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Tiles on a Patio

How many ways can you find of tiling the square patio, using square tiles of different sizes?

Square Corners

Stage: 2 Challenge Level: Challenge Level:2 Challenge Level:2
We had a lot of solutions sent in for this challenge, but unfortunately most of them contained a square that had not been noticed - it's so easily done. These are the two variations that people slipped up on.

Firstly from Year 6 Numeracy Group at New City Primary School:

We think that the answer is 10 counters.
We used squared paper to plot examples of how the counters could be placed and asked members of the group to check.  We had to be careful to check for squares that were placed diagonally.  We worked as a group and tested out as many variations as we could and compared these.  We thought carefully about how we should begin placing the counters.

Jack and Holly from North Molton Primary School wrote;

First we carefully read the explanation a few times to properly understand what we had to do. We then drew a 4 by 4 square grid (shown on the screen) and tried many different patterns following the rules that you are not allowed to make the four corners of a square. After many tries we worked out the answer is 11.

Nathan and Charles from Greentrees Primary School sent in solutions, here is what Charles wrote;

Why I believe the formula for the square counters problem is N(N+1)/2 (where N is the number of squares along one side of the grid)
I started off the journey with a 1x1 square.
Then I did 2x2, I still hadn't found a formula yet I carried on re- cording my results.
Then I did 3x3 and my fellow mathematician and I started to look at the amount in between them.
This was when we cracked the case, or should I say my friend cracked the case, we knew that our next box would be able to hold 15 yet we still had to check this.
Yes we had done it! We know knew that these were triangle numbers, so we looked up what the formula was on the internet and then we checked it. All of the numbers were triangle and the formula worked perfectly, and so we had cracked it and went on to submitting a solution (which is what we are doing right now!) I hope you enjoy this and will share this with other people!!

Maths Challenge Groups at Lyneham Primary School in Australia sent in the following;

If we made a five by five grid the max would be fifteen counters because if we did a two by two grid you can only get three counters ...

Gina's comments follow :

The students did not have time to write up their solutions in full. The text above is one student's attempt to begin a solution.

Some of the students placed counters on the grid and moved them around to "fit them in" without making a square. Others filled up the grid and then strategically removed counters until there were no squares.

The students found that they could fit a maximum of 6 counters on a 3x3 grid, and a maximum of 10 counters on a 4x4 grid. They initially thought that, since the maximum changed by 4, it would mean that going to a 5x5 grid would increase the maximum by 4 again, from 10 to 14. But then someone pointed out that on a 2x2 grid you could fit a maximum of 3 counters, which doesn't fit the +4 pattern. They thought then that maybe the difference between the maximum increases by 1 with every increase in grid size. They speculate that the maximum number of counters on a 5x5 grid would thus be 10+5=15, but they have not had time to check.

Thank you for sending your ideas in.