# Route Product

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

## Problem

There are lots of different routes from $A$ to $B$ in this diagram:

The idea is to work out the product of the numbers on these different routes from $A$ to $B$. Let's say that in a route you are not allowed to visit a point more than once.

For example, we could have $3\times0.5$ but we couldn't have $3\times2\times5\times4\times1\times 0.1$ because that route passes through $A$ twice.

Which route or routes give the largest product?

Which route or routes give the smallest product?

Do you have any quick ways of working out the products each time?

*[This problem is adapted from a SMILE Centre card.]*

## Getting Started

Can you see immediately which routes might have large products and which ones small products?

How will you know that you have checked all the routes?

How many routes can you find which use the $5$? What is the product of each one?

How many routes use the $3$? And the $4$?

## Student Solutions

Neil from Kells Lande Primary School wrote to say:

The route with the largest product is going up from A along the $5$ line, and down along the $2$ line, and then along the $0.5$ line to B, which has a product of $5$.

The route with the smallest product is to go horizontally along the $3$ line from A, then down the first $0.5$ line, along the $1$ line going horizontally, and then up to B along the $0.1$ line, which has a product of $0.15$.

You're right, well done, Neil. Class K's Magic Mathematicians at Charter Primary School told us how they found out the solution:

We worked in a group of four and found out all the routes from A to B, then we did all the sums and worked out all the answers.

Many of you also said that to find the largest and smallest products, it helped to look for the largest and smallest numbers. That's a good strategy, well done.

## Teachers' Resources

**Why do this problem?**

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