We are going to use Cuisenaire rods. Here is the start of a (1,5) puzzle.

We win if we can reverse this (1,5) into a (5,1). Each time we add a new row underneath we can split a Cuisenaire rod into two. In this puzzle (1,5) has become (1,3,2).

The other thing you can do is join two rods together. (1,5) => (1,3,2) => (4,2) 

(1,5) => (1,3,2) => (4,2) => (3,1,2)

(1,5) => (1,3,2) => (4,2) => (3,1,2) => (3,3)… No you can never use two identical rods in a row. We have failed to turn (1,5) into (5,1). There is one other rule…

(1,5) => (6). Why do you think this might be a mistake? Because we can never use a longer Cuisenaire rod than what we started with in the top row. The longest one in the top row is a 5, so we can never use a 6.

(1,5) is impossible. That makes it a good one to use in front of the class when explaining the rules. Try to solve (2,7) before you continue. It is possible.

Done! Can you find a better solution? What do you think a “better” solution would look like? I am going to suggest that it means a shorter solution. Maybe your students will have a different definition of “better.”

This is a little better because it is a shorter route to get to a roll reversal.

One of the things I particularly enjoy about these puzzles is plotting all the possible directions a solution can take. I don’t know how to show this beautifully, but here is one idea, before we thought about using Cuisenaire rods.