Crossing the Bridge

Why do this problem?

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.

Possible approach

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.

Key question

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

Possible extension

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

Possible support

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