Mary and Lucy’s Coin

Lucy has forty-eight identical coins. She places half of them, with heads facing up, in a 4 by 6 rectangle. She uses the other twenty-four coins, with tails facing up, to form a frame around the rectangle. Mary has a different number of coins. She too, can use half of her coins to form a rectangle and the rest to precisely surround that rectangle with a frame. How many coins does Mary have ?

Let be the number of coins in the row and be the number of coins in column. Thus there are . Now this should be same as the number of coins in the frame . We can write a small sagemath program to check

for x in range(1,1000):
    for y in range(1,1000):
        if (x*y==(2*x+2*y+4)):
            print(x,y)

3 10
4 6
6 4
10 3 

Thus there is just one more rectangle that exists and that rectangle is Quite amazing.