Enigma 1324: Rhombic squares
From New Scientist #2483, 22nd January 2005
Imagine you have two identical isosceles right-angled triangles. Lay one down so that one short side is horizontal and one is vertical. Lay the second one down so that a short side of the second is against a short side of the first, the two hypotenuses being parallel, so forming a rhombus, one example being as shown.
You are asked to place a digit at each of nine points, the four corners of the rhombus, the midpoint of each of the four sides, and the midpoint of the shorter diagonal such that eight numbers read horizontally and vertically from the top are all different perfect squares.
Alison and Bertha have each found a different solution. One of Alison’s two-digit squares is 81.
Which two did Bertha find?