How Many Routes: Pass the Lantern
Read the puzzle
A square grid is made of 4 by 4 small squares, so it has 5 by 5 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 lantern sits on the centre corner, 2 steps right and 2 steps up from the start. Moving only right or up along the lines, how many routes from the start to the gem pass the lantern?
Moving only right or up, how many routes from start to the gem pass the lantern?
Count the ways to the lantern, then the ways from it to the gem, and multiply.
36
6 ways to the lantern, then 6 ways on: 6 × 6 = 36.
How to solve it
Split every route at the lantern and count the two halves.
Start to lantern: 2 moves right and 2 moves up, in any order. The orders are RRUU, RURU, RUUR, URRU, URUR, UURR, so 6 ways.
Lantern to gem: again 2 right and 2 up, so again 6 ways.
Each first half can be joined to each second half, so the total is 6 × 6 = 36 routes.
The common wrong answer is 12, from adding 6 + 6. Adding would only be right if you picked one half or the other; a route uses both halves, so the counts multiply. Another slip is 70, which is every route on the grid, with or without the lantern.
Takeaway: for routes forced through one point, multiply the ways in by the ways out.
Learn the method: Counting and Combination Puzzles: How to Count Every Way
