Triangle numbers are :

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

Somebody tells you a random positive integer of any size. Is there a quick way to work out if it is a triangle number or not (in other words, without going through all of the triangle numbers untill you get to a number as high or higher as the integer you've been told)?

(In reply to

Puzzle Solution: Method I by K Sengupta)

Let the random positive integer be P.

If P is a triangle number, then by definition P = m(m+1)/2, for some positive integer m.

Then, 2*P = m(m+1) = m^2 + m

But for any positive integer m, we must have:

m^2 < m^2 + m < (m+1)^2

Thus, if V(2*P) lies between two consecutive integers and if by multiplying these consecutive positive integers and dividing by 2, we get P then we will know for certain that P is a triangular number.

*Edited on ***May 8, 2008, 3:37 pm**