When you have a \(7 \times 11\) chocolate bar, the answer will definitely be No if any of the following conditions are met. Please fill in the blanks.

• You need to give chocolate bars that are at least \(1 \times 1\) with a total area of \(??\) or more
• You need to give chocolate bars that are at least \(2 \times 2\) with a total area of \(??\) or more
• You need to give chocolate bars that are at least \(4 \times 4\) with a total area of \(??\) or more

Consider the case where you have a \(7 \times 11\) chocolate bar. What will be the answer if none of the three conditions in Hint 1 is met?

Let’s apply the considerations from Hints 1 and 2 to general \(H, W\).

Consider what the answer would be when \(K = 1\).

Consider what the answer would be when \(K = 2\).

Consider what the answer would be when \(K = 3\).

Let’s apply the considerations from Hints 1, 2, and 3 to a general \(K\).

Let’s say voter \(1\) is the \(t\)-th voter. When \(t\) changes as \(t = 1, 2, \dots, N\), how will the following change?

• The first character of the preliminary result string
• The second character of the preliminary result string
• The third character of the preliminary result string (and so on)

Instead of solving the original problem right away, let’s first think about the case where we want all pairwise Euclidean distances to be the same. Specifically, consider the following problem.

• \(N = 3\)
• \(d(p_1, p_2) = d(p_1, p_3) = d(p_2, p_3)\)

By slightly shifting the answer from Hint 1, let’s meet the following conditions.

• \(N = 3\)
• \(d(p_2, p_3) < d(p_1, p_3) < d(p_1, p_2)\)

Let’s apply the considerations from Hints 1 and 2 to the case where \(N \geq 4\).

In fact, for any \(X\) that satisfies the constraints, there is only one integer \(n\) such that \(0 \leq n < 10^9\) and \(n^n \mod 10^9 = X\).

In fact, not only Hint 1, but also the following properties hold.

• For any \(X\) \((0 \leq X < 10^2)\) that satisfies \(\mathrm{gcd}(X, 10^2) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^2\) and \(n^n \mod 10^2 = X\)
• For any \(X\) \((0 \leq X < 10^3)\) that satisfies \(\mathrm{gcd}(X, 10^3) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^3\) and \(n^n \mod 10^3 = X\)
• For any \(X\) \((0 \leq X < 10^4)\) that satisfies \(\mathrm{gcd}(X, 10^4) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^4\) and \(n^n \mod 10^4 = X\)
• For any \(X\) \((0 \leq X < 10^5)\) that satisfies \(\mathrm{gcd}(X, 10^5) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^5\) and \(n^n \mod 10^5 = X\)
• For any \(X\) \((0 \leq X < 10^6)\) that satisfies \(\mathrm{gcd}(X, 10^6) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^6\) and \(n^n \mod 10^6 = X\)
• For any \(X\) \((0 \leq X < 10^7)\) that satisfies \(\mathrm{gcd}(X, 10^7) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^7\) and \(n^n \mod 10^7 = X\)
• For any \(X\) \((0 \leq X < 10^8)\) that satisfies \(\mathrm{gcd}(X, 10^8) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^8\) and \(n^n \mod 10^8 = X\)
• For any \(X\) \((0 \leq X < 10^9)\) that satisfies \(\mathrm{gcd}(X, 10^9) = 1\), there is only one integer \(n\) such that \(0 \leq n < 10^9\) and \(n^n \mod 10^9 = X\)

Based on Hint 2, let’s determine the integer \(n\) starting from the lower digits.

Let \(N = 20\) and solve the following two problems.

• What is the state after walking from intersection \(1\) to \(39\) from the state XXXYXYYXXXYXXXXYYXXY?
• What is the state after walking from intersection \(13\) to \(28\) from the state XXXYXYYXXXYXXXXYYXXY?

For the two problems in Hint 1, compare the starting and ending state strings. What is happening?

Remember the famous Edit Distance Problem. Doesn’t it look similar?