Math Help - A regular problem

I need help finding an equation to solve the problem found here: aha! : paul scott : a regular problem

i have worked it out using trail and error and stuffing around with match sticks but i have know idea how to make an equation to solve it.

Do i need to use simultaneous equations?

Thanks in advance,
Fuzzy

The key is that each side of a figure must be an integer number of matches. And, of course, with "triangle", "square", "pentagon", "hexagon", "octagon", there are 10 pairs. To make "triangle and square" from n matches we must have 3i+ 4j= n, for some integers i and j. To make "triangle and pentagon" we must have 3u+ 5v= n for integers u and v. To make "triangle and hexagon" we must have 3p+ 6q= n, etc.

I just have a homework question I've been stuck on and hoping you guys can help me out a bit.
COnsider the number 48. If you add 1 to it, you get 49, which is a perfect square. If you add 1 to its (1/2), you get 25, which is also a perfect square. Please find the next 2 numbers with the same properties. like 48+1=49 (perfect square)
48/2=24, 24+1=25 (perfect square)

Thank you so much!

thanks for that hallsofive i got it all figured out