### How it works

The “ladders” in this activity produce permutations (reorderings) of letters. Each ladder has a set of posts (vertical lines) and rungs (horizontal lines between adjacent posts). To start, there is a different letter at the top of each post (in alphabetical order). Letters then slide down the posts (no going up), and when they reach a rung, they must turn and cross that rung to the other post, then continue sliding down. The order the letters are in at the bottom of the ladder is the permutation the ladder produces.

For example, in the image above, the ladder produces the permutation DCBA.

In this activity, students start by learning about ladders and practicing figuring out which permutations various ladders produce. Then they are given specific permutations and they try to find ladders that produce these permutations. They explore which permutations require the most rungs (and how many rungs these permutations require), and try to find a general rule to help predict how many rungs a given permutation will require.

### Why we like this activity:

• It’s fun! Students enjoy exploring and making ladders.

• It helps to develop algorithmic reasoning.

• It requires students to engage in mathematical habits of mind:

• Finding and using strategies to solve various ladder puzzles.

• Making observations / looking for patterns / making and testing predictions / finding similarities and differences / understanding and explaining when exploring which permutations require the most rungs.

• It has a low floor and high ceiling: Students can get started exploring ladders and solving ladder puzzles by trial and error, but there is a lot to explore and discover.