Tag Archives: Puzzles

Magic Squares 4 – Polo Squares

A while ago I pointed out that while I’ve never found creating a simple 3×3 magic square very useful in the classroom there are plenty of spinoff activities that really do work well.  I devised Polo Squares when I was giving a session for 300 children at the Royal Institution in the middle of London.  The children were from a dozen or more schools and I wanted to offer something that was easily accessible and could be instantly usable no matter how big the groups were or whether they arrived early or with just five minutes to spare.

Every child had a set of ten digit cards, from 0 to 9.  I asked them to make a hollow square using eight of the cards, so the cards on each side of the square add up to the same total.  I provided lots of blank record sheets; clearly a lot of work was done at home after the session, because the postman was delivering solutions for weeks afterwards.

Polo Purple

I’ve no idea how many different Polo Squares there are, but I know it’s several dozen, so no-one’s going to say “I’m finished, what do I do next?”  Each solution uses eight of the ten cards, so there’s an element of Trial and Improvement.

And there are plenty of challenges – “What’s the highest / lowest Polo total?”, “Can you make all totals in between?”, “Which totals have the highest number of solutions?”, …..

Polo Squares offer a nice challenge – even wider than I’d realised; George was just five, and the younger brother of someone who’d attended the session.

Polo George

Using digit cards helps the activity and also makes it feel like fun.  The cards invite the use of some problem-solving skills (look at L & A’s) , and give some meaty mental arithmetic practice.

Polo L&A

And in just the same way I could use them with children waiting for my session to start, teachers can have some digit cards and answer blanks available on parents’ evenings!



There Are Polar Bears Round The Icehole

It’s good to know that something you’ve done has stuck with someone; out of the blue I had a call from someone I hadn’t seen for ten years or more.  “Hello Alan, can you remind me about the Polar Bears?”

So here’s the activity that Gaynor wanted to talk about.


I roll five dice: POLAR 13456

“There are six polar bears round the ice-hole.”


Again:POLAR 23446

“There are two polar bears round the ice-hole.”


Once more:POLAR 12226

“There are no polar bears round the ice-hole.”


It’s much the best fun to do this informally when you can test out all sorts of ideas, but I’ll have to tell you that the colours of the dice don’t matter, nor their size, nor their positions.  (Generally speaking, I’ll use five dice, but we could use a different number.)  We’re left with just one factor to consider, that it’s the numbers showing on the top faces of the dice.


Here are a few more:POLAR 12334

“There are four polar bears round the ice-hole.”


POLAR 14446“There are no polar bears round the ice-hole.”


POLAR 23345“There are eight polar bears round the ice-hole.”


POLAR 13556

“There are ten polar bears round the ice-hole.”

Any ideas?

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Number Riddles and Mystery Numbers

Amina was in Y3 when she composed this puzzle:

My number:

  • is bigger than 289
  • is smaller than 310
  • is a multiple of 10
  • is not a multiple of 100

 It was Colleen Young, in her blog “Mathematics, Learning and Web 2.0”   (  http://colleenyoung.wordpress.com/  )  who made me think of Amina.  Unlike this one, Colleen’s blog is obviously created by someone who knows just what they’re doing; she pointed out that there are a number of books of interest to maths teachers which are available as free or very cheap Kindle downloads.

One of these is Mind Hurdles: Mystery Numbers (Cool Math Games For Mathematically Gifted Kids).  L L Kross’ Mystery Numbers are just the same as what Amina and I called Number Riddles.   I’m afraid that what was a free download when Colleen spotted it now costs you £1.93, but it’s still little enough for a useful resource you can get plenty of mileage from.  There are 36 puzzles in all, ranging from simple clues to identify two-digit numbers, up to as many as ten clues for numbers of five and even six digits.

Of course, creating these puzzles and checking that they’re watertight and contain the right amount of information is a lot more demanding than solving them – which is why I was so pleased with Amina’s effort, and why it’s handy to have plenty of examples available.

Incidentally, when I tried a few with two ten year-olds at least a couple of the puzzles seemed insecure.  Children rarely object to proving adults wrong, so I don’t think it matters too much, but you may find it helpful to know of the possibility in advance.