Triangle numbers are calculated taking each integer plus all the ones before it. The first triangle number is 1, the second is 1+2 or 3, and the third is 1+2+3 or 6.

If you take 8 times a triangle number plus 1, the result will be a perfect square. This number also will be the square of the triangle number's place doubled, plus one.

For example, 6 is third in the triangle number sequence. (1, 3, 6...) This means 8 times 6 plus 1 = 49 equals 3 times 2 plus 1, squared, or 7 squared.

Prove why this works.

yes yes, another old problem that has been solved for a while.  But here is a different approach that I thought was worth mentioning:

A triangle number can be geometrically represented by a triangular stack of items, so that each row has one more item than the row before it. The triangle number’s place is n. This is also the total number of rows in the triangle representation, and the total number of items in the last row. For my purposes, I will show the triangular stack in the form of a right triangle. Here, n=4:

M
MM
MMM
MMMM

Now, let’s make an identical right triangle with n=4. If we rotate it and move it around, we can use these two right triangles to make a rectangle. Notice that this rectangle has dimensions 4x5. Generally speaking, this will be always be an n x (n+1) rectangle.

MOOOO        OOOO
MMOOO   or   MOOO
MMMOO        MMOO
MMMMO        MMMO
             MMMM

Now let’s make 3 more rectangles and rotate them so they alternate from being n x (n+1) and (n+1) x n either clockwise or CCW.

MOOOOWWWW
MMOOOWWWG
MMMOOWWGG
MMMMOWGGG
CCCC GGGG
CCCXQNNNN
CCXXQQNNN
CXXXQQQNN
XXXXQQQQN

Now we have a perfect square, but with a hole in the middle.

Well, this is the geometric representation of 8 times a triangle number (because there is a total of 8 triangles in the final square) plus 1 (for the hole that is left in the middle).

The edge length of this square is 9, but generally speaking it will always be the sum of the two different dimensions of the first rectangle. That is n+(n+1) = 2n+1.

Posted by nikki on 2005-01-27 21:23:40