Enrichment in Mathematics – Algebra
Prime or out!
Teachers Notes
This activity focuses on:
 devising a strategy for winning a game
 the sequence of prime numbers
It could be used as a starter or plenary activity and needs to be played in pairs.
Pupils should be encouraged to compare their winning strategies and discuss how they
devised them.
Start by playing Twenty-One and ensure pupils have found winning strategies before
moving onto Prime or Out.
Questions
What happens if you start with …?
What must the second player do next?
Is there a key number you need the second player to say so that you win?
Can you follow a chain to get to this key number?
Which helps you work out your strategy, thinking about the starting numbers or about
the finishing numbers?
Solutions
Twenty-One
If you leave a running total greater than or equal to 16 you will always lose.
If you leave a running total between 10 and 14 you will lose 3 goes later, since your
opponent can then move the total on to 16-1=15, and then force you to leave a total
of 16 or more.
If you leave a total between 4 and 8, you will lose 5 goes later, as your opponent will
move the total to 10-1=9, and then force you to leave a total between 10 and 14.
Hence the first player must start by choosing 4-1: 3.
Whenever the second player chooses 1, the first responds by choosing 5, whenever
the second player chooses 2, the first responds by choosing 4 etc to always make 6.
Prime or Out
The first player must put up 2, 3 or 5 fingers (primes only).
For there to be no winner, it would have to be always possible to get from one prime
number to the next by adding 5 or less. However, the gap between the sequence of
prime numbers prevents this (e.g. the jump from 23 to 29 is 6, 31 to 37 is 6 etc)
If a player can get his total up to 23, his/her opponent will lose.
Thinking backwards:
23  19  17  13  11  7  5 or 3 or 2
The only way to force the second player to choose 7 is to start with 5.
This game is adapted from “Vingt et un” – Maths Challenge 1- Tony Gardiner.
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Algebra
Prime or Out!
You need to be in pairs for each game.
Think about strategies to either
win every time or to avoid losing.
Twenty-One
Player 1 holds up any number of fingers from 1 to 5.
The next player then holds up fingers from 1 to 5 and says the total so
far.
The winner is the first person to get the total to 21.
Play the game several times, taking turns to go first.
Can you work out a strategy so that the first person to go always wins?
Or can the game go on for ever?
Prime or out!
Player 1 holds up any number of fingers from 1 to 5.
The next player then holds up fingers from 1 to 5 and says the total so
far.
The total must always be a prime number.
The loser is the first person to give a total that is not a prime number.
Play the game several times, taking turns to go first.
Can you work out a strategy so that the first person to go always wins?
Or can you work out a strategy so that the second person to go always
wins?
Or can the game go on for ever?
Worcestershire Numeracy Team
Enrichment Activities