Balanced Steinhaus triangles

A familiar fact from the game of pool is that 15 objects can be arranged in an equilateral triangle of side length 5. This is because 15=1+2+3+4+5 is the fifth triangular number, T(5), where T(n)= n(n+1)/2. If we have balls of five different colours, it is possible to create an arrangement like this where there are equally many balls of each colour, for…