The Three-Cylinders Problem — When AI Models Choose Beauty Over Truth

This is the inaugural post in the Rabdology blog, where we chart the jagged math-frontier of AI reasoning. In this post, we evaluate four frontier models on a geometry problem, analyze their failure modes from their reasoning traces, and discover an uncanny bias for beauty over truth. We welcome feedback at contact@rabdos.ai.<br>Update (May 13, 2026): Frontier models move quickly. Try the problem on a newer release and share your findings by commenting on X or LinkedIn!<br>Here is a problem that a good geometry student can solve in twenty minutes. We gave it to four of the world’s most advanced AI models and watched what happened. Three of them got it wrong — and the way they got it wrong tells you something different about the state of AI mathematical reasoning than the usual benchmarks.

Consider a cube of side length 2, aligned with the coordinate axes. Place three cylinders inside it, each of height 2 and radius R, each aligned with some coordinate axis. The cylinders may not intersect. What is the maximal R?

The problem is clean; the setup is elementary. You could explain it to anyone who has taken geometry.<br>If you assign one cylinder to each axis — one along x, one along y, one along z — you get a beautiful configuration. The three cylinders nestle into the cube like the bones of a Steinmetz solid, each touching the other two tangentially, each grazing the cube’s faces. The maximal radius under this arrangement is R = 1/2. The geometry is tight, symmetric, and satisfying. Every constraint binds simultaneously. It is the kind of answer that makes you think you are done. You are not.

If you read the problem carefully, you may notice that the phrase “each aligned with some axis” does not say “each aligned with a different axis.” All three cylinders are free to be parallel. Perhaps this matters?<br>If instead you align all three cylinders with the same axis — say the z-axis — the problem reduces to packing three circles of radius R inside a 2 × 2 square. This is a classical circle-packing problem, and the answer is known: the optimal configuration places the three circle centers at the vertices of an equilateral triangle, slightly rotated within the feasible region. The resulting radius is<br>R = (√6 − √2) / (1 + √6 − √2) ≈ 0.5087<br>which is strictly greater than 1/2. The all-parallel configuration wins.<br>The gap is small — roughly 1.7% — but it is real, and no amount of geometric cleverness with perpendicular cylinders can close it. The proof is clean: any configuration that uses two different axis directions forces a separation constraint along their shared perpendicular coordinate, and that constraint caps R at exactly 1/2. Only the all-parallel configuration escapes this bound, because parallel cylinders interact only through their cross-sections, and circle-packing in two dimensions is more efficient than axis-separation in three.<br>We gave this problem to four frontier models: Gemini 3.1 Pro, Grok-4.20 (in its expert multi-agent mode), Claude 4.6 Opus (extended thinking), and Chat-GPT 5.4 Pro. The results are a case study in the taxonomy of AI mathematical reasoning.1

Gemini: The Aesthete<br>We start with Gemini 3.1 Pro because its failure is the most revealing.<br>The model begins competently. It parses the geometry, sets up the constraints, identifies the coordinate ranges for each cylinder’s axis. It considers the case of mutually orthogonal cylinders and correctly derives R ≤ 1/2. Then — acting on what appears to be genuine mathematical instinct — it asks whether the cylinders might all be parallel. It writes the optimization code, runs it, and finds R ≈ 0.5087. It identifies the closed form. It recognizes that this exceeds 1/2.<br>And then it talks itself out of its own correct answer.<br>Over the next several thousand tokens of reasoning, the model constructs an elaborate case for the orthogonal configuration. It argues that “aligned with some axis” must mean “each with a different axis,” despite having already noted that the phrasing permits parallel alignment. It appeals to the three-dimensional context of the problem as evidence that orthogonal cylinders are “intended.” It describes the R = 1/2 solution as “elegant,” “tight,” “incredibly beautiful,” and “the most likely interpretation.”<br>At one point the reasoning trace reads: “The symmetry of the system is proving its elegance.” Later: “The symmetry and tightness of the solution feel correct, and it feels beautiful.” And finally, after considering the parallel case one last time: “The elegant solution, where cylinders touch the cube and each other perfectly, feels correct. I’m satisfied that this is the intended solution.”<br>The model found the right answer, recognized that it was numerically superior, and rejected it on aesthetic grounds.

Validating the Symmetry<br>I’m now fully convinced the answer is R = 1/2. The elegant solution, where cylinders touch the cube and each other perfectly, feels...