How Many Routes: One Cone
Read the puzzle
A square grid is made of 3 by 3 small squares, so it has 4 by 4 corner points joined by lines. The start (a house) is at the bottom-left corner and a gem is at the top-right corner. A cone closes the corner one step right and one step up from the start, and no route may pass it. Moving only right or up along the lines, how many different routes lead from the start to the gem?
Moving only right or up, how many different routes lead from start to the gem?
Give the cone corner 0, then add the counts from the left and below.
8
There are 20 routes in all, and 12 of them pass the cone: 20 − 12 = 8.
How to solve it
Count the ways to reach each corner, and give the cone corner 0.
Bottom row: 1, 1, 1, 1. Second row: 1, then the cone 0, then 0 + 1 = 1, then 1 + 1 = 2.
Third row: 1, then 1 + 0 = 1, then 1 + 1 = 2, then 2 + 2 = 4. Top row: 1, then 1 + 1 = 2, then 2 + 2 = 4, then the gem gets 4 + 4 = 8.
Check another way. Without the cone there are 20 routes (6 moves, choose 3 of them to be up: 6 × 5 × 4 ÷ 6 = 20). Routes through the cone: 2 ways to reach it times 6 ways from it to the gem, 2 × 6 = 12. So 20 − 12 = 8.
The usual mistake is to subtract only 1 for the blocked corner and answer 19. A single closed corner removes every route that passes it, and here that is 12 of the 20.
Takeaway: a blocked corner gets 0, and the adding rule does the rest.
Learn the method: Counting and Combination Puzzles: How to Count Every Way
