is 5. So if the third group is at 5, then it is meta to both 1 and 3. Therefore, the three groups are at positions Answer: C (Incorrect) Chain of Thought:Okay, so I need to figure out the correct IUPAC name for this com- plex benzene ring molecule. Let me start by breaking down the information given. First, the molecule...

work page

[18] [18]

But the addition is between the product and 1, so how can we group it differently? Wait, for example, maybe insert- ing parentheses between the multiplications and the addition

So, to have the addition happen earlier, we need to have parentheses that include the ad- dition operator. But the addition is between the product and 1, so how can we group it differently? Wait, for example, maybe insert- ing parentheses between the multiplications and the addition. For example, 2·3·(4·(5 + 1)) which is same as before. Or 2·(3·4·(5 + 1))...

work page 2048