Freaky Factorials

Since n can only be up to 9. Then it would be easy to try the numbers $n$ from 1-9 if $\sqrt{n!+1}$ is an integer.

Values of $n$ could be $4$ , $5$ and $7$ . Using these numbers we could get $m$ as $5$ , $11$ , and $71$ . $5+11+71=$ $\boxed{87}$ .

According to the question, $\sqrt{n!+1}$ is an integer m such that $m \leq 999.$ In order to find m, we need to find the value of $n!+1 \leq 999^{2}$ The value of n which suffice this inequality are limited, i.e. $n \epsilon\{1,2,3,4,5,6,7,8,9\}$ $\Rightarrow (n!+1) \epsilon\{1!,2!,3!,4!,5!,6!,7!,8!,9!\}$ Now, taking case by case, we find that only three cases suffice that m is an integer less than 1000 and they are, $\sqrt{4!+1}=\sqrt{25}=5$ $\sqrt{5!+1}=\sqrt{121}=11$ $\sqrt{7!+1}=\sqrt{5041}=71$ Thus the values of m are $5,11,71$ Hence, sum of all values of m, $5+11+71$ $=\boxed{87}$

Clearly $n > 0$ and $m < 1000$

So, $m = \sqrt(n!+1) < 1000 \Rightarrow n! < 1000^2-1$

Hence, $n < 10$ because $10! = 3628800 > 1000^2-1$

Since $m$ and $n$ both should be integers, we find that only for $n = 4,5,7$ , $m$ is an integer.

for $n=4$ , $m=\sqrt(4!+1) = 5$

for $n=5$ , $m=\sqrt(5!+1) = 11$

for $n=7$ , $m=\sqrt(7!+1) = 71$

Summing up, all the values of $m = 5+11+71=\boxed{87}$

we observe that here root over of 9!+1 exceeds 1000 so the value of n must be less than 9 we have thus the following results

\sqrt{4!+1}=5

\sqrt{5!+1}=11

\sqrt{7!+1}=71

Required sum is \boxed{87}

:

We just need to check for $n=1$ to $n=9$

( because $10! + 1 = 3,628,801 > 1,000,000 = 1000^2$

so for $n \geq 10, \sqrt{n!+1} > 1000$ )

Now

$1! + 1 = 2 \rightarrow$ Not a perfect square

$2! + 1 = 3 \rightarrow$ Not a perfect square

$3! + 1 = 7 \rightarrow$ Not a perfect square

$4! + 1 = 25 = 5^2$

$5! + 1 = 121 = 11^2$

$6! + 1 = 721 \rightarrow$ Not a perfect square

$7! + 1 = 5041 = 71^2$

$8! + 1 = 40,321 \rightarrow$ Not a perfect square

$9! + 1 = 362,881 \rightarrow$ Not a perfect square

So possible values of m are $5, 11, 71$

Answer: $5 + 11 + 71 = \boxed{87}$

n n! n!+1 perfect square?
1...1....2....no
2...2....3....no
3...6....7....no
4..24..25...yes -- 5
5..120.121..yes -- 11
6..720..721..no
7..5040..5041..yes -- 71
8..40320..40321...no
9...362880..362881..no
10...3628800 greater than 1 million sqrt(10!) is greater than 1000

5+11+71