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Spot Thirteen

Age 7 to 11 Challenge Level:

This problem was perhaps more challenging than it looks!  Sophie from King's Junior School in Chester told us:

If you have three in a row you get three points, if you get four in a row you get four points etc.
I think the highest possible score is $40$.


The Super Six at Coldean agreed with Sophie and clarified that the line could be horizontal, vertical or diagonal.  Angus from Australia also articulated the rules very clearly.

Will from Holmfirth High School suggests a way of making a score of $40$:

$5$ in a row first, which gives you $5$ points.
Make two rows of $4$ which gives $4$ points each.
This will give you $13$ points altogether so far.
There are nine rows and diagonals of three which give $3$ points each so altogether you get $40$!!

Here is a picture of Will's description:

Phoebe from Walton High School described a different way:

If you place $3$ dots in a row you will score $3$ points, if you place $4$ dots in a row you will score $4$ points and so on.

The shape you need to make to score the highest result consists of a $3 \times 4$ rectangle on the grid, with the final dot above the top right dot on the rectangle. This should leave you with the highest score of $40$.

Here is a picture of Phoebe's arrangement:


Are Phoebe's and Will's ways different from each other, do you think?

Millie and Charlotte from Princes Risborough School sent a very comprehensive solution.  They sent this series of pictures to illustrate the scoring system:

What do you think about their way of getting $40$ compared with Pheobe's and Will's?

Millie and Charlotte, Hannah, Pete and Will, and Krystof from Prague all noticed a mistake in the Poly Plug Rectangle interactivity.  They sent in this screenshot:

With the scoring system you have identified, we would expect this arrangement to score $3$ points but it doesn't.  This was completely our fault - sorry if this caused you confusion.  It should score $3$ points.  Thank you for pointing out our mistake - we will put it right ASAP!  Hopefully by the time you read this, it will be correct.

Lucy from King Athelstan Primary School found two other ways of getting $40$ points:

I wonder whether there are other ways of getting $40$ points?

How do we know that it is impossible to get more points?