mlsanityData to 7 Oct 2026

Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics

arXiv 2410.21272 ↗First posted 28 Oct 2024cs.CLcited by 94

On the map: LLM reasoning → Mathematical reasoning

Abstract (arXiv)

Do large language models (LLMs) solve reasoning tasks by learning robust generalizable algorithms, or do they memorize training data? To investigate this question, we use arithmetic reasoning as a representative task. Using causal analysis, we identify a subset of the model (a circuit) that explains most of the model's behavior for basic arithmetic logic and examine its functionality. By zooming in on the level of individual circuit neurons, we discover a sparse set of important neurons that implement simple heuristics. Each heuristic identifies a numerical input pattern and outputs corresponding answers. We hypothesize that the combination of these heuristic neurons is the mechanism used to produce correct arithmetic answers. To test this, we categorize each neuron into several heuristic types-such as neurons that activate when an operand falls within a certain range-and find that the unordered combination of these heuristic types is the mechanism that explains most of the model's accuracy on arithmetic prompts. Finally, we demonstrate that this mechanism appears as the main source of arithmetic accuracy early in training. Overall, our experimental results across several LLMs show that LLMs perform arithmetic using neither robust algorithms nor memorization; rather, they rely on a "bag of heuristics".

Key facts. Cited by 94 papers in the map (data through 7 Oct 2026). Placed in Mathematical reasoning, one of 17 directions in LLM reasoning. Builds on 31 of its 40 references that are in the map. First posted on arXiv 28 Oct 2024.

Builds on

31 of 40 references.

Closest ancestors · 2

Same direction, newest first

  1. 2024Pre-trained Large Language Models Use Fourier Features to Compute Addition5 Jun
  2. 2023A Mechanistic Interpretation of Arithmetic Reasoning in Language Models using Causal Mediation Analysis24 May

Other references · 29

  1. 2024The Llama 3 Herd of Modelscited by 14,192
  2. 2023Pythia: A Suite for Analyzing Large Language Models Across Training and Scalingcited by 1,734
  3. 2020Transformer Feed-Forward Layers Are Key-Value Memoriescited by 1,307
  4. 2022Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 smallcited by 1,130
  5. 2021Probing Classifiers: Promises, Shortcomings, and Advancescited by 831
  6. 2023Progress measures for grokking via mechanistic interpretabilitycited by 822
  7. 2022Quantifying Memorization Across Neural Language Modelscited by 755
  8. 2022Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Taskcited by 417
  9. 2021Causal Abstractions of Neural Networkscited by 401
  10. 2021Pay Attention to MLPscited by 367
  11. 2013Direct and Indirect Effectscited by 276
  12. 2022Extracting Latent Steering Vectors from Pretrained Language Modelscited by 217
17 more
  1. 2023How does GPT-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language modelcited by 199
  2. 2023Attribution Patching Outperforms Automated Circuit Discoverycited by 196
  3. 2023The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networkscited by 153
  4. 2024Have Faith in Faithfulness: Going Beyond Circuit Overlap When Finding Model Mechanismscited by 144
  5. 2024Fine-Tuning Enhances Existing Mechanisms: A Case Study on Entity Trackingcited by 140
  6. 2022Analyzing Transformers in Embedding Spacecited by 131
  7. 2023Explaining grokking through circuit efficiencycited by 105
  8. 2024Generalization v.s. Memorization: Tracing Language Models' Capabilities Back to Pretraining Datacited by 83
  9. 2023Successor Heads: Recurring, Interpretable Attention Heads In The Wildcited by 77
  10. 2023Copy Suppression: Comprehensively Understanding an Attention Headcited by 73
  11. 2023Towards a Mechanistic Interpretation of Multi-Step Reasoning Capabilities of Language Modelscited by 50
  12. 2024Interpreting and Improving Large Language Models in Arithmetic Calculationcited by 43
  13. 2021Memorisation versus Generalisation in Pre-trained Language Modelscited by 32
  14. 2022Measures of Information Reflect Memorization Patternscited by 12
  15. 2023Arithmetic with Language Models: from Memorization to Computationcited by 12
  16. 2024Generalisation First, Memorisation Second? Memorisation Localisation for Natural Language Classification Taskscited by 9
  17. 2024Survival of the Fittest Representation: A Case Study with Modular Additioncited by 7

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 56; 2026: 37.

94 papers.

Sort:

By area:

