#2 51. COUNT DOWN

Numbers such as 2, 3, 5, 7 ... are said to be prime – they have no positive integer divisors other than themselves and 1. Here is a game you can play singlehanded with someone else!

Two players alternately hold up any number of fingers from one to five. The cumulative total is kept, and the object is for the total to be a prime number each time. This means that the first player would hold up two, three or five fingers. During play, the first player who is unable to raise the total to a higher prime is the loser.

Now suppose you were to play first. Would you hold up two, three or five fingers?

Hints and strategies

Hint 1
Hint 2

Solution

Extensions

HINT 1

Make a list of the first primes.

Thinking about the game, at what stage in the list do you think there is going to be a problem?

HINT 2

Starting from the prime number you found in Hint 1, work backwards to find the best starting number.

SOLUTION

The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... . Since the gap between 23 and 29 is six, clearly the winner is the one obtaining the total 23. Hence you want your opponent to reach 19. This will happen if you reach 17, your opponent reaches 13, you reach 11, your opponent reaches 7, you start with 5.

EXTENSIONS

1. We have seen that first player wins if the starting number is 5. Are there other possibilities?

2. Explore how the game would go if you could use both hands.

3. We could vary the rules, for example by using Fibonacci numbers in place of primes.