X is called a triangular number if X = 1 + 2 + ... + n for

a number that satisfies both requirements can be expressed as n(n+1)/2 and k^2.

Since it is the same number:

n(n+1)/2=k^2
=> n(n+1) = 2 * k^2

careful observation reveals that we are looking for a positive integer that can be expressed as a product of two consecutive positive integers and also as double of a perfect square.

quick things to realize:
1. two consecutive positive integers would never have a common factor (except 1 which we can ignore)
2. two consecutive positive integers would have one odd and one even integer
3. odd integer has to be a perfect square
4. 2 is a factor of only the even integer which would cancel out the 2 from the other side of the equation
5. after the factor 2 is taken out of the even integer, it is also a perfect square

now lets start looking for odd perfect squares whose adjacent even integer is double of a perfect square.

1 and 2 - our number is 1 * 2 /2 = 1
9 and 8 - our number is 9 * 8 /2 = 36
25 - reject, 24 or 26 do not satisfy
49 and 50 - our number is 49 * 50 /2 = 1225