How Many Routes: 2 × 2 Grid
Read the puzzle
A square grid is made of 2 by 2 small squares, so it has 3 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.
6
Add the ways from the left and from below: 3 + 3 = 6.
How to solve it
Write a number on every corner: the number of ways to reach it.
Every corner on the bottom edge and the left edge can be reached in only 1 way, because you can only keep going straight. The centre corner can be reached from the left (1 way) or from below (1 way), so it gets 1 + 1 = 2.
The top middle corner gets 1 from its left plus 2 from below: 3. The right middle corner gets 2 from its left plus 1 from below: 3. The gem gets 3 + 3 = 6.
Check by listing. Every route is 4 moves, 2 right (R) and 2 up (U): RRUU, RURU, RUUR, URRU, URUR, UURR. That is 6.
A common wrong answer is 4, from thinking 2 choices at each of 2 steps gives 2 × 2. The choices are not independent: some routes turn once, some turn three times, and counting corner by corner catches them all.
Takeaway: each corner's count is the count to its left plus the count below it.
Learn the method: Counting and Combination Puzzles: How to Count Every Way
