Henry Ernest Dudeney/Puzzles and Curious Problems/247 - The Counter Cross/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $247$

Arrange twenty counters in the form of a cross, in the manner shown in the diagram.

Now, in how many different ways can you point out four counters that will form a perfect square if considered alone?

Thus the four counters composing each arm of the cross, and also the four in the centre, form squares.

Squares are also formed by the four counters marked $\text A$, the four marked $\text B$, and so on.

And in how many ways can you remove six counters so that not a single square can be so indicated from those that remain?

There are $21$ squares:

$9$ of them will be of the size indicated by the four instances of $\text a$

$4$ of them will be of the size indicated by the four instances of $\text b$

$4$ of them will be of the size indicated by the four instances of $\text c$

$2$ of them will be of the size indicated by the four instances of $\text d$

$2$ of them will be of the size indicated by the four instances of $\text f$.

These are indicated by the blue dotted squares in the above diagram.

If you now remove all counters marked $\text e$, no squares can be formed from the remaining counters.