Puzzle| 53 bricks of size 1x1x4 in 6x6x6 sized cube

Question:

Is it possible to fit 53 bricks of size 1×1×4 into a 6×6×6 box? Why or why not?

53 bricks of size 1x1x4 in a 6x6x6 sized box

Solution: 

If we look at the total volume of the larger cube, 53 cubes have a volume of 212 which is smaller than the box’s volume of 216. So we might be thinking, that the answer to this problem is yes. Yet, we can show that it is not, i.e. it is impossible to pack all the bricks into the box using a similar approach as the chessboard problem. 

Follow the given train of thought: 

• Let us imagine that the 6x6x6 box, is comprised of 2x2x2 boxes. Clearly, there will be 27 small cubes (2x2x2 cubes).
• For each 1x1x4 brick, 2 cubes of size 2x2x2 are required and
• 2 such 2x2x2 cubes are completely filled by 4 bricks. 
• So we can use 26 cubes of size 2x2x2 to accommodate 4*13 = 52 bricks.
• There will be 1 cube of size 2x2x2 remaining and 1 brick.

Remaining space of size 2x2x2

No matter how we arrange, we will not be able to arrange this brick in the remaining empty block. See the images below to understand it more clearly.

The brick cannot be arranged horizontally

The brick cannot also fit if it is arranged vertically.

The brick cannot be arranged vertically

Conclusion: We cannot fit 53 bricks of size 1x1x4 inside a cube of size 6x6x6.