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Picturing Triangle Numbers

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Forgotten Number

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

The factorials are:

2 x 1 = 2,
3 x 2 x 1 = 6,
4 x 3 x 2 x 1 = 24,
5 x 4 x 3 x 2 x 1 = 120,
6 x 5 x 4 x 3 x 2 x 1 = 720....

The 3 digit numbers less than 500 which are sums of consecutive factorials are:
144, 150, 152 and 153.

The only one of these numbers which satisfies the condition that it is equal to the sum of the cubes of its digits is 153 because 153 = 1 + 125 + 27 so this must be the forgotten number. The information about triangular numbers was not needed but it is easily checked that 153 is a triangular number.

An excellent solution was recieved from Ben from Methwold High School.