## problem

L,Gからなる長さ$K$の文字列に対し、以下の操作を$C$回行う。

• 文字列中のLを最初の長さ$K$の文字列で置き換え、文字列中のGを$K$のGからなる文字列で置き換える。

こうしてできた文字列があるとする。 その文字列中の高々$S$個の位置の文字(のみ)を(同時に)見て、最初の長さ$K$の文字列に文字Gが含まれていたかどうか判定したい。 これは可能か、可能ならどうのように$S$個の位置を選べばよいか答えよ。

## solution

Dive a tree. $O(TSC)$.

Think the tree of the replacement. For example, the original string is LGL, the LGL is root, another LGL is the 1st child of the root, GGG is the 2nd, LGL is the 3rd, and LGL is the 1st child of the 1st child of the root, and so on. This tree has such a property: the $i$-th character of last string is G if and only if $j$-th children/edges of some nodes are used in the unique path from root to the $i$-th leaf, and $j$-th character of original string is G. So you need to use all of $1, 2, \dots K$-th edges at least once, and it is enough.

## implementation

#include <iostream>
#include <vector>
#define repeat(i,n) for (int i = 0; (i) < (n); ++(i))
typedef long long ll;
using namespace std;
template <class T> ostream & operator << (ostream & out, vector<T> const & a) { bool i = false; for (T const & it : a) { if (i) out << ' '; else i = true; out << it; } return out; }
void solve() {
int k, c, s; cin >> k >> c >> s;
vector<ll> ans;
int cnt = 0;
repeat (s_,s) {
if (cnt >= k) break;
ll p = 0;
repeat (i,c) {
p *= k;
p += min(k-1, cnt ++);
}
ans.push_back(p + 1);
}
if (cnt < k) {
cout << "IMPOSSIBLE" << endl;
} else {
cout << ans << endl;
}
}
int main() {
int t; cin >> t;
repeat (i,t) {
cout << "Case #" << i+1 << ": ";
solve();
}
return 0;
}