  1. 2025MATH-Perturb: Benchmarking LLMs' Math Reasoning Abilities against Hard Perturbationscited by 88
  2. 2025Challenging the Boundaries of Reasoning: An Olympiad-Level Math Benchmark for Large Language Modelscited by 65
  3. 2025MIB: A Mechanistic Interpretability Benchmarkcited by 49
  4. 2025What Has a Foundation Model Found? Using Inductive Bias to Probe for World Modelscited by 46
  5. 2026When Models Manipulate Manifolds: The Geometry of a Counting Taskcited by 43
  6. 2025Same Task, Different Circuits: Disentangling Modality-Specific Mechanisms in VLMscited by 35
  7. 2025Detection and Mitigation of Hallucination in Large Reasoning Models: A Mechanistic Perspectivecited by 32
  8. 2025Back Attention: Understanding and Enhancing Multi-Hop Reasoning in Large Language Modelscited by 23
  9. 2026Locate, Steer, and Improve: A Practical Survey of Actionable Mechanistic Interpretability in Large Language Modelscited by 22
  10. 2026Large Language Model Reasoning Failurescited by 20
  11. 2025Questioning Representational Optimism in Deep Learning: The Fractured Entangled Representation Hypothesiscited by 19
  12. 2026Language Model Circuits Are Sparse in the Neuron Basiscited by 18
38 more
  1. 2025Finite State Automata Inside Transformers with Chain-of-Thought: A Mechanistic Study on State Trackingcited by 17
  2. 2025Probing for Arithmetic Errors in Language Modelscited by 17
  3. 2025Understanding and Mitigating Gender Bias in LLMs via Interpretable Neuron Editingcited by 14
  4. 2025Implicit Reasoning in Transformers is Reasoning through Shortcutscited by 14
  5. 2025The Lookahead Limitation: Why Multi-Operand Addition is Hard for LLMscited by 13
  6. 2025Guiding Reasoning in Small Language Models with LLM Assistancecited by 13
  7. 2025Are formal and functional linguistic mechanisms dissociated in language models?cited by 11
  8. 2025The Serial Scaling Hypothesiscited by 11
  9. 2025The Validation Gap: A Mechanistic Analysis of How Language Models Compute Arithmetic but Fail to Validate Itcited by 10
  10. 2026Procedural Pretraining: Warming Up Language Models with Abstract Datacited by 10
  11. 2025Towards Understanding the Nature of Attention with Low-Rank Sparse Decompositioncited by 9
  12. 2025Do Large Language Models Truly Grasp Addition? A Rule-Focused Diagnostic Using Two-Integer Arithmeticcited by 8
  13. 2025Understanding In-context Learning of Addition via Activation Subspacescited by 8
  14. 2025Can LLMs Reason Abstractly Over Math Word Problems Without CoT? Disentangling Abstract Formulation From Arithmetic Computationcited by 8
  15. 2025Thinking Sparks!: Emergent Attention Heads in Reasoning Models During Post Trainingcited by 8
  16. 2025Locate-then-Merge: Neuron-Level Parameter Fusion for Mitigating Catastrophic Forgetting in Multimodal LLMscited by 7
  17. 2025Comprehension Without Competence: Architectural Limits of LLMs in Symbolic Computation and Reasoningcited by 7
  18. 2025Circuit Stability Characterizes Language Model Generalizationcited by 6
  19. 2025Benford's Curse: Tracing Digit Bias to Numerical Hallucination in LLMscited by 6
  20. 2025Mechanistic Indicators of Understanding in Large Language Modelscited by 6
  21. 2025Modular Arithmetic: Language Models Solve Math Digit by Digitcited by 6
  22. 2025Bridging Kolmogorov Complexity and Deep Learning: Asymptotically Optimal Description Length Objectives for Transformerscited by 6
  23. 2025Why Can't Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfallscited by 6
  24. 2025Large Language Models for In-File Vulnerability Localization Can Be "Lost in the End"cited by 5
  25. 2025Trustworthy AI: Safety, Bias, and Privacy -- A Surveycited by 5
  26. 2025None of the Others: a General Technique to Distinguish Reasoning from Memorization in Multiple-Choice LLM Evaluation Benchmarkscited by 5
  27. 2025Bigram Subnetworks: Mapping to Next Tokens in Transformer Language Modelscited by 5
  28. 2025All for One: LLMs Solve Mental Math at the Last Token With Information Transferred From Other Tokenscited by 5
  29. 2025Large Language Model Sourcing: A Surveycited by 5
  30. 2026Reward Modeling for Reinforcement Learning-Based LLM Reasoning: Design, Challenges, and Evaluationcited by 5
  31. 2025CoCo-CoLa: Evaluating and Improving Language Adherence in Multilingual LLMscited by 4
  32. 2025Brain-Inspired Exploration of Functional Networks and Key Neurons in Large Language Modelscited by 4
  33. 2025Failure by Interference: Language Models Make Balanced Parentheses Errors When Faulty Mechanisms Overshadow Sound Onescited by 4
  34. 2025Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Constructioncited by 4
  35. 2025In-Context Algebracited by 4
  36. 2025Large Language and Reasoning Models are Shallow Disjunctive Reasonerscited by 3
  37. 2025Can Global XAI Methods Reveal Injected Behaviours in LLMs? SHAP vs Rule Extraction vs RuleSHAPcited by 3
  38. 2025Addition in Four Movements: Mapping Layer-wise Information Trajectories in LLMscited by 3

50 most cited of 94 All 94 (JSON)

Similar, not cited

Close in wording, no citation either way

  1. 2024An Investigation of Neuron Activation as a Unified Lens to Explain Chain-of-Thought Eliciting Arithmetic Reasoning of LLMs
  2. 2025Language Models Use Trigonometry to Do Addition
  3. 2023Frugal LMs Trained to Invoke Symbolic Solvers Achieve Parameter-Efficient Arithmetic Reasoning
  4. 2024Language Models Do Hard Arithmetic Tasks Easily and Hardly Do Easy Arithmetic Tasks
  5. 2026Shared circuits predict whether LLMs generalize across formats in arithmetic reasoning
  6. 2025A Survey on Mathematical Reasoning and Optimization with Large Language Models
  7. 2026Artificial Intelligence for Mathematical Reasoning: An Integrated Survey of Language Models, Neuro-symbolic Systems, and Verified Discovery
  8. 2026Mathematical Reasoning in Large Language Models: Benchmarks, Architectures, Evaluation, and Open Challenges