How Many Routes: 3 × 2 Grid
Read the puzzle
A grid is 3 small squares wide and 2 small squares tall, so it has 4 by 3 corner points joined by lines. The start (a house) is at the bottom-left corner and a gem is at the top-right corner. 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?
Each corner's count is the count to its left plus the count below it.
10
The two corners next to the gem can be reached in 6 and 4 ways: 6 + 4 = 10.
How to solve it
Write on each corner the number of ways to reach it, working from the start.
The bottom row and the left column are all 1s, since there is only one straight way along an edge.
Middle row, left to right: 1, then 1 + 1 = 2, then 2 + 1 = 3, then 3 + 1 = 4.
Top row, left to right: 1, then 1 + 2 = 3, then 3 + 3 = 6, then the gem gets 6 + 4 = 10. The 6 comes from the corner to its left and the 4 from the corner below it.
Check with letters: every route is 5 moves, 3 right and 2 up. Choosing which 2 of the 5 moves are up gives 5 × 4 ÷ 2 = 10 routes.
An easy slip is to answer 6, the number of small squares, or to list routes by hand and miss some. The corner-by-corner count cannot miss any.
Takeaway: add the count from the left and the count from below at every corner.
Learn the method: Counting and Combination Puzzles: How to Count Every Way
