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You are given two identical standard dice as shown below. You can stack them one on top of the other, or place them touching side by side. In all cases the face of one die must fully touch the face of the other die (square to square). Also the face of at least one die must lie flat on the table. How many distinct ways are there to place them on a table? Two combinations are considered distinct if they visually look different, ignoring the viewer's position relative to the dice and their location on the table. Note that faces 2, 3 and 6 have multiple unique rotations. For example, two adjacent 3s can look like "//" or "\/". Similarly two adjacent 6s can look like "==", "=||" or "|| ||".

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  • $\begingroup$ Does that include rotation of the dice, or location on the table? If so, there are a vast multitude of positions for even one die including rotation, orientation, and location on the table. $\endgroup$
    – Blue Herring
    Commented May 12, 2023 at 16:30
  • $\begingroup$ @TylerSelden good question. I tried to answer it in the puzzle. And yes I think there are MANY combinations, but it must be a finite number and I wonder what it is. I don't know the exact answer myself, but I have some bounds. $\endgroup$
    – Dmitry Kamenetsky
    Commented May 12, 2023 at 16:36
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    $\begingroup$ I think the asymmetric patterns on certain numbers are a bit of a red herring, as every rotation about a particular face can be inferred from the other sides of the dice. I don't think there's anything special about the asymmetric faces, since the dice as a whole are asymmetric. Two 1's side-by-side still have as many unique rotations as two 3's, which can be determined by the other sides, even if the 1 faces themselves don't look any different. $\endgroup$
    – Nuclear Hoagie
    Commented May 12, 2023 at 17:10

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Let's start with cases where both dice are on the table.

There are 36 face-to-face combinations, but only 21 are unique since the dice are indistinguishable. For each of those 21 combinations with both dice are on the table, you can rotate each of the dice into one of 4 positions (turning it so that different faces of the die touch the table while the same face remains touching the other die), so there are 16 rotations for each face-to-face combination. There are 336 ways to put two dice on the table touching each other that are unique.

We also need the cases where the dice are stacked.

We now get to use all 36 face-to-face combinations since the dice now are distinguishable (one is on the table, one is not), but we now only have 4 rotations of each one, since the dice can only rotate relative to each other but not the table. There are 144 ways to stack the two dice that are unique.

In total, I think there are

480

ways to have the two dice touching face-to-face that look different.

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  • $\begingroup$ Can you show or describe those 21 unique combinations? $\endgroup$
    – Dmitry Kamenetsky
    Commented May 12, 2023 at 23:53
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    $\begingroup$ @DmitryKamenetsky It's just the $\binom62=15$ ways of choosing two distinct numbers plus the $6$ ways of choosing two identical numbers. $\endgroup$
    – Jaap Scherphuis
    Commented May 13, 2023 at 6:12
  • $\begingroup$ @DmitryKamenetsky As another way of thinking about it, visualize the 6x6 grid of all possible two-dice combinations. Everything below the diagonal is a "repeat" (e.g. 1-6 vs. 6-1). Counting down the rows, there are 6+5+4+3+2+1 unique combinations. $\endgroup