Years 5 - 8

A game for two players using one pack of cards.
This investigation has several levels of challenge. You don't have to do it all. Like a video game, you can stop at any level then return later for the next level. Keep good journal notes so you know the level you reached.


  • Find a pack of cards and, for now, use all the cards Ace - 6. Put the others away until later. Ace = 1.
  • Write the title of this challenge and today's date on a fresh page in your maths journal.

Investigating Thirty-One

  • Open this Thirty-One Starter.
    You can read it on screen or print it.
  • Play the game a few times with your partner.
  • After a while stop playing against each other and instead work together to try to discover a winning strategy.
    Have fun exploring Thirty-One.
  • Make notes and diagrams in your journal to explain your strategy.

Questions To Help You Hunt

If you need a little help to find a winning strategy, try asking yourself these questions:
  • Do I only know when someone wins if they get to 31? Can I sometimes tell before that?
  • Is there a number I can be on before 31 and know that I will win?
When you are analysing a game you must assume it is being played by the world's best. There is no point creating a strategy that will only work if the opponent makes a mistake.

In this game there is a special number where you can be certain that someone will win. But once you find it you have to ask the next question:

  • Is there a number before that where I can be certain that I will get to the special number?
and then, when you find more numbers that lead to the special number...
  • Is there a pattern in these numbers?
  • If there is, how is the pattern connected to the structure of the game?

Digging Deeper

You can find out much more about Thirty-One, perhaps even become an undefeated champion, if you use these questions to help you dig deeper. You don't have to try them all and you don't even have to start now.

A mathematician is never finished with a problem. They are only finished with it for now, until the next interesting questions comes along.

  • What happens if the winning number is 28 instead of 31?
  • What happens if we change the range of cards used?
  • What happens if we change the winning total?
  • For a given range of cards, what is the highest possible (lowest possible) winning total?
  • If I give you any range of cards 1 to N and any (allowable) winning total can you tell me how to play to win?
  • Can you tell me which combinations of range and winning total will allow a counter-strategy?

Just Before You Finish

For this part you need your maths journal and your Working Like A Mathematician page.
  • Read again what it means to work like a mathematician.
  • In your journal, finish this paragraph:
    Working with Thirty-One I was a successful mathematician because...
Note: If your school is a member of Maths300, your teacher can provide software that will help you explore Thirty-One
and test hypotheses against the computer.


This report was prepared by Jess, a Year 5 student at Regency Park Primary School.


Answers & Discussion

These notes were originally written for teachers. We have included them to support parents to help their child learn from Thirty-One.

Send any comments or photos about this activity and we can start a gallery here.


Maths At Home is a division of Mathematics Centre