Question

$abcd$ is a square. how many squares have two or more vertices in the set $\{a, b, c, d\}$?

Answer 1

There are 4 squares having a side corresponding to a side of square abcd.

There are 4 squares having a side corresponding to the diagonals of square abcd.

And then the original square

Therefore, there are 4 + 4 + 1 = 9 squares having two or more vertices in the set {a, b, c, d}.

Answer 2

Answer:

13

Step-by-step explanation:

There are (4 choose 2) = 6 ways to choose two points P and Q from the set of four points {A, B, C, D}. Two squares can be formed with P and Q as adjacent vertices (one on each side of PQ), and one square can be formed with P and Q as opposite vertices. This gives us 6 x 3 = 18 squares total.

However, we must subtract repeats. We see that the original square ABCD is counted 6 times, so the number of actual squares is 18 - 5 = 13.