aime26-16

Proposed solution

To solve this problem, we need to determine the number of paths the bug can take from point \( A \) to point \( B \) while following the given constraints. We start by analyzing the grid and the allowed movements. The grid consists of 10 squares in a row, and the bug can move either right, up-right, or down-right along the diagonals.

We can represent the grid as a series of vertices and edges, where each square's vertices are connected by edges. The vertices are labeled as follows:
- \( A \) at \((0, 0)\)
- \( B \) at \((10, 1)\)

The allowed movements are:
1. Right: from \((i, j)\) to \((i, j+1)\)
2. Up-right: from \((i, j)\) to \((i+1, j+1)\)
3. Down-right: from \((i, j)\) to \((i+1, j)\)

We need to find the total number of paths from \( A \) to \( B \) without traversing the same segment twice and without moving left. We can use a recursive approach to count the number of valid paths.

Define \( f(i, j) \) as the number of ways to reach vertex \((i, j)\) from \( A \). The initial condition is \( f(0, 0) = 1 \) since there is one way to start at \( A \).

The recurrence relations for the movements are:
- \( f(i, j) = f(i-1, j) + f(i, j-1) + f(i-1, j-1) \) for \( j \geq 1 \) and \( i \geq 1 \)
- \( f(i, 0) = f(i-1, 0) + f(i-1, -1) \) for \( i \geq 1 \)

We need to compute \( f(10, 1) \) using the above recurrence relations.

Let's start with the base cases:
- \( f(0, 0) = 1 \)
- \( f(1, 0) = 1 \)
- \( f(0, 1) = 1 \)

Now we compute the values step by step:

\[
\begin{aligned}
f(1, 1) &= f(0, 1) + f(1, 0) + f(0, 0) = 1 + 1 + 1 = 3 \\
f(2, 1) &= f(1, 1) + f(2, 0) + f(1, 0) = 3 + 1 + 1 = 5 \\
f(1, 2) &= f(0, 2) + f(1, 1) + f(0, 1) = 1 + 3 + 1 = 5 \\
f(2, 2) &= f(1, 2) + f(2, 1) + f(1, 1) = 5 + 5 + 3 = 13 \\
f(3, 1) &= f(2, 1) + f(3, 0) + f(2, 0) = 5 + 1 + 1 = 7 \\
f(1, 3) &= f(0, 3) + f(1, 2) + f(0, 2) = 1 + 5 + 1 = 7 \\
f(3, 2) &= f(2, 2) + f(3, 1) + f(2, 1) = 13 + 7 + 5 = 25 \\
f(2, 3) &= f(1, 3) + f(2, 2) + f(1, 2) = 7 + 13 + 5 = 25 \\
f(3, 3) &= f(2, 3) + f(3, 2) + f(2, 2) = 25 + 25 + 13 = 63 \\
f(4, 1) &= f(3, 1) + f(4, 0) + f(3, 0) = 7 + 1 + 1 = 9 \\
f(1, 4) &= f(0, 4) + f(1, 3) + f(0, 3) = 1 + 7 + 1 = 9 \\
f(4, 2) &= f(3, 2) + f(4, 1) + f(3, 1) = 25 + 9 + 7 = 41 \\
f(2, 4) &= f(1, 4) + f(2, 3) + f(1, 3) = 9 + 25 + 7 = 41 \\
f(4, 3) &= f(3, 3) + f(4, 2) + f(3, 2) = 63 + 41 + 25 = 129 \\
f(3, 4) &= f(2, 4) + f(3, 3) + f(2, 3) = 41 + 63 + 25 = 129 \\
f(4, 4) &= f(3, 4) + f(4, 3) + f(3, 3) = 129 + 129 + 63 = 321 \\
f(5, 1) &= f(4, 1) + f(5, 0) + f(4, 0) = 9 + 1 + 1 = 11 \\
f(1, 5) &= f(0, 5) + f(1, 4) + f(0, 4) = 1 + 9 + 1 = 11 \\
f(5, 2) &= f(4, 2) + f(5, 1) + f(4, 1) = 41 + 11 + 9 = 61 \\
f(2, 5) &= f(1, 5) + f(2, 4) + f(1, 4) = 11 + 41 + 9 = 61 \\
f(5, 3) &= f(4, 3) + f(5, 2) + f(4, 2) = 129 + 61 + 41 = 231 \\
f(3, 5) &= f(2, 5) + f(3, 4) + f(2, 4) = 61 + 129 + 41 = 231 \\
f(5, 4) &= f(4, 4) + f(5, 3) + f(4, 3) = 321 + 231 + 129 = 681 \\
f(4, 5) &= f(3, 5) + f(4, 4) + f(3, 4) = 231 + 321 + 129 = 681 \\
f(5, 5) &= f(4, 5) + f(5, 4) + f(4, 4) = 681 + 681 + 321 = 1683 \\
\end{aligned}
\]

Continuing this process until we reach \( f(10, 1) \):

\[
\begin{aligned}
f(6, 1) &= f(5, 1) + f(6, 0) + f(5, 0) = 11 + 1 + 1 = 13 \\
f(1, 6) &= f(0, 6) + f(1, 5) + f(0, 5) = 1 + 11 + 1 = 13 \\
f(6, 2) &= f(5, 2) + f(6, 1) + f(5, 1) = 61 + 13 + 11 = 85 \\
f(2, 6) &= f(1, 6) + f(2, 5) + f(1, 5) = 13 + 61 + 11 = 85 \\
f(6, 3) &= f(5, 3) + f(6, 2) + f(5, 2) = 231 + 85 + 61 = 377 \\
f(3, 6) &= f(2, 6) + f(3, 5) + f(2, 5) = 85 + 231 + 61 = 377 \\
f(6, 4) &= f(5, 4) + f(6, 3) + f(5, 3) = 681 + 377 + 231 = 1289 \\
f(4, 6) &= f(3, 6) + f(4, 5) + f(3, 5) = 377 + 681 + 231 = 1289 \\
f(6, 5) &= f(5, 5) + f(6, 4) + f(5, 4) = 1683 + 1289 + 681 = 3653 \\
f(5, 6) &= f(4, 6) + f(5, 5) + f(4, 5) = 1289 + 1683 + 681 = 3653 \\
f(6, 6) &= f(5, 6) + f(6, 5) + f(5, 5) = 3653 + 3653 + 1683 = 8990 \\
f(7, 1) &= f(6, 1) + f(7, 0) + f(6, 0) = 13 + 1 + 1 = 15 \\
f(1, 7) &= f(0, 7) + f(1, 6) + f(0, 6) = 1 + 13 + 1 = 15 \\
f(7, 2) &= f(6, 2) + f(7, 1) + f(6, 1) = 85