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Progressive hints first, then the full explanation and implementation when you're ready to cash out.
Review status
AI-generated and still unreviewed. Double-check the details before internalizing them.
Hints
Open only as much as you need to keep the solve alive.
Ignore the exact bracket shape for a second. A match always creates exactly one loser, so try counting what each kind of loss does.
A winners' group match does not eliminate anyone. It only moves exactly one team out of the winners' group and into the losers' group for a later round.
The winners' group starts with teams and ends with . Every winners' match decreases that count by exactly , so there must be winners' matches.
A losers' group match eliminates exactly one team. Right before the final, only two teams are still alive, one in each group, so teams have already been eliminated.
Add the forced pieces: winners' matches, losers' matches, and final match. Total: .
The tournament looks like it wants a simulation: odd team counts, byes, winners dropping down one round later, all that bracket spaghetti. But that is mostly noise. The exact schedule changes, but the total number of matches is forced.
Key idea
Do not track which team wins. Track what each match changes.
There are only two types of pre-final matches:
The final is just one extra match at the end.
Winners' group matches
Initially, all teams are in the winners' group.
In one winners' group match, two teams play. One wins and stays in the winners' group. The other loses and leaves the winners' group, dropping to the losers' group next round.
So every winners' group match decreases the number of teams in the winners' group by exactly .
Byes do nothing. They just let a team sit there awkwardly for a round. No count changes.
The tournament reaches the final stage when the winners' group has exactly team. Since it started with teams and each winners' match reduces that count by , the number of winners' group matches is:
Losers' group matches
A losers' group match is different: the loser gets eliminated.
So every losers' group match eliminates exactly one team from the tournament. Also, nothing else eliminates teams. Losing in the winners' group only sends you down; it does not kick you out.
Right before the final match, there are exactly two teams still alive: one in the winners' group and one in the losers' group.
Since we started with teams, the number of teams eliminated before the final is:
Therefore, the number of losers' group matches is also:
The annoying detail that winners' group losers drop down only in the next round does not change this. It only affects timing, not the total count. Scheduling nonsense, not math magic.
Final match
After both groups have one team, they play one final match. That adds:
Total
Add everything:
So for every test case, the answer is simply:
Complexity
Each test case is answered in time. Memory usage is too. No simulation needed, thankfully.
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
void setIO() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
}
int main() {
setIO();
int t;
cin >> t;
while (t--) {
ll n;
cin >> n;
cout << 2 * n - 2 << '\n';
}
return 0;
}#include <bits/stdc++.h>
using namespace std;
using ll = long long;
void setIO() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
}
int main() {
setIO();
int t;
cin >> t;
while (t--) {
ll n;
cin >> n;
cout << 2 * n - 2 << '\n';
}
return 0;
}