Walking the squares

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Age

7 to 11

Challenge level

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Problem

So here's a special square of tiles to walk on!

The black square in the middle contains some very special prizes.

You can get the prizes by collecting tokens which are on each tile. So you have to step on as many tiles as possible.

BUT

You cannot go onto any tile more than once.

You are not allowed to step on more than two tiles of the same colour one after another.

So this path would be OK.

But this path is not allowed;

Why? Because the path goes along $3$ blues which is not allowed, the blue to green is OK but then there are $4$ green tiles one this path next to each other and that is also not allowed.

Your challenge is to find a path from anywhere on the outside of the square to the black square in the middle, following the rules above and trying to step on as many tiles as possible.

Student Solutions

Thank you to everybody who sent in their solutions to this activity.

Haaniah from Woodthorpe JI School sent in this good solution:

I first thought about how to get to the middle using any numbers of squares, for example, clockwise. Then I decided to go clockwise, using two squares and two rows at a time. When I got to the corner I decided to enter the corner diagonally. When I got to the final three rows, I decided to still go clockwise using two squares at a time, except this time I used three rows at a time.

Later she added:

First, I imagined how you could use the most number of tiles up using more than two of the same colour; you would follow it all the way around, clockwise. Then I decided to try two rows at a time for example: You start at the top right orange corner. Go right 1 then down 1, right 1 and up 1.

Continue until you reach the corner. Enter the corner diagonally then 1 across and exit diagonally as demonstrated in the picture when you have gone the whole way round enter the next rows diagonally and repeat, this time three rows at a time. When there is only three squares left, enter straight into the middle.

Swasti from Maybury Primary School, Woking also used diagonal lines to walk on as many squares as possible:

So, firstly, I worked out how many tiles the square has (121) as it would be the maximum number of tiles to walk on.

Then, I used a zig-zag pattern (as shown below) to walk on the square until I met a corner.

When I met a corner, I just realised that in order to walk on the maximum number of tiles, I couldn't use the zig-zag pattern as I did earlier because then I would have to walk on a same coloured tile more than twice.

Well done to both of you for filling in all of the tiles! Have a look at Swasti's full explanation to see how she did it.

Some children didn't allow diagonal lines in their solutions. Megan from St. Oswald's Primary School managed to miss out only 10 tiles in her solution:

Daya from Brackenbury Primary School sent in these three pictures to show their progression, missing out only 9 tiles:

Orla from All Saints Primary School in the UK found a solution which only missed out 8 tiles:

Thank you all for sending in these pictures! I wonder if anybody can miss out fewer than 8 tiles without using any diagonal lines? If you manage this, please email us to let us know.

Teachers' Resources

Walking the Squares

So here's a special square of tiles to walk on!

The black square in the middle contains some very special prizes.

You can get the prizes by collecting tokens which are on each tile. So you have to step on as many tiles as possible.

BUT

You cannot go onto any tile more than once.

You are not allowed to step on more than two tiles of the same colour one after another.

So this path would be OK.

But this path is not allowed;

Why? Because the path goes along $3$ blues which is not allowed, the blue to green is OK but then tere are $4$ green tiles one this path next to each other and that is also not allowed.

Your challenge is to find a path from anywhere on the outside of the square to the black square in the middle, following the rules above and trying to step on as many tiles as possible.

Why do this problem?

This activity gives pupils an opportunity to explore a scenario and gradually start using a strategy while learning to overcome difficulties. It also gives pupils an opportunity to use their curiosity to pursue many different associated questions. Their perseverance can be tested alongside their resilience in answering their own questions of "I wonder what would happen if we . . .?

Possible approach

It would be good to give pupils a sheet with several copies of the square on it (see here for doc or pdf).

The pupils can then work on the first one. If (and when!) they come to a stop because they realise it is going wrong, encourage them not to rub it out but write about what they are thinking - in particular what they did and what they should now do - on the right and start again underneath.

Here is a page (doc, pdf) that gives you an opportunity to print out four different sizes.

Key questions

Tell me about what happened here that made you start again.

You seen to be using a system (if it looks like it!) can you tell me about it?

Is there anything you would like to change about the way that you did it?

Have you covered as many pairs as possible?

Possible extension

Here is a page (doc, pdf) that gives you an opportunity to print out four different sizes of squares for the pupils to have a go at and compare, and perhaps then they can be encouraged to make some predictions.

Possible support

Some pupils might want to use some drawing software.

Or search by topic

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners