MathDB

An ant is walking on a sidewalk and discovers

12

sidewalk panels with leaves inscribed in them, as shown below. Find the number of ways in which the ant can traverse from point

A

to point

B

if it can only move
[*] up, down, or right (along the border of a sidewalk panel), or [*] up-right (along one of two margin halves of a leaf)
and cannot visit any border or margin half more than once (an example path is highlighted in red). [asy] unitsize(1cm); int r = 4; int c = 5;
for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { pair A = (j,i); } } for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { if (j != c-1) { draw((j,i)--(j+1,i)); } if (i != r-1) { draw((j,i)--(j,i+1)); } } }
for (int i = 1; i < r+1; ++i) { for (int j = 0; j < c-2; ++j) { fill(arc((i,j),1,90,180)--cycle,deepgreen); fill(arc((i-1,j+1),1,270,360)--cycle,deepgreen); draw((i-1,j)--(i,j+1), heavygreen+linewidth(0.5)); draw((i-2/3,j+1/3)--(i-2/3,j+1/3+0.1), heavygreen); draw((i-1/3,j+2/3)--(i-1/3,j+2/3+0.1), heavygreen); draw((i-2/3,j+1/3)--(i-2/3+0.1,j+1/3), heavygreen); draw((i-1/3,j+2/3)--(i-1/3+0.1,j+2/3), heavygreen); draw(arc((i,j),1,90,180)); draw(arc((i-1,j+1),1,270,360)); } } draw((0,3)--(0,1), red+linewidth(1.5)); draw((0,3)--(0,1), red+linewidth(1.5)); draw(arc((1,1),1,90,180), red+linewidth(1.5)); draw((1,2)--(1,1)--(2,1), red+linewidth(1.5)); draw(arc((2,2),1,270,360), red+linewidth(1.5)); draw(arc((4,2),1,90,180), red+linewidth(1.5)); draw((4,3)--(4,0), red+linewidth(1.5)); dot((0,3)); dot((4,0)); label("

A

", (0,3), NW); label("

B

", (4,0), SE); [/asy]

Problem Statement