Coloured Edges
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
Age
11 to 14
Challenge level
Problem
| There is a set of tiles which are all different and coloured just on the edges. Each edge on a tile is a different colour. For example: |
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Image
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| Altogether $10$ different colours are used for the edges, and there is an equal number of edges of each colour used throughout the set. |
| The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. |
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Image
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| How many tiles are there in the set? |
Getting Started
| What sort of number of tiles must you use to make a square? |
| What sort of number of tile edges must fill the perimeter of the square? |
Student Solutions
Laura Turner and Laura Malarkey from the Mount School have explained how they worked out the answer to this problem:
n is equal to the number of tiles along one side.
We can calculate the number of edges in two different ways:
Method 1 - In total there are $n ²$ tiles on
$4n ²$ edges.
Method 2 - There are a total of $2n$ green edges which implies
there are a total of $20n$ edges of all colours.
Therefore:
$20n = 4n ²$
$5n = n ²$ (divide by $4$)
$5 = n$ (divide by $n$)
So there are $25$ tiles in the set.
Teachers' Resources
Why do this problem?
This problem requires learners to work logically with numbers and shapes at the same time. The best ways of finding a solution include using a letter to represent the length of the side of the square.Key questions
What sort of number of tiles must you use to make a
square?
What sort of number of tile edges will make up the perimeter
of the square?
Have you thought of using a letter to represent the length of
the side of the square?