Editorial for An Animal Contest 2 P2 - Koala Party


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Author: samliu12

Given that each a_i is unique, we realize that the answer is always 1, 2, or 3.

If N = 2, the answer is 1 if the total number of cookies among all bowls is odd and 2 if the total is even.

For all N \ge 3, the answer is 3 if we can find three bowls a_i, a_j, and a_k that form an arithmetic sequence, and 2 otherwise.

Subtask 1

We loop possible values of i, j, and k, and check if the elements form an arithmetic sequence.

Time Complexity: \mathcal{O}(N^3)

Subtask 2

To optimize, we add all a_i into a set and loop possible values of i and j, checking if \frac{a_i + a_j}{2} is contained.

Time Complexity: \mathcal{O}(N^2)


Comments


  • 3
    suguruchhaya  commented on May 13, 2021, 5:40 p.m.

    Kind of an explicit intuitive proof about why for n>=3, if an arithmetic sequence doesn't exist, the answer is 2. For the answer to be 2, there must be 2 bowls with the sum of their cookies being an even number. An even number can be reached either by even+even or odd+odd. In 3 or more bowls, intuitively, there must be at least one pair of bowls that have their cookies adding to an even number. For example, when n==3, possible unordered combinations are (odd, odd, odd), (odd, odd, even), (odd, even, even), and (even, even, even). In all of these combinations, there is at least 1 pair adding up to an even number (either odd+odd or even+even).

    This also explains why we have to explicitly check for cases when n==2. Unlike >=3, for n==2, it isn't guaranteed that a pair in which the sum is even exists. (even, even) and (odd, odd) has satisfying pair but (odd, even) doesn't.