Nice Dice: Students' Notes
Introduction
These notes are designed to provide you with hints to help you while you work towards solving this problem and also to help you see a much larger picture of both the problem and of mathematics itself. The activity is available online at:
The notes below are split into four phases:
- Understanding and solving specific problems
- Generalising
- Proving
- Extending
The four phases are the same as those used by mathematicians when they are working on research problems. Of course, if you only want to go as far as finding a set of nice dice, you won’t need to go through all of these phases. However, the problem is certainly set up for you to do more than just find one pair of nice dice.
Below we have provided a number of hints for you to use if you need help. Try to do as much of the problem as you can yourself, and only use the hints if you get stuck.
Phase 1: Understanding and solving specific problems
This activity asks you to design two dice so that when they are rolled and the numbers on top added together, the sum is equally likely to be any number from 1 to 12. We call such dice, ‘nice’ dice. Just to be sure we remind you that a dice is in the shape of a cube and one number is assigned to each face.
To add numbers to the dice, click on the arrows on a face till you get to the number you want. Then rotate the cube to another face and add a number there. This way you can put any number from 0 to 12 on any face of both dice.
Hint 1
If you are not sure how to solve the problem, the first thing you might try is just play. First add numbers at random and then adjust things to make the dice as nice as possible.
Hint 2
This problem is not about normal dice (with the numbers 1, 2, 3, 4, 5 and 6). It is easier to start out thinking of blank dice and deciding what to put on each side than it is to start out thinking of normal dice and changing the numbers.
Hint 3
Think about the number of possible results that you can get when you roll two dice. No matter whether the dice are normal or nice, how many are there?
Hint 4
Because there are six possible results for each dice there are six times six (36) possible results from rolling two dice. How many of these need to sum to each of the numbers from 1 to 12?
Hint 5
There are 36 possible results and 12 numbers to sum to. If we want each of the numbers 1 to 12 to be equally probable, each of the possible sums needs to be able to be made three different ways.
Hint 6
This is a problem where using a piece of paper may be very helpful. Try drawing the sample space for the problem as a grid like the one below.
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Hint 7
When you roll the dice there needs to be three ways that the sum of the two numbers can be 1. How is this possible? Note that the computer will only let you put one of the numbers from 0 to 12 on any face.
Hint 8
The only way you can make a sum of 1 is with a 1 and a 0. You cannot have a 0 on both dice or you would be able to roll a sum of 0, so only one dice has a 0. Because you need to be able to make 1 in three ways either one dice has three 0s and the other has a 1 or one dice has three 1s and the other has a 0. Either will work, but we have started by putting three 0s on one dice (see below).
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Now what can you do to make sure that there are three ways to roll a sum of 2?
Hint 9
There are a lot of different ways you could make three possible sums of 2, but the two simplest are:
- Add three 1s to the dice that already has three 0s.
- Add a 2 to the dice that already has a 1.
Each of these options will lead to a solution to the problem. See if you can find both.
Hint 10
If you followed option 1 from Hint 10 you have put numbers on all six sides of one dice. There is only one possible way to number the other dice so that it works, and if you work your way systematically through the required sums you should find it.
Hint 11
Here is the solution from option 1.
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If you followed option 2 then you still have decisions to make, but you have started a pattern that seems to work pretty well. Putting a 2 below the 1 gives you three ways to sum to 2, so to make three ways to sum to 3 why not add a 3, and then for 4 …
Hint 12
Once you put the numbers 1, 2, 3, 4, 5, and 6 on one dice and three 0s on the other you should have been able to see that you needed three 6s on the dice with the 0s to make all the required sums.
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So now you have two solutions, but are there any more?
Phase 2: Generalising
What we are trying to do here is to take you through a process that will give you nice dice with any numbers on them. In other words we are going to remove the restriction posed by the problem, of having only the numbers 0 to 12 on the faces of the dice. But to get there we need to produce all possible nice dice that use the numbers 0 to 12.
Hint 13
What do you notice about the two solutions so far? In both solutions one of the dice has three each of two different numbers. Can you find any other solutions like that?
Hint 14
Both of the solutions above also have one of the sets of three numbers being 0. Can you find another number that could go with the three 0s other than three 1s or three 6s? There are four possible solutions altogether.
Hint 15
The other two solutions that include three 0s on one dice are shown below.
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Are there any other different solutions than the four described so far?
Hint 16
You may have tried to enter what you thought was a new pair of nice dice and been told by the computer that it was not essentially different from a previous pair. Why do you think that was? Try to produce as many nice dice as you can that will be accepted by the computer as different.
Hint 17
Among all of the solutions you have found, including those that the computer did not recognise as different, are there any that might be related?
Hint 18
It is possible to make variations on solutions by adding a fixed amount to every side on one dice and subtracting the same fixed amount from every side of the other dice. This could produce dice with negative numbers on them. It might also produce dice that have numbers other than whole numbers on them.
Hint 19
Let’s say that two pairs of nice dice are related if you can get from one of them to the other by adding a fixed amount. And let’s say that a set of pairs of dice that are all related to each other are a related set. Can you produce a related set that is infinite?
Hint 20
How many related sets are there?
Phase 3: Proving
So what is your conjecture for the number of related sets? And what is your conjecture for the form of each related set? In this section we try to find a conjecture and a proof in reply to each of these questions. Where you start here very much depends on how many pairs of nice dice you found in phase 1.
Conjecture: There are less than 4, 4, 5, 6, 7, 8, 9, more, related sets.
Choose the number that your work so far suggests.
Hint 21
A proof here will mean that you can go through all of the steps that you took using the numbers 0 to 12 on the faces of the dice but you will have to bring in a variable. ‘x’ is a good choice for a variable. If you put x on one face of a dice what do you have to put on the other to get a sum of zero?
Hint 22
Is there any difference between these two things? Put x on one face of one dice and 1 – x on three faces of the other; put 1 – x on one face of one dice and x on three faces of the other.
Hint 23
Now just follow the steps you used earlier but be systematic. With x on one dice and three lots of 1 – x on the other, you now have to make 2. How can that be done? Follow each possibility.
When you have made 1 and 2, how do you make 3? How can that be done from where you got with the 2? (Every now and again you will get to dead ends that don’t work but that’s life.)
How many related sets did you find?
Phase 4: Extending the problem
The question now is can we take the problem any further. What Nice Dice-like problems can we build around this one? Extending problems is a fundamental way of working for mathematicians. Let us suggest some related problems.
- Can we make nice dice that are tetrahedral (four-sided) in shape? Or eight-sided? Or …?
- Can we make a pair of nice dice that give only a sum of 1? Or sums of 1 and 2? Or sums of 1, 2, 3?
- What sums that are produced with equal probability can be obtained using just two dice?
- Can we make three ‘nice dice’ that produce only the sums 1 to 12 and do so with equal probability? Or sums of 1 to 18? Or sums of …?
- Can we make n ‘nice dice’ that produce equally likely sums of 1 to m? What restrictions are there on m?
- What are the related (infinite) sets that are involved in all of these?
