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Here are two of my favorite answers. Do you have another way to get to 30? Put it in the comments!

## Question

How many ways can a die be numbered, using the numbers 1, 2, 3, 4, 5, 6 exactly one time each?

## Term: “Same Dice”

Two dice are “the same” if they can be rotated until their numbers match up exactly. For example, if you take a standard die, and place it with the 1 on top, the 6 will be on the bottom. If you rotate it so the 2 is in front, the 5 will be in back. And then you might notice the 4 is on the left, and the 3 is on the right. All standard dice are numbered this way.

## Solution 1 – way overcount

Consider one die, with any numbering. There are six sides that could be face down. And then there are four rotations with that face down. Six times four is twenty-four (24) orientations for each numbering.

Now let’s think about a blank die. How many numbers can we put on the bottom? Six. How many are left over to put on the front? Five. How many left to put on the left (ha ha)? Four. How many to place on the right? Three. In the back? Two. On top? Only one number left. That’s 6×5×4×3×2×1=720 ways to put 6 numbers on a die.

But hold on. Each numbering has 24 orientations. So those 720 numberings – 24 of them are the standard die, 24 are the standard but with 3 and 4 swapped, etc. To get the actual number of ways to number a die, divide 720 by 24. That’s 30. (And for each of those 30 ways, there are 24 numberings. 30×24=720)

## Solution 2 – constructive

This time we will avoid overcounting.

We will imagine that the “1” has been put on the bottom of the die (and if it hasn’t, flip the die around so the “1” is on the bottom, which is always possible).

What number is on the top? We have five choices.

Now, we have four numbers left to wrap around the middle. How many are there?

Well, to put four things in order we have twenty-four ways. Twenty-four is 4×3×2×1. But that’s not the answer. Let’s look at some letters instead of numbers, for the moment:

1. ARST
2. ARTS
3. ASRT
4. ASTR
5. ATRS
6. ATSR
7. RAST
8. RATS
9. RSAT
10. RSTA
11. RTAS
12. RTSA
13. SART
14. SATR
15. SRAT
16. SRTA
17. STAR
18. STRA
19. TARS
20. TASR
21. TRAS
22. TRSA
23. TSAR
24. TSRA

Looks good, right? 24? Nope. Because these wrap around the middle of the die, ARTS and RTSA are the same. ARTS is really ARTS and then A and then R like this: ARTSARTSARTSAR and as you keep going around there’s not a beginning or an end, just a sequence.

So how many ways are there to wrap four things in a circle? Let’s list them:

1. ARST RSTA STAR TARS
2. ARTS RTSA TSAR SART
3. ASRT SRTA RTAS TASR
4. ASTR STRA TRAS RAST
5. ATRS TRSA RSAT SATR
6. ATSR TSRA SRAT RATS

Six ways. And that kind of makes sense. Each of these ways can be written with four different starting points. So our list of twenty-four including four copies of each arrangement, and twenty-four divided by four is six.

As an aside, arranging objects in a circle I call a “circle permutation” – and if the items are unique, and there are n of them, then there are n×(n-1)×(n-2)×…×2×1 ways to list them, but then we have to divide by n, so we get (n-1)×(n-2)×…×2×1circle permutations.

Let’s get back to our problem.

We place “1” on the bottom face.

We have five choices for the top face.

We have four numbers left to put in a circle, which gives us 3×2×1=6 circle permutations.

1×5×6= 30 ways to number a die.

1. December 6, 2022 am31 8:00 am 8:00 am

Those are the two ways that I would do it! (Of course you can fancy them up with words from abstract algebra like “group action” and “orbit”, but that’s just vocabulary, not different ideas.)