A frog needs to cross a pond by stepping on all, some or none of nine lily pads floating across the pond. How many different ways are there that the frog could cross the pond? Use our activity sheets to investigate this problem with your students.

## Curriculum Topics Covered

Combinations, Counting Problems, Being Systematic, Powers, Pascal’s Triangle

## More information

**Matt’s Frog Problem video:**

In Matt’s Frog Problem video he poses and attempts to solve a complex problem about frogs. The problem we have made (see **Frog Jumps Task**) is a simplification of the problem in the video.

**Activity documents: **

**Frog Jumps Task – **Students are challenged to find the number of ways the frog can cross a pond with nine lily pads. Students are advised to solve the problem for fewer lily pads first. In the document we show all the ways the pond can be crossed when there are only two lily pads as an example.

**Frog Jumps – Three Lily Pads** – This document is designed to help students record all the ways of crossing the pond if there are three lily pads. It is a set of empty lily pad diagrams that students can draw the jumps on to.

**Frog Jumps Results** – This is a table where students can record their results for each number of lily pads. Recording the results like this may help them spot patterns in their results. (Spoiler: it’s Pascal’s Triangle).

**Frog Jumps Solutions** – find solutions and further teachers’ notes here.

Zoe also wrote a blog about the different ways of explaining the appearance of Pascal’s Triangle in this puzzle and related puzzles: https://zoelgriffiths.co.uk/index.php/2019/08/28/pascals-puzzles/

**Matt’s Frog Problem:**

The more complex problem in Matt’s video asks: what is the expected number of jumps the frog makes?

You can assume that at any point in its journey, each of the possible jump sizes the frog could make on its next go are equally likely. See the video for more details.

Can you solve it?