Mark read that the ancient Greeks use to arrange pebbles to represent numbers. He used square on a grid instead of pebbles to model both triangular and square numbers

Question

How can you divide a square number into q two triangular numbers?
(6 triangular number) and (9 sqaure)

Answer 1

Say represent 5² as a square:

OOOOO
OOOOO
OOOOO
OOOOO
OOOOO

and split it up into two triangles next to the diagonal:

O
OO
OOO
OOOO
OOOOO

and (it should be the mirror image)
OOOO
OOO
OO
O

So in general, for an n×n square, we can split it into the larger triangle of base n using n(n+1)/2 pebbles, and the smaller of side (n-1) using n(n-1)/2 pebbles.

The total:
n(n+1)/2+n(n-1)/2
=(n²+n +n²-n)/2
=n²

Answer 2

Thanks :)

Answer 3

You're welcome!