abcd is a square. How many squares in the same plane have two or more vertices in the set {a,b,c,d} ?

I'm counting 8.

(a,b), (b, c), (c, d), (d, a) each with two other vertices not in {a,b,c,d}

(a,c) to the ne and sw

(b,d) to the nw and the se

I counted 8 too.

1 square share the vertices a and b.

1 square share the vertices c and b.

1 square share the vertices c and d.

1 square share the vertices a and d.

2 square share the vertices c and a.

2 square share the vertices b and d.