SIG Painted Cube Probability 2027: Question & Solution

The answer is 1/18, or about 5.6%. Count unpainted faces: 162 faces total minus 54 painted leaves 108 unpainted faces. Only the single fully interior cube is completely unpainted, contributing 6 of those faces — so 6/108 = 1/18. This SIG painted cube probability question is commonly reported by candidates as a conditional-probability trap.

SIG Painted Cube Probability: What This Question Assesses

The question tests conditional probability: you are not picking a random cube, you are picking a random cube given that the face you see is unpainted. Candidates who answer 1/27 (one unpainted cube out of 27) ignore the conditioning. SIG wants you to update on the observation — the exact instinct behind Bayesian thinking in trading.

SIG Painted Cube Probability: How to Answer

• Classify the 27 small cubes. Corners: 8 cubes with 3 painted faces. Edges: 12 cubes with 2 painted faces. Face centers: 6 cubes with 1 painted face. Interior: 1 cube with 0 painted faces.
• Count total faces and painted faces. 27 cubes × 6 faces = 162 faces. Painted faces = 6 large faces × 9 small squares = 54.
• Count unpainted faces. 162 − 54 = 108 unpainted faces — these are the only faces you could be observing.
• Apply the condition. The completely unpainted cube shows 6 unpainted faces. So P(completely unpainted | observed face unpainted) = 6/108 = 1/18 ≈ 5.6%.

Sample line: "Conditioning on seeing an unpainted face, there are 108 candidate faces and only 6 belong to the fully unpainted cube — so 6 over 108, or 1/18."

Common Mistakes

• Answering 1/27 — that ignores the conditioning on the observed unpainted face.
• Counting cubes instead of faces — the observation is a face, so the denominator must be faces (108), not cubes.
• Miscounting painted faces as 27 × something instead of 6 × 9 = 54.

If the conditioning step did not jump out at you, practice "given that you observe X" problems — they are everywhere in SIG's later rounds. Question styles may vary by role and region; confirm on SIG's official careers page.

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FAQ

What is the probability the cube is completely unpainted given an unpainted face? 1/18, about 5.6% — 6 unpainted faces on the interior cube out of 108 total unpainted faces.

Why isn't the answer 1/27? Because the question conditions on observing an unpainted face. Cubes with more unpainted faces are more likely to produce that observation, which changes the odds.

How many small cubes have exactly two painted faces? 12 — the edge (non-corner) cubes of the 3×3×3 cube.

Is this a real SIG interview question? Painted-cube and conditional probability puzzles are commonly reported by candidates in SIG interviews, though exact versions may vary by role and region.

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