Magic Trick IOI 2025

Alicia and Beatriz are preparing a magic trick for the IOI Talent Show. The trick works as follows:

• A volunteer selects a permutation $P$ of length $N$ and places $N$ cards on the table. The cards are numbered from $0$ to $N − 1$, with card $i$ displaying the value $P[i]$.
• Alicia enters the room, observes the cards, and selects $K$ of them to flip face down, hiding their values.
• Beatriz then enters the room, sees the current arrangement of the cards (including which ones are face down), and magically determines the values of all $K$ hidden cards!

Your task is to devise and implement a strategy for Alicia and Beatriz. The more impressive the trick, the better your score: the objective is to maximize $K$, the number of hidden cards Beatriz can correctly reveal.

Implementation details

The first procedure that you have to implement:

std::vector<int> Alicia(std::vector<int> P)

• $P$: array of length $N$, representing the selected permutation.

This procedure should return an array $Q$ of length $N$, representing the card flips Alicia performs. For each index $i$ ($0 ≤ i ≤ N − 1$), the values in $Q$ must be set as follows:

• $Q[i] = −1$ if Alicia flips card $i$, and
• $Q[i] = P[i]$ otherwise.

The second procedure that you have to implement:

std::vector<int> Beatriz(std::vector<int> Q)

• $Q$: array of length $N$, as returned by Alicia. This array specifies the configuration of the cards when Beatriz enters.

This procedure should return an array $B$ of length $N$, representing the original permutation $P$, that is, $B[i] = P[i]$ should hold for each $i$ ($0 ≤ i ≤ N − 1$).

In each test case, the two procedures are called exactly once, as follows:

• During the first run of the program:

  • Alicia is called with the original permutation $P$.

  • For the array $Q$ returned by Alicia:

    • If the array does not conform to the constraints described above, you receive an Wrong Answer verdict.
    • Otherwise, array $Q$ is stored by the grading system.

• During the second run of the program:

  • Beatriz is called with the array $Q$.

Limit

• $N = 256$
• $1 ≤ P[i] ≤ N$ for each $i$ such that $0 ≤ i < N$.
• All values in $P$ are distinct.

Scoring

Let $M$ be the minimum value of $K$ for which your solution successfully performs the trick across all test cases.

• If $M = 0$ or your solution fails to correctly reconstruct the permutation $P$ in any test case, you receive $0$ points.
• Otherwise, your score is: $\min(20 + 5 \cdot M, 100)$

In particular, a full score is achieved if $M = 16$.

Example

Consider a scenario where $N = 6$ and $P = [2, 4, 3, 1, 5, 6]$. The procedure Alicia is called as:

Alicia([2, 4, 3, 1, 5, 6])

Suppose Alicia uses the following strategy: flip every card $i$ such that $P[i] = i + 1$. In this case, the condition holds for$i = 2, 4$, and $5$. Hence, the procedure returns the array $[2, 4,−1, 1,−1,−1]$.

Now, the procedure Beatriz is called as:

Beatriz([2, 4, -1, 1, -1, -1])

Knowing the strategy of Alice, she finds and returns the original array $P = [2, 4, 3, 1, 5, 6]$.

In this case, $K = 3$, as three cards were flipped. However, if submitted, this strategy would receive a score of $0$, because there exist permutations where no index $i$ satisfies $P[i] = i + 1$.

Note that this example does not satisfy the constraint $N = 256$ and therefore will not be used during grading. The downloadable attachment for this task includes a sample input for the grader with $N = 256$. The same permutation $P$ is used in Subtask 0 during evaluation.

Sample grader

Input format:

N
P[0] P[1] ... P[N-1]

Output format:

S
Q[0] Q[1] ... Q[S-1]
T
B[0] B[1] ... B[T-1]

Here:

• $S$ is the length of the array $Q$ returned by Alicia.
• $T$ is the length of the array $B$ returned by Beatriz.

Note that while $N = 256$ holds for all testcases in this task, you may use the sample grader with any value of $N$.

Attachments

• magic.zip

채점 및 기타 정보

• 100점 이상을 획득해야 를 받는다.

N
P[0] P[1] ... P[N-1]

S
Q[0] Q[1] ... Q[S-1]
T
B[0] B[1] ... B[T-1]

Here:

• $S$ is the length of the array $Q$ returned by Alicia.
• $T$ is the length of the array $B$ returned by Beatriz.

Note that while $N = 256$ holds for all testcases in this task, you may use the sample grader with any value of $N$.