aime26-16

Proposed solution

I need to find the number of paths from point A to point B in a grid where each square has a diagonal from lower left to upper right, with restrictions on movement.

Let me first understand the structure. I have 10 squares in a row, so 11 vertical lines (labeled 0 through 10) and 2 horizontal lines (at y=0 and y=1). Each square has a diagonal from lower left to upper right, which means from vertex (i,0) to vertex (i+1,1).

The key constraints are:
1. Never traverse the same segment twice
2. Never move from right to left along horizontal or diagonal segments

Let me think of this systematically. I'll label the vertices as (i,j) where i goes from 0 to 10 and j is either 0 or 1.

The starting point is A = (0,0) and ending point is B = (10,1).

At each vertex, the bug can move:
- Right along horizontal segments: from (i,0) to (i+1,0) or from (i,1) to (i+1,1)
- Up along vertical segments: from (i,0) to (i,1)
- Diagonally up-right: from (i,0) to (i+1,1)

But there's a constraint that we cannot move from right to left on horizontal or diagonal segments.

Let me reconsider the structure. Looking at the Asymptote code more carefully:
- Each square has vertices at (i,0), (i,1), (i+1,0), (i+1,1)
- The diagonal connects (i,0) to (i+1,1)
- There are horizontal segments: (i,0) to (i+1,0) and (i,1) to (i+1,1)

So from each vertex, I can move:
- From (i,0): to (i+1,0) [horizontal], to (i+1,1) [diagonal]
- From (i,1): to (i+1,1) [horizontal], to (i,0) [vertical] - but this would be moving left!

Wait, I need to be more careful about the constraint. The bug cannot move from right to left along horizontal or diagonal segments.

Let me re-read: "never moving from right to left along a horizontal or diagonal segment."

This means:
- Horizontal segments: (i,j) to (i+1,j) - this is always left-to-right, so OK
- Diagonal segments: (i,j) to (i+1,j+1) - this is always left-to-right, so OK  
- Vertical segments: (i,j) to (i,j+1) - this is always bottom-to-top, so OK

The constraint is that we cannot move from right to left on horizontal or diagonal segments.

Actually, let me re-read the problem. The bug starts at lower left corner A and ends at upper right corner B.

Looking at the figure more carefully:
- From (0,0) I can go to (1,0) or (1,1)
- From (1,0) I can go to (2,0) or (2,1) 
- From (1,1) I can go to (2,1) or (2,0) - but wait, that would be going left!

No, from (1,1) I can go to (2,1) [horizontal] or to (2,0) [diagonal], but the diagonal from (1,1) to (2,0) is going right-to-left in terms of x-coordinates? No, that's wrong.

Let me be systematic. The vertices are:
- Horizontal segments: (i,0) to (i+1,0) and (i,1) to (i+1,1)
- Vertical segments: (i,0) to (i,1) 
- Diagonal segments: (i,0) to (i+1,1)

The constraint is that we cannot move from right to left on horizontal or diagonal segments.

So from vertex (i,j):
- Can go to (i+1,j) [horizontal] - always left-to-right
- Can go to (i,j+1) [vertical] - always bottom-to-top  
- Can go to (i+1,j+1) [diagonal] - always left-to-right

But wait, the constraint says we cannot move from right to left along horizontal or diagonal segments. 

Actually, let me think about what "from right to left" means:
- Horizontal segment from (i,j) to (i+1,j): this is moving left-to-right (increasing x)
- Diagonal segment from (i,j) to (i+1,j+1): this is also moving left-to-right

So the constraint doesn't actually restrict any valid moves in this setup.

But looking at the example path, I see that it goes:
(0,0) → (0,1) → (2,1) → (2,0) → (3,1) → ...

Let me trace this more carefully. The bug starts at A=(0,0).

From (0,0), it can go to:
- (1,0) [horizontal]
- (1,1) [diagonal]

The path shown goes: (0,0) → (0,1) → (2,1) → (2,0) → ...

Wait, that's not right. Let me re-read the Asymptote code.

Looking at the red path:
(0,0) to (0,1) [vertical]
(0,1) to (2,1) [horizontal] 
(2,1) to (2,0) [vertical] 
(2,0) to (3,1) [diagonal]
(3,1) to (3,0) [vertical]
(3,0) to (4,1) [diagonal]
...

So the path is: (0,0) → (0,1) → (2,1) →