Forgotten number
Problem
I have forgotten the number of the combination of the lock on my
briefcase. I did have a method for remembering it...
It went as follows:
The number is equal to the sum of the cubes of its own digits.
It is a three digit number less than 500
It is the sum of consecutive factorials.
It is a triangle number.
Getting Started
1 factorial = 1
2 factorial = 1 x 2 = 2
3 factorial = 1 x 2 x 3 = 6
4 factorial = 1 x 2 x 3 x 4 = 24
5 factorial = 1 x 2 x 3 x 4 x 5 = 120...
The triangle numbers are 1, 3, 6, 10, 15, 21...
Student Solutions
The factorials are:
1,
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.