Q: FUN is a three digit number where F, U, N are single digits (not including 0).If FUN = F! + U! + N!, then what is the number FUN?
(where ! = factorial).

A: 145 = 1! + 4! + 5!

Because it is a three digit number, none of the digits can be more than 6 as 7! is 5040 and is therefore four digits. This means that we only need to try 1 2 3 4 5 and 6. But 6! is 720 and we only have 1 2 3 4 5 6 so now 6 is disqualified.
This leaves us with only 1 2 3 4 5.

5! is 120 (5 x 4 x 3 x 2 x 1)
4! is 24 (4 x 3 x 2 x 1)
3! is 6 (3 x 2 x 1)
2! is 2 (2 x 1) and
1! is 1

In order for the number to add up to three digits, 5! has to be one of the components. Our answer will definitely start with F=1 and therefore one of the other digits is 1.

Now we have two out of 3 digits, F = 1 and N = 5 and 1! + 5! = 121 (our subtotal)

Now we will try adding U! 2!, 3! and 4! to our subtotal.

1! + 2! + 5! = 1 + 2 + 120 = 123 not equal to FUN of 125
1! + 3! + 5! = 1 + 6 + 120 = 127 not equal to FUN of 135
1! + 4! +5! = 1 + 24 + 120 = 145 and is EQUAL to FUN of 145.

A shoutout to Colin Porteus who amended the solution for July’s Brain Puzzler — thanks for your sharp mind and eyes.
https://mathconfidence.com/2020/07/27/july-2020-brain-puzzler-solution/