Henry Ernest Dudeney/Puzzles and Curious Problems/345 - The Egg Cabinet/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $345$

A man has a cabinet for holding birds' eggs.

There are $12$ drawers, and all -- except the first drawer, which holds the catalogue -- are divided into cells by intersecting wooden strips,

running the entire length or width of a drawer.

The number of cells in any drawer is greater than that of the drawer above.

The bottom drawer, No. $12$, has $12$ times as many cells as strips,

No. $11$ has $11$ times as many cells as strips, and so on.

Can you show how the drawers were divided -- how many cells and strips in each drawer?

Give the smallest possible number in each case.

Let the drawer number be $n$.

Then there will be $2 n - 1$ strips one way and $2 n - 3$ strips the other way.

This gives $4 n^2 - 4 n$ cells and $4 n - 4$ strips.

So in the $12$th drawer we get $23$ and $21$ strips, or $44$ altogether, and hence $528$ cells.

This applies to all drawers except the second, where we may have any number of strips one way, and one strip the other way.

So $1$ and $1$ will serve, but because all cells have intersecting strips, it cannot have just one.

Hence there are $262$ strips in all, and $2284$ cells.