Playing Games to Help with Maths

Me

Teach maths at secondary school at The High School of Glasgow

Volunteer for UKMT – mentoring and write questions for JMC/IMC

Teach all levels including top sets – spend lots of time on extension material

Help prepare maths students for University including Oxbridge

Playing Games to Help with Maths

Revision

Simple games can be used to deliver revision content

Playing Games to Help with Maths

Bingo

Playing Games to Help with Maths

Nine numbers from 1 to 20

Games

Use for investigation

Play against a pupil first to demonstrate

Don’t try too hard to link to maths

Wait for strategies to emerge

Doing anything well is maths

Playing Games to Help with Maths

21

Start counting at 1

Playing Games to Help with Maths

Say either one, two or three numbers, taking turns

Whoever says 21 loses

High-Low

Guess if the next card is higher or lower

Playing Games to Help with Maths

Ace is highest, then King, Queen, Jack, Ten

Also lose if you get a pair

Unlucky 6

Stand up

Playing Games to Help with Maths

Every time the dice is rolled add that number to your score

Sit down when you feel you have enough points

If a 6 is rolled and you are still standing you lose all your points

Nim

Make at least three piles of at least three objects

Playing Games to Help with Maths

On your turn take some or all objects – must all be from the same pile

Whoever takes the last object loses

Each card has eight symbols on it

Dobble

Every pair of cards has exactly one pair of symbols that matches

Just call it out to win!

Dobble – Rules

Have another go

Dobble – Rules

Dobble – Rules

Here is one variation of the game

Play in groups of three. �For each round one person is the referee and doesn’t play

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1. The referee turns over two cards at the same time

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2. The first person to name the matching symbol takes both cards

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3. The referee replaces both cards

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4. The person with the most cards at the end is winner and becomes the� referee next time

Dobble

Playing Dobble is easy – for ages 4+

Constructing a Dobble set is what is difficult

(and interesting mathematically)

Dobble – Golden Rule

The challenge is to arrange symbols so that they follow the Golden Rule

Every pair of cards matches with one and only one symbol

Dobble – Bad Sets

This Dobble set follows the Golden rule, but wouldn’t be much fun to play!

Every pair of cards matches with one and only one symbol

A

A

A

A

A

Dobble – Bad Sets

This Dobble set also follows the Golden rule, but wouldn’t work either�(each pair of cards has it’s own matching symbol)

Every pair of cards matches with one and only one symbol

A B�C D

B F �I J

A F �G H

C G �I K

D H �J K

A Dobble set with 57 cards would need over 1000 symbols

For a Dobble set with 5 cards you already need 10 different symbols.

Dobble – Perfect Sets

A good set has different matches between each pair, and not many symbols

Every pair of cards matches with one and only one symbol

In fact, a ‘perfect set’ has as many different symbols as there are cards

Pictured is a perfect Dobble set with three cards and three symbols �(but still too small to be playable)

Dobble – Seven card set

The next perfect Dobble set is seven cards.

Every card has three symbols

There are seven symbols in total

Dobble – Seven card set

Example solution

A B C

A D E

A F G

B D F

B E G

C D G

C E F

It is possible to make a seven card set

just by writing down symbols

Dobble – Fano Plane

An alternative approach uses this �Fano Plane

Dobble – Fano Plane

Every circle is on three lines and��every line goes through three circles

A banana �B crab�C dog

D moon�E spiral�F candle

G cloud

1

2

3

4

5

7

6

A

Name Example

D

E

B

C

F

G

Fano Plane

A �B �C

D�E�F

G

1

2

3

4

5

7

6

A

1

2

3

7

4

5

6

Name

Fano Plane

1. Add letters B-G to circles in any order

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2. Draw pictures for A-G at the top

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3. Copy three pictures in to each numbered square based on what circles that numbered line is on in the Fano Plane

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4. Cut out the squares and play Dobble

Dobble – Perfect Sets

Perfect Dobble sets

Can you spot the pattern?

In fact normal Dobble has 55 cards, not 57. Can anyone work out why?

Dobble – Larger Sets

Larger Dobble sets need a method to form them. �For example:

1. Extending the Fano Plane

2. A grid method

3. Double Latin Squares method

4. A projective geometry method

5. Computer Programming method

Dobble – Larger Sets #1 (Fano Plane)

Dobble – Larger Sets #2 (Grid Method)

Dobble – Larger Sets #3 (Double Latin Squares method)

Dobble – Larger Sets #4 (Projective geometry method)

d

d

d

e

e

e

f

f

f

d e f x

1

1

1

123x

2

2

2

3

3

3

G

G

G

H

H

H

I

I

I

GHIx

Dobble – Larger Sets #5 (Computer Programming method)

Output for N=4 (13 card set)

Dobble – Larger Sets #5 (Computer Programming method)

Output for N=5 (21 card set)

Dobble – Larger Sets #5 (Computer Programming method)

Output for N=6 (31 cards)

Dobble – Perfect Sets

Perfect Dobble sets

Minibridge

Played in pairs, with tricks and trumps

Optimal strategy a long way off, hope for sensible play only

Basic mnemonics – win with cheapest card that will win, lose with worst card

Some arithmetic, mostly strategy

Playing Games to Help with Maths