Possible Pairs

For this challenge, you will need to print out a set of triangle cards, or scroll down to use the interactivity.

The first part is a game for two (or more) players; then there is a question you can think about.

This type of game is played with lots of different sorts of cards; you might have heard it called Matching Pairs.

To play this game:

• Shuffle the cards and then lay them face down on the table, arranged in rows.
• Players take it in turns to turn over two cards. If the player can draw a triangle with the two properties shown, then s/he takes the cards. If not, once all the players have had a chance to look at the two cards and see where they are placed, the cards are turned back over.
• The game finishes when no matter which two cards are turned over, there is no triangle with both of those properties.
• The winner is the person with the most cards at the end of the game. Of course it will help you if you can remember where the cards are!

Play the game several times to get a feel for it.

Are there some cards that are particularly 'good'? Why?
Are there some cards that are particularly 'bad'? Why?

Now, suppose instead of having the cards face down we have them all face up.

If it's your turn first, how many possible pairs of cards are there that you could choose and win (that is, in how many ways could you choose a pair so that there is a triangle with both the properties)?

Can you list all the possible pairs? How do you know you have found them all?

If you would prefer not to print out cards, here is an interactivity. 

Player 1 clicks on a card to turn it over, then clicks on another card. If the player can draw a triangle with the two properties shown, then s/he can press 'Accept'. The two cards will remain face-up.
If a triangle cannot be drawn, once players have had a chance to look at the two cards and see where they are placed, s/he presses 'Don't accept'. 

You will need to keep track of the number of pairs you have 'collected'.

You may like to explore alternative versions of the interactivity by clicking on the 'Settings' icon (the purple cog) in the top right-hand corner.

You may like to have a go at Name That Triangle! as a follow-up to this task.

This problem is based on the Triangle Property Game from "Geometry Games", a photocopiable resource produced by Gillian Hatch and available from the Association of Teachers of Mathematics.