Can you fit 53 bricks, each of size 1×1×4, into a 6×6×6 cube?

This spatial puzzle blends volume computation with parity and tiling logic. Let’s break it down.

Step 1: Volume Check

  • Cube volume: (6 × 6 × 6 = 216) unit cubes
  • Each brick occupies: (1 × 1 × 4 = 4) unit cubes
  • 53 bricks cover: (53 × 4 = 212) unit cubes

So, the total volume used would be 212, leaving exactly 4 unit cubes empty.

At first glance, this seems possible.

Step 2: Parity Insight

To go further, we analyze the structure of the cube and the constraints of packing.

Color the cube in a 3D checkerboard pattern: alternate black and white unit cubes such that adjacent cubes (in any direction) have opposite colors.

In a (6×6×6) cube, this gives:

  • 108 black and 108 white cubes.

Step 3: Brick Coverage Pattern

Each 1×1×4 brick occupies 4 consecutive cubes in a straight line. No matter how it’s placed (along x, y, or z direction), it will cover:

  • 2 black and 2 white cubes.

Why? Because starting on black leads to black-white-black-white or vice versa—4 in total, equally split.

Step 4: Total Balance

53 bricks × (2 black + 2 white) = 106 black + 106 white

But the cube has 108 of each.

That means the remaining 4 unit cubes must be made up of 2 black and 2 white cubes—perfectly balanced.

So the parity check passes.

Step 5: Can You Actually Pack Them?

Despite parity and volume working out, no known arrangement can fit 53 such bricks into a 6×6×6 cube.

This was a research-level puzzle, and the key insight comes from modulo arithmetic.

Mod-4 Grid Analysis

Imagine labeling all unit cubes with coordinates (x, y, z). Now sum all coordinates mod 4.

Each brick always covers 4 distinct values of x+y+z mod 4. Thus, the sum mod 4 of all positions used by bricks must be divisible by 4.

When applied to the cube and 53 bricks, this leads to a modulo contradiction.

Final Verdict

Though volume and parity seem to allow a solution, deeper structure-based arguments prove it’s not possible.

Final Answer

No, you cannot pack 53 bricks of size 1×1×4 into a 6×6×6 cube.

Reference