## Wednesday, 2 August 2017

### 5x5 Puzzle

I have seen something like this before somewhere, so it is unlikely that it is original.

But since we were becoming experts at patterns in Year 2, I thought I would throw it out to the kids and see what they came up with.

## The Challenge

Can you make a 5x5 array using 5 different coloured blocks, so that each colour appears once in each row and once in each column?

Seemed easy enough.

Here is what the kids did:

So I was pleased to see some cognitive struggle happening. It wasn't going to be so easy to solve this puzzle. This student has produced some very nice symmetry but their array doesn't have one of each colour appearing in each row and column.

After allowing some time to play and explore, we slowly started to see some solutions appear.

The first one looked like this:

Having started off trying random arrangements of blocks, this student decided to be systematic and was the first to find a solution.

Of course, I then asked, "Can you do it another way?"

He groaned at me and started all over again.

More solutions started to appear - some who had copied the original solution, but others who had been less than methodical and who had used trial and error to move their Unifix cubes around until they worked it out.

A duplicate of the original solution.

Similar to the original solution but the array is rotated 90 degrees.

The first of the "random" solutions. Trial and error was an effective strategy but not necessarily the most efficient. It seemed to take these students longer to get an array that fitted the requirements.

This was the pattern a student showed me. I swear that only a minute or two previously I had looked at the pattern and it was like the first example with the diagonal lines running top left to bottom right. When I looked back, it had been moved around.

What was going on?

"Well," the student explained patiently. "Once you have made the 5 columns of blocks with each colour in a different position, you can move them around into any order and your new square will have one of each colour in every row and column."

Yes indeed you will.

And then we recorded our experiences.

I also asked the question, "How many ways do you think you can solve this problem?"

There were some interesting answers -
• 3
• 5
• 25 because 5x5 = 25
You may not be able to read the writing in the workbook example above, but this student has written, "There is probably 100 because I know there is more than 3."

Seems a bit of a leap from 3 to 100.

I wonder how we can narrow this down?

I wonder how many ways there are?