This problem solving activity has a number focus.
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.
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?
This game challenges students to calculate and find squares, cubes and primes below 200. Through this game, students are introduced to investigations.
Working backwards is one strategy that is applied. Students begin by thinking of numbers they want to end up with and then working to make them.
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.
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?
There are many possible variations. These include:
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). 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 is 125 – 44 = 81. So one way of getting 12 points is to have chosen the numbers 1, 7, 20, 44 and 81.
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.
The solution to extension number 5 is as follows:
Possible point scores are 0 (examples of this have already been discussed): 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. It is possible to produce many examples of each of these.
Only 1 point can’t be obtained.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/five-number-game at 8:58pm on the 26th February 2024