# There are 77 right-angled blocks of dimensions 3xx3xx1. Is it possible to place all these blocks in a closed rectangular block of dimensions 7xx9xx11?

Nov 11, 2016

No.

#### Explanation:

Call a $1 \times 1 \times 1$ component a cell.

Consider a $7 \times 11 \times 1$ slice of the rectangular block.

Each $3 \times 3 \times 1$ block intersects it in one of three ways:

• Not at all. In which case it contributes $0$ cells.

• Edgewise. In which case it contributes $3$ cells.

• Full. In which case it contributes $9$ cells.

In any of these three cases, the number of cells in the slice contributed by each $3 \times 3 \times 1$ block is a multiple of $3$.

In order to fill the slice, we need $7 \times 11 = 77$ cells filled, which is not a multiple of $3$.