Problem
2026 AIME II Problem 2
The figure below shows a grid of 10 squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments from vertex to vertex, never traversing the same segment twice and never moving from right to left along a horizontal or diagonal segment. Let N be the number of paths the bug can take from the lower left corner (A) to the upper right corner (B). One such path from A to B is shown by the thick line segments in the figure. Find \sqrt{N}.
![[asy] size(12cm); draw((0, 0)--(0, 1)); for (int i = 0; i <= 9; i += 1) { draw((i, 1)--(i + 1, 1)--(i + 1, 0)--(i, 0)--(i + 1, 1)); } draw((0, 0)--(0, 1)--(2, 1)--(2, 0)--(3, 1)--(3, 0)--(4, 1)--(4, 0)--(5, 0)--(5, 1)--(6, 1)--(6, 0)--(7, 1)--(8, 1)--(8, 0)--(10, 0)--(10, 1), red + linewidth(2)); label("$A$", (0, 0), dir(180)); label("$B$", (10, 1), dir(0)); [/asy]](/static/aops/543dec7ec1ca466ca0413a9e2d5ecb9e763fede7.png)
Leading zeroes must be inputted, so if your answer is 34, then input 034
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