# hexagons and squares

iamatiger
Posers and Puzzles 26 Nov '12 00:17
1. 26 Nov '12 00:173 edits
Prove that if a hexagon is dissected into identical equilateral triangles, and these triangles are rearranged into smallr hexagons (6 triangles to a small hexagon) then you always get a square number of small hexagons.
2. forkedknight
Defend the Universe
26 Nov '12 17:05
As far as I know, the only way to disect an equilateral triangle into smaller equilateral triangles is to cut it into a triforce, resulting in 4 smaller triangles.

4 is a square number.

The simplest way to dissect a hexagon into equilateral triangles results in 6 triangles.
3. wolfgang59
26 Nov '12 19:11
Originally posted by forkedknight
As far as I know, the only way to disect an equilateral triangle into smaller equilateral triangles is to cut it into a triforce, resulting in 4 smaller triangles.

4 is a square number.

The simplest way to dissect a hexagon into equilateral triangles results in 6 triangles.
An equilateral triangle also divides into 9 smaller ones, ... or 16, ... or 25.

How to prove?

Consider an equilateral triangle made up of rows of smaller triangles. Any row will have as many triangles as the one above plus 2. So the rows (satrting at top) contain 1,3,5,7,9 etc triangles.

The sum of any number of odd numbers is always a square so any equilateral triangle can be made from a square number of smaller triangles.

nth odd number is (2n=1)

S = 1+3+5+ .... +(2n+1)
S= (2n+1) + (2n-1) + (2n-3) + .....+ 5+3+1

simply add the two lines above

2S = 2n + 2n + 2n + ... +2n
there are n terms so

2S =2n^2
S=n^2

Rubbish proof .... sorry ðŸ˜³