The Five Number Game

The Problem

Take any five numbers between 1 and 99 inclusive. Write them in ascending order. Now add the first and the second; the second and the third; the third and the fourth; and the fourth and the fifth.

Now score points as follows: every square number is worth two points; every cube number is worth 3 points; every prime number is worth 2 points. (Numbers that are both squares and cubes only get 3 points.)

What is the highest score that you can get? In how many ways can you get it?

What is the lowest score that you can get? In how many ways can you get it?

Teaching Sequence

1. Introduce the problem by playing one game as a class.
2. Ask the students to select 5 numbers from 1 to 99.
3. Write these on the board in ascending order.
4. Now add the first and the second; the second and the third, the third and the fourth and the fourth and the fifth.
5. Ask the students to look at the 4 resulting numbers and make statements about them, for example, odd, even, prime, multiple of 5.
6. Share the scoring system for the game and apply to the 4 numbers from the class game.
7. Let the students play the games in pairs.
   As the students play the game ask questions that focus on their understanding of square numbers, cubes and primes.
   How do you know that the number is a square?
   Can you convince me that this number is a cube?
   Are you sure that this number is a prime? Why?
8. Share solutions for the highest and lowest scores.

Other Contexts

This game can be played as a whole class activity. Get the class to choose the numbers. Write them on the board and then allow the students to do the arithmetic mentally, including calculating their points scores.

Extensions to the problem

This game is a source of endless variations. We’d like to hear from you or your students about which variations you tried, which you liked and which were the hardest.

Here are some suggestions.

1. Change the number of numbers that you use.
2. Change the number of points.
3. Change the range of numbers that you use.
4. Use other number properties like even numbers, numbers divisible by 9, and so on.
5. The highest number of points that you can get is 12. Is it possible to get every number from 0 to 12 as the result of choosing some five-number set?
6. Change the operation from addition to multiplication, subtraction or even division.

Solution

It should quickly become clear that the highest number of points that is available for each of the four numbers is 3. Hence the highest total possible is 12. To get 12 all of the numbers have to be cubes. The only cubes less than 200 are 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64 and 5³ = 125. (6³ = 216, which is too big.) Now 1³ = 1 cannot be obtained by adding two numbers between 1 and 99 inclusive. Hence we would try to get 8, 27, 64 and 125.

One way of getting 8 is by adding 1 and 7. If you went this way, then to get 27 you would have had to have 20 (= 27 – 7) as the third number of the original five. It then follows that the fourth number had to be 64 – 20 = 44. Finally the fifth number was 125 – 44 = 81. So one way of getting 12 points is to have chosen the numbers 1, 7, 20, 44 and 81.

Now 8 can only be obtained in three ways: 1 + 7, 2 + 6 and 3 + 5. You can probably now see that there are only three ways of getting 12.

The smallest sum is zero. There must be many ways of getting zero. One example is by choosing, 1, 5, 7, 8, and 10. The sums here are 6, 12, 15 and 18, none of which are powers or primes.

Mrs Oliver’s class from St Margaret’s School has suggested that cubes that are squares should get 5 points and not just 3. If you use their suggestion, what is now the highest score that you can get?

They also suggest that, if the numbers from 0 to 99 inclusive are allowed, you can get as high as 16 points. They also suggest that there are exactly three ways of doing this. Do you agree?

Solutions to the extension

Of the six extensions that we suggested above, we can really only tackle one and that is number 5.

Possible point scores are 0 (we have already seen examples of that), 2, 3, 4 (= 2 + 2), 5 (= 2 + 3), 6 (= 3 + 3), 7 (= 2 + 2 + 3), 8 (= 2 + 3 + 3), 9 (= 3 + 3 + 3), 10 (= 2 + 2 + 3 + 3), 11 (= 2 + 3 + 3 + 3) and 12 (and we have seen examples of this too). We’re sure that you can produce lots of examples of each of these.

Clearly now, only 1 point can’t be obtained, and for obvious reasons!