"Close enough?" An investigation into multiplication errors in LLMs

Multimodal LLMs can accurately perceive numerical content across modalities yet fail to perform exact multi-digit multiplication when the identical underlying arithmetic problem is presented as numerals, number words, images, or in audio form. Because existing benchmarks often lack systematically paired instances across modalities, it remains difficult to compare genuine arithmetic limits within and across model families. We therefore introduce a controlled multimodal multiplication benchmark that factorially varies digit length, digit sparsity, representation (e.g., numerals vs. number words), and modality (text, rendered images, audio), with paired instances from a reproducible generator. We also define arithmetic load, C, as the product of the total and non-zero digit count as a compact, mechanistically motivated proxy for operation count. Across evaluations, accuracy falls sharply as C grows, often nearing zero by C > 100. Indeed, C remains predictive of performance across modalities and models, with R-squared often > 0.5, nearing the value from more complex measures of arithmetic load that count the number of intermediate arithmetic steps. A separate perception-versus-computation decomposition shows that multimodal degradation is primarily computational rather than perceptual: on matched-perception checks, models are near-perfect (> 99%) across modalities, even when multiplication accuracy drops. Beyond measuring when models fail, we ask which procedures they are predisposed to follow. We introduce a forced-completion loss probe that scores heuristic-specific reasoning prefixes--including columnar multiplication, distributive decomposition, and rounding/compensation. Here, decomposition is favored in both text and vision modalities; heuristic-specific LoRA adapters produce near-orthogonal updates yet degrade accuracy, indicating the base model maintains a well-tuned internal router.

Autoregressive Large Language Models (LLMs) frequently struggle with deterministic multi-step algorithmic tasks such as multi-digit multiplication and long division. In this paper, we investigate the mechanics of multi-step arithmetic in compact "Tiny" Transformers (~10.6M non-embedding parameters, 49.3M total) trained on synthetic data across four basic operations (+, -, *, /) unrolled as step-by-step scratchpads. First, we establish the necessary training foundations: (1) dataloader sequence padding creates an 83% gradient starvation artifact that collapses accuracy from 40% to 1%, remediated via continuous sequence packing; (2) linguistic pretraining is an essential prerequisite (<= 2.0% without it); and (3) modern architectural primitives (RoPE, RMSNorm, SwiGLU) and Sparse Mixture of Experts (MoE) substantially improve additive reasoning over baseline GPT-2. Second, we demonstrate that algorithmic scratchpad formulation directly dictates success. Introducing a deterministic Digit-by-Digit Long Division scratchpad within a 4-stage Hierarchical Developmental Curriculum dramatically elevates single-digit division from 4.0% to 86.7% accuracy on a 4,000-problem held-out benchmark. In contrast, multi-digit multiplication remained challenging: detailed error analysis revealed that while the model correctly computed single-digit sub-products and place-value zeros, our FOIL scratchpad failed because it forced a simultaneous summation of up to nine multi-digit terms in a single step without pairwise intermediate accumulation. Finally, we identify two key boundaries: performance collapses to 0.00% on unseen 4-digit operands, and unbuffered training induces catastrophic forgetting, collapsing division accuracy from 86.7% down to 0.00%.

Transformers asked to multiply multi-digit numbers in a single forward pass often fail, and interpretability studies of pretrained language models find arithmetic solved by input-range heuristics rather than by an explicit carry. We train small Llama-style transformers from scratch on 4x4 multiplication without chain of thought and find that the input format is decisive: inserting a space token between digits raises exact-match accuracy from 1% to 89%. Output positions are learned in carry-chain order, with the middle digits, which have the longest-range dependencies, learned last. Inside the model, the separator token that predicts each digit (its prediction slot) encodes the carry-in as an angle on a ring in the residual stream; examples with more distinct carry values fill more of the ring. Activation patching between examples matched on the column sum shows that this state is causally used before the last layer: patching the prediction slot alone transfers the source carry in up to 84% of cases after block 4 for one middle column of our best model, while for other columns the carry is first assembled at the neighboring answer slot before reaching its own. Remaining errors are almost always off by one, consistent with a small error on the carry or on the circular digit code.