Slime slayer

Let's first prove that an $n$ -level slime will take $4^n$ hits total to be killed. By induction:

When $n=0$ , the $n$ -level slime takes $1=4^0$ hits and the statement holds true.

Then, let's suppose the statement holds true for $n=m$ and prove it holds true for $n=m+1$ : The $m$ -level slime takes $4^m$ hits to be destroyed. The $m+1$ -level slime takes $4^m$ hits to divide into three level $m$ slimes. The total number of hits for slime level $m+1$ is $4^m+3 \times 4^m = 4 \times 4^{m} = 4^{m+1}$ , and the statement is proven.

This means we can represent the number of slimes of each level by defining $a_0, a_1, ...$ for each $n$ where $a_n$ is the number of $n$ -level slimes in the army. By the fact proven above and the fact tha the whole army takes 10 000 to destroy, we get

$a_0+4a_1+4^2a_2+...+4^na_n=10000$

where $n$ is the level of the highest-level slime.

If we present the number $10 000$ as a number in 4-base number system, i.e. $10 000=(a_na_{n-1}...a_1a_0)_4$ the equation above holds true which means this combination of slimes is one of the possible ones. Let's show it is the one with the least slimes.

It is a known fact that each integer $n$ has an unambiguous presentation in $k$ -base number system (and thus, the number 10 000 in 4-base system), i.e. $n=a_0+ka_1+k^2a_2+...$ where $a_0,a_1,...$ are integers between $0$ and $k-1$ . That means, all other combinations of slimes has at least one of numbers $a_0,a_1...$ with a value greater than $3$ . If we have one combination, and $a_i$ is such number in it, we can form another combination of slimes by replacing four $i$ -level slimes with one $i+1$ -level slime. This combination has less slimes, and we have proven that the combination of slimes formed by the 4-base presentation of 10 000 has the least slimes.

Let's transform 10 000 into base 4:

$10 000=2\times4096+1024+3\times256+16=2\times2^6+1\times2^5+3\times2^4+1\times2^2$

The combination has 2 level 6 slimes, 1 level 5 slime, 3 level 4 slimes and 1 level 2 slime which makes $2+1+3+1=\boxed{7}$ total.