Sicherman Dice

This week we will put my adventure on the Camino del Norte on pause. Instead, the puzzle I present I feel is worthy of a newsletter to itself. Here is the puzzle:

How can you number two dice such that the probability of any given total is the same as with two convention six-sided dice? One of the dice must have one to four dots on each side. The other die may have any number per side.

First, let’s review the number of combinations of each total with two conventional dice.

Here are the totals according to the roll of both dice.

Die 1	Die 2
	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

 

Here is how often each total from 2 to 12 is observed.

Total	Combinations
2	1
3	2
4	3
5	4
6	5
7	6
8	5
9	4
10	3
11	2
12	1

 

Let’s say that die 1 is the one that is limited to numbers 1 to 4. It’s safe to assume there is at least one 1 and at least one 4. Otherwise, the range given would probably have been narrower.

In order to achieve one combination of a total of 2, there must be exactly one 1 on die 2.

In order to achieve one combination of 12, there must be exactly one 8 on die 2.

We can’t have more than one 1 and one 4 on die 1, otherwise we would have more than one total of 2 or 12. Here is where we are at now is:

Die 1	Die 2
	1	?	?	?	?	8
1	2	?	?	?	?	9
2 or 3	?	?	?	?	?	?
2 or 3	?	?	?	?	?	?
2 or 3	?	?	?	?	?	?
2 or 3	?	?	?	?	?	?
4	5	?	?	?	?	12

 

If die 1 had four 2’s, then there would be at least 4 ways to roll a total of 3. We need exactly two ways. So, at least one face must have a 3.

If die 2 were 1-2-2-2-3-4, then there would be at least 3 ways to roll a total 3. Again, we need exactly two ways.

So, dice 2 must be 1-2-2-3-3-4. That gives us 3 ways to roll a 3 so far.

Given that configuration of die 1, die 2 can’t have any 2’s. Otherwise, there would be too many ways to roll a total of 3. Let’s try 3 as the next number on die 2. That gives us the following totals between the two dice:

Die 1	Die 2
	1	3	?	?	?	8
1	2	4	?	?	?	9
2	3	5	?	?	?	10
2	3	5	?	?	?	10
3	4	6	?	?	?	11
3	4	6	?	?	?	11
4	5	7	?	?	?	12

 

Notice there are three ways to roll a total of 5. We need a total of four ways. Recall with two dice we can roll a total of 5 with 1+4, 4+1, 2+3, or 3+2.

If we add another 3 on die 2 then we will have five ways to roll a total of 5, which is too many. So, the next number of die 2 must be a 4 to match with the one 1 on die 1 to make one more total of 5. So, now we are at:

Die 1	Die 2
	1	3	4	?	?	8
1	2	4	5	?	?	9
2	3	5	6	?	?	10
2	3	5	6	?	?	10
3	4	6	7	?	?	11
3	4	6	7	?	?	11
4	5	7	8	?	?	12

 

Note we are four ways to make a total of six. We need five ways. If we added another 4 to die 2, then we would have too many ways to make a total of 5. So, the other way to make a total of 5 must come from a 5 on die 2, to combine with the 1 on die 1. That gives us:

Die 1	Die 2
	1	3	4	5	?	8
1	2	4	5	6	?	9
2	3	5	6	7	?	10
2	3	5	6	7	?	10
3	4	6	7	8	?	11
3	4	6	7	8	?	11
4	5	7	8	9	?	12

 

The next total to look at is 7. So far we have five ways and we need six. That can come only by adding a 6 to die 2. That will combine with the 1 on die 1 to give us one more total of 7. That will give us:

Die 1	Die 2
	1	3	4	5	6	8
1	2	4	5	6	7	9
2	3	5	6	7	8	10
2	3	5	6	7	8	10
3	4	6	7	8	9	11
3	4	6	7	8	9	11
4	5	7	8	9	10	12

 

The total combinations for each total matches those with two conventional dice. So, we are done.By the way, these are called Sicherman Dice.

Credit

I have seen this problem several times over the decades. I think it is too hard if you’re not given the hint that one die has a range from 1 to 4.

A good video where I got the logic for this solution is titled “Can you solve the cursed dice riddle? - Dan Finkel” on the Ted-Ed YouTube channel.