**From New Scientist #2595, 17th March 2007**

A triangular number is an integer that fits the formula n(n+1)/2; such as 1, 3, 6, 10, 15.

45 is not only a triangular number (9 × 10)/2, but also the product of three different factors (1 × 3 × 15) each of which is itself a triangular number.

But Harry, Tom and I don’t regard that as a satisfactory example since one of those factors is 1; so we have been looking for triangular numbers that are the product of three different factors each of which is a triangular number other than 1.

We have each found a different 3-digit triangular number that provides a solution. Of the factors we have used, one appears only in Harry’s solution and another only in Tom’s solution.

What are the three triangular factors of my solution?

[enigma1434]

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The generation of the candidate triangular numbers is straightforward. But the code to chose the right three numbers is messier than I’d hoped – especially as it’s easy to find the solution by inspection of the candidates. This Python program runs in 50ms.

Solution:The three factors in your solution are 3, 6 and 21.Written in python 3,

solve()

378 3 6 21

990 3 6 55

630 3 10 21