Scoring with Dice
Problem
I have a handful of dice:
I throw three of them and get 5, 3 and 2:
Add the scores on the three dice. What do you get? Now multiply the scores. What do you get?
The next time I throw four of the dice and add and multiply the score as before.
This time the product is exactly 4 times the sum. What scores could I have thrown?
Again I throw four of the dice, adding and multiplying the scores as above. This time the product is exactly 5 times the sum. What scores could I have thrown?
For the last go I throw five of the dice and add and multiply the scores. The product is exactly 6 times the sum. What scores could I have thrown this time?
Getting Started
Try numbers which have products with suitable factors. For example $21$ would not be a good choice as one of its factors is $7$.
Student Solutions
Clement from Sha Tin College in Hong Kong has sent in this solution for Scoring with Dice:
For the first question, I had to find a combination using 4 numbers, and its product had to be 4 times its sum. I first tried to find a suitable number, with many factors. And it had to be big enough in order to fit the sum of the combination in.
I thought: 40 would probably be too small to fit the sum of the combination in and 44 only had 4 factors, so I tried 48. I listed all the factors: 2*24, 3*16, 4*12 and 6*8 and tried splitting each one of them into 4 steps, such as: 2*6*4*1 or 3*4*2*2 and I eventually came up with one: 6*2*2*2, the sum being 12.
I used the same method for the next two questions and came up with 5*3*2*2 which made 60, the sum being 12 and 3*3*2*2*2 which made 72, the sum being 12.
Thank you for this clearly explained answer Clement.
Teachers' Resources
Scoring with Dice
I have a handful of dice:
I throw three of them and get 5, 3 and 2:
Add the scores on the three dice. What do you get? Now multiply the scores. What do you get?
The next time I throw four of the dice and add and multiply the score as before.
This time the product is exactly 4 times the sum. What scores could I have thrown?
Again I throw four of the dice, adding and multiplying the scores as above. This time the product is exactly 5 times the sum. What scores could I have thrown?
For the last go I throw five of the dice and add and multiply the scores. The product is exactly 6 times the sum. What scores could I have thrown this time?