Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### Advanced mathematics

### For younger learners

# Crossing the Bridge

### Why do this problem?

### Possible approach

### Key question

### Possible extension

### Possible support

## You may also like

### Consecutive Numbers

### Tea Cups

### Counting on Letters

Or search by topic

Age 11 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

This problem offers an interesting challenge which can be used to develop mathematical thinking.

Students reaction to the problem has often been "It's impossible!". Once the problem has been solved it can provided a useful vehicle for discussing what mathematicians do when they are stuck: experiment, explore dead ends, discuss with friends, walk away from the problem and return to it later... Teachers may want to use this problem to help students think about what they do when they want
to give up.

*This problem featured in an NRICH Secondary webinar in June 2021.*

You could use the interactivity to introduce the problem and make sure that students understand the constraints.

Ask the students to tackle the problem in pairs, either at computers or on paper. Challenge them to find a strategy for getting the friends across in just 17 minutes. As they experiment, circulate around the classroom and listen in on conversations. Some might suggest that it's impossible...

We experience pleasure and satisfaction when solving problems, so you may want to ensure that those who solve this problem quickly, do not deny the rest of their classmates that satisfaction ("let's not spoil anyone's fun").

Consider giving everyone as much time as they need to work on the first part of this problem, and rather than bringing the whole class together and inviting the fastest pairs to demonstrate their strategy, introduce the second part of the problem which challenges students to determine the criteria for selecting when to use each of the two optimal strategies. If students are using the interactivity they can click on the purple cog to enter the settings menu and change how long it takes each person to cross the bridge.

As students go solving the first part of the problem they can then progress onto the second part.

The second part of the problem will require clear recording of results and careful analysis of the different possibilities.

To help with this analysis you may like to suggest that students consider the time taken by each strategy when A is the quickest to cross, followed by B, then C, then D. So for example, if A and C cross together, the time taken would be determined by C.

So Strategy 2 would have a total crossing time of B + A + C + A + D.

Bring the class together at the end of the lesson to share discoveries and explanations.

If one person takes 10 minutes to cross the bridge, and another takes 7 minutes, how can they all cross in just 17 minutes?

The second part of this problem challenges students to provide clear and convincing justifications.

Students could have a go at River Crossing before embarking on this problem.

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?