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Painted Cube Puzzle

Solve five sourced painted-cube counting questions with a repeatable position-class method and exact 1,000-point scoring.

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How to use

  1. 1.Read the grid size and identify whether the question asks for corner, edge-interior, face-interior, core, or any painted unit cubes.
  2. 2.Classify positions before calculating; use the arrow keys or pointer to move among the four possible counts.
  3. 3.Press Enter, Space, or a choice button. Complete all five questions for 1,000 points before two distinct misses lock a round.

About Painted Cube Puzzle

Painted Cube Puzzle is a five-round spatial counting game built around one precise setup: every outside face of a large cube is painted, and the cube is then cut into an n × n × n grid of equal unit cubes. Each round asks for the number of small cubes in one position class. You may need to count corner cubes with three painted faces, edge-interior cubes with two, face-interior cubes with one, hidden core cubes with none, or every cube touched by paint. The questions are fixed and deterministic, so a repeated run is a chance to improve your method rather than hope for easier random numbers.

The central idea is to classify positions before doing arithmetic. A cube has exactly three painted faces only when it sits at a corner of the original solid. A cube with exactly two painted faces lies along an edge but not at either corner. A cube with exactly one painted face lies inside one of the six outer faces while avoiding its boundary. A cube with no painted faces belongs to the interior core. These four classes do not overlap, and together they account for all n³ unit cubes. Keeping the classes separate prevents the most common error: counting corners again while counting edges or face interiors.

The standard formulas follow directly from that classification. There are always 8 corner positions. After removing the two corners from each of 12 edges, the two-painted count is 12(n − 2). Each of the 6 faces has an interior square of side n − 2, so the one-painted count is 6(n − 2)². Removing the outer layer leaves an unpainted cube of side n − 2, giving (n − 2)³ cubes with no painted face. The number with at least one painted face can be found either by adding the first three painted classes or by subtracting the unpainted core from the total: n³ − (n − 2)³.

Use the visual grid as a scale cue, then solve on a mental worksheet. First identify the requested class. Second write only the factors that belong to that class: corners, edge interiors, face interiors, or core. Third check whether the prompt says exactly a number of painted faces or at least one. “Exactly one” excludes all edge and corner cubes, while “at least one” includes every painted class. Finally compare your result with the four choices. The game never marks a correct option in advance, and the same logic works without relying on the drawing.

Each correct answer awards 200 points, and all five correct answers produce the exact maximum score of 1,000. One distinct incorrect answer can be repaired on the same round. Repeating that same wrong option does not consume another mistake, which prevents accidental double taps from ending a run. A second different wrong option locks the run, and Restart resets the puzzle completely. After completion or deadlock, scoring and navigation are frozen. This makes touch, pointer, and keyboard play behave consistently.

Keyboard players can Tab into the answer grid, use the arrow keys to move through its two-by-two layout, and press Enter or Space to choose. Each answer button has a minimum touch-friendly height, and the layout remains two columns on narrow screens so the directional model stays stable. Score, progress, misses, feedback, and terminal states are conveyed as text rather than color alone. The isometric diagram has an accessible label describing the cube dimensions and the fact that all six outside faces were painted.

The formulas and external checks are documented rather than invented for the quiz. NRICH at the University of Cambridge provides the position-class formulas and worked tables for several cube sizes. A Snow College mathematics contest answer key independently states the corner, edge, face-interior, and core expressions. Math Is Fun supplies an independent worked distribution for a 4 × 4 × 4 cut cube. Automated tests lock those published examples, verify conservation across sizes 2 through 20, confirm that every question has one correct option, and exercise scoring, repair, deduplication, deadlock, Restart, cursor boundaries, and terminal immutability.

This puzzle is useful as a compact lesson in combinatorial classification, surface area intuition, and three-dimensional visualization. It is recreational and educational practice only. It is not an intelligence test, a diagnostic assessment, or a claim about a player’s general mathematical ability.

Methodology & sources

For integer n >= 2, derive disjoint position classes after all six outside faces are painted: three painted faces = 8 corners; two = 12(n - 2) edge-interior positions; one = 6(n - 2)^2 face-interior positions; zero = (n - 2)^3 core positions; any painted = n^3 - (n - 2)^3. Five fixed questions award 200 points each. One distinct wrong option is repairable, the same wrong option is deduplicated, a second distinct wrong option deadlocks the run, and terminal states are immutable. External fixtures from NRICH, Snow College, and Math Is Fun plus conservation checks for n = 2 through 20 validate the formulas independently of the option order.

Frequently asked questions

What is a painted cube puzzle?
All six outside faces of a large cube are painted before it is cut into equal unit cubes. The puzzle asks how many small cubes have a specified number of painted faces.
How are painted unit cubes counted?
Separate the original cube into corners, edge interiors, face interiors, and its hidden core. Those disjoint position classes correspond to three, two, one, and zero painted faces.
Can I play with a keyboard or on a phone?
Yes. The answer buttons support touch and pointer input. Keyboard players can Tab into the grid, move with arrow keys, and choose with Enter or Space.

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