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: The Clever Coloring

To prove it’s impossible, we use a special coloring scheme. Instead of coloring individual \(1×1×1\) cubes, imagine dividing the \(6×6×6\) cube into 27 smaller \(2×2×2\) sub-cubes.

Now, color these 27 sub-cubes in a 3D checkerboard pattern (alternating Black and White). Since 27 is an odd number, there won’t be an equal amount of Black and White sub-cubes. Let’s say we end up with:

  • 14 Black sub-cubes
  • 13 White sub-cubes

Since each sub-cube contains exactly 8 unit cubes, our entire \(6×6×6\) cube has:

  • \(14 × 8 = \mathbf{112}\) Black unit cubes
  • \(13 × 8 = \mathbf{104}\) White unit cubes

Step 3: Brick Coverage Parity

Now, think about placing a single \(1×1×4\) brick anywhere in this grid. Along its length, the axis is divided into three 2-unit segments (the sub-cubes). A brick of length 4 must cover 4 consecutive units. It can either align perfectly with two segments (taking 2 units from each), or it can sit in the middle (taking 1 unit from the first segment, 2 from the middle, and 1 from the third).

Because the alternating sub-cubes have opposite colors, the outer segments in the middle scenario share the same color.

  • Aligned: 2 Black, 2 White.
  • Middle: 1 Black + 2 White + 1 Black = 2 Black, 2 White.

Therefore, no matter where it’s placed, every single brick covers exactly 2 Black and 2 White unit cubes.

Step 4: The Contradiction

If we were to pack 53 bricks, we would need:

  • \(53 × 2 = \mathbf{106}\) Black unit cubes
  • \(53 × 2 = \mathbf{106}\) White unit cubes

But look at our cube: there are only 104 White unit cubes available!

Since 106 > 104, it is mathematically impossible to fit 53 bricks into the cube.

Final Answer

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

Reference