How Many Routes: Two Roadblocks
Read the puzzle
A street grid has 5 corners across and 4 corners up (4 blocks wide, 3 blocks tall). Start is the bottom-left corner and the gem is the top-right corner. You may only move right or up along the streets. Number columns 0 to 4 from the left and rows 0 to 3 from the bottom: a cone closes the corner at column 2, row 1, and the street from column 1, row 2 up to column 1, row 3 is closed. How many different routes reach the gem?
Only right or up. The cone corner and the striped street are closed. How many different routes from start to the gem?
Add the counts from the left and below, but skip the cone and the closed street.
14
Skip the cone and the closed street, add left + below: 14.
How to solve it
Step 1: write a count on every corner. A corner's count is the count of the corner to its left plus the count of the corner below it, because those are the only ways in. The bottom row and the left column are all 1.
Step 2: apply the blocks. The cone corner gets 0. Row 1 then reads 1, 2, 0, 1, 2. Row 2 reads 1, 3, 3, 4, 6 (for example 3 = 3 + 0 and 6 = 4 + 2). In row 3 the corner at column 1 cannot be entered from below, so it gets only the 1 from its left. Row 3 reads 1, 1, 4, 8, 14 (4 = 1 + 3, 8 = 4 + 4, 14 = 8 + 6).
Step 3: the gem corner shows 14.
The usual wrong answer is 17, which removes only the cone. With no blocks there are 35 routes (choose which 3 of the 7 moves go up). Routes through the cone: 3 ways in times 6 ways out = 18, leaving 35 - 18 = 17. The closed street removes 3 more routes (3 ways to reach its bottom end, 1 way on from its top), so 17 - 3 = 14.
The answer is 14.
Learn the method: Counting and Combination Puzzles: How to Count Every Way
