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### Number and algebra

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### For younger learners

# Xavi's T-shirt

You should have found only two:

2x2x2x2x2x2 = $2^6$ = 64

2x2x2x2x2x3 = $2^5 \times 3$ = 96

You should have found four:

2x2x2x2x2 = $2^5$ = 32

2x2x2x2x3 = $2^4 \times 3$ = 48

2x2x2x2x5 = $2^4 \times 5$ = 80

2x2x2x3x3 = $2^3 \times 3^2$ = 72

You should have found twelve.

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Age 7 to 16

Challenge Level

Xavi, a friend of NRICH, was showing off his T-shirt, with circles representing the numbers from 2 to 100.

We thought we could turn it into a problem...

Here is a clearer image (using different colours) and here is the same image with some of the numbers identified.

Here are downloadable sheets of the two images:

one without numbers and one with some of the numbers

**What do you notice?**

Here are some questions you may like to consider:

What is special about the circles that are not split up?

Can you explain what is happening in the top row?

What do you notice about the colours in the fifth column?

What is happening in the far right column?

Take a look at the colours of the first two circles (2 and 3).

What is special about the circles/numbers that have these two colours?

What is special about the circles/numbers that have *only* these two colours?

Why do multiples of 11 appear on a diagonal line?

Why do multiples of 9 appear on a diagonal line?

Look at the circle that represents 8. The three parts are all the same colour.

How many other circles/numbers will also be split into three identical colours?

On the bottom row, 93, 94 and 95 appear as three consecutive circles/numbers, each split into two, and no colour is repeated.

Can you find a similar set of four consecutive circles/numbers where no colour is repeated? If not, why not?

What other patterns can you see? Can you explain why they occur?

**Imagine you are wanting to create your own version of this T-shirt.**

Before worrying about the colours, you need to decide how the circles are going to be split...

Using either a sheet of blank circles, a sheet with the prime numbers identified, or a sheet with all the numbers identified, sketch how you would need to split the circles.
Start at the top and work your way downwards. Stop when you get to 40.

Can you anticipate how many circles in the whole T-shirt will be split into six sections?

You should have found only two:

2x2x2x2x2x2 = $2^6$ = 64

2x2x2x2x2x3 = $2^5 \times 3$ = 96

Can you anticipate how many circles will be split into five sections?

You should have found four:

2x2x2x2x2 = $2^5$ = 32

2x2x2x2x3 = $2^4 \times 3$ = 48

2x2x2x2x5 = $2^4 \times 5$ = 80

2x2x2x3x3 = $2^3 \times 3^2$ = 72

Can you anticipate how many circles will be split into four sections?

You should have found twelve.

You may want to continue creating your version of the T-shirt, but we think that the insights you have gained will prepare you for some more interesting challenges...

**If we 'zoom in' on part of the T-shirt, do six circles offer you sufficient clues to figure out where you are on the T-shirt?**

**Can you figure out where you are on the T-shirt with fewer clues...?**

Either take a look at these eight challenges or have a go at the T-shirt interactivity below, which gives you feedback.

**Click on the purple cog (Settings) and try each of the games in turn.**

**In each case, you will be offered an image of part of the T-shirt.**

Can you figure out which numbers the circles represent?

*We hope many of you will share screenshots of the challenges you have worked on, along with your reasoning of how you used the 'clues' to solve the problem.*

**Imagine Xavi also had a scarf, with all the numbers from 2 to 100 in a straight line.**

**Click on the purple cog (Settings) of the scarf interactivity below and try each of the games in turn.**

**In each case, you will be offered an image of part of the scarf.**

Can you figure out which numbers the circles represent?

*We hope many of you will share screenshots of the challenges you have worked on, along with your reasoning of how you used the 'clues' to solve the problem.*

These two group activities use mathematical reasoning - one is numerical, one geometric.

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.