Question

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`?

Answer 1

No.

Explanation:

Call a `1 xx 1 xx 1` component a cell.

Consider a `7 xx 11 xx 1` slice of the rectangular block.

Each `3 xx 3 xx 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 xx 3 xx 1` block is a multiple of `3`.

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