### Brain Teasers

# Triangle Numbers

Triangle numbers are formed by adding successive integers, i.e.

1 = 1+0,

3 = 1+2,

6 = 1+2+3,

10 = 1+2+3+4,

etc.

So the first few triangle numbers would be:

1,3,6,10,15,21,28, ...

Suppose somebody tells you a random integer. How can you quickly find out whether or not it is a triangle number?

1 = 1+0,

3 = 1+2,

6 = 1+2+3,

10 = 1+2+3+4,

etc.

So the first few triangle numbers would be:

1,3,6,10,15,21,28, ...

Suppose somebody tells you a random integer. How can you quickly find out whether or not it is a triangle number?

### Hint

The method is linked to another common series.### Answer

Multiply the number by 8 then add 1. If the result is a perfect square, then the original number is a triangle number.For example: 10*8+1 = 81. 81 = 9^2. So 10 is a triangle number.

11*8+1 = 89. 89 is not a perfect square, so 11 is not a triangle number.

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