Problem 34

145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145.

Find the sum of all numbers which are equal to the sum of the factorial of their digits.

Note: as 1! = 1 and 2! = 2 are not sums they are not included.

Solution

This is a pretty easy problem since we can brute force it. The first thing that comes to mind is, that we must calculate factorials up to 9 over and over again, which makes them ideal to pre-calculate them:

var FACT = [1];
for (var i = 1; i < 10; i++)
    FACT[i] = FACT[i - 1] * i;<br />

Which results in the following list:

FACT = [1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880]

If we brute-force the task, the question for a good upper and lower bound arises. The problem description excludes 1! and 2! to be valid, since they are not sums. 3 is a reasonable lower bound, but since the factorial of a one-digit number - except 3 itself - has always more than one digit, we can start with 10.

For the upper bound it's a bit trickier to find a reasonable value. If we take a number \(n\), we can bound it's number of digits \(d\) to

\[10^{d-1} \leq n < 10^d\]

Let's say we form a \(d\) digit long number of all 9 as the maximum, the sum of it's factorial digits would be \(9!d\) and hence

\[\begin{array}{rl} & 10^{d-1} \leq d\cdot 9! < 10^d\\ \Leftrightarrow & (d-1)\log(10) \leq \log(d) + \log(9!) < d\log(10)\\ \Leftrightarrow & d-1 \leq \log_{10}(d) + \log_{10}(9!) < d\\ \Leftrightarrow & d-1 \leq \log_{10}(d) + 5.56 < d\\ \Leftrightarrow & d - \log_{10}(d) \leq 6.56 < d - \log_{10}(d)+1\\ \Rightarrow & d - \log_{10}(6) \leq 6.56 < d - \log_{10}(6)+1\\ \Leftrightarrow & d \leq 7.33 < d+1\\ \end{array}\]

From this derivation follows, that a maximum upper bound can have at most 7 digits and will fail for any \(d> 7\), which gives an upper bound of 9999999, a 7 digit number of all 9. But we know, that \(7\cdot 9!=2540160\), so we have a better upper bound. Okay, we know the upper bound must be 7 digit long and the first digit of the upper bound is at most 2, resulting in two possible ways a 6 digit number of 9 can be formed, so the new upper bound becomes \(2!+6\cdot 9!\). This implies that if \(n\) is a 7 digit number, either the second digit must be 0 or 1 or the last digit is 1. 

If the first digit is 2 and thus the second digit is 0 or 1, the numbers are limited by \(2!+1!+5\cdot 9!\) = 1814403 - which is a contradiction to the first digit being 2. Thus, a 7-digit number can be at most 1999999.

Another observation is that all factorials of digits above 4 will have 2 and 5 as a factor and thus end with 0. If all but the first digit of the 7 digit number are at least 5, the last digit will be 1, coming from \(1!\) of the first digit, which is a contradiction to the statement that all 6 digits are at least 5. This means that at least one of the 6 remaining digits can be at most 4, which reduces the upper bound again: \(1!+4!+5\cdot 9!=1814425\). If we assume \(n\) is a 7 digit number, the second digit is at most 8. If the second digit is now 5, one of the remaining digits has to be at most 4. This implies an upper bound of \(1!+8!+4!+4\cdot 9!=1491865\), which is a contradiction that the second digit has to be at least 5. Therefore the second digit can be at most 4 and the new upper bound is 1499999.

Okay, let's come to the code and begin with a helper function which calculates the sum of the factorial of their digits:

function factorialSum(n) {

    var sum = 0;
    while (n > 0) {
        sum+= FACT[n % 10];
        n = n / 10 | 0;
    }
    return sum;
}

Now calculating final solution is pretty straightforward:

function solution() {
    var sum = 0;
    for (var i = 10; i <= 1499999; i++) {
        if (i === factorialSum(i)) {
            sum+= i;
        }
    }
    return sum;
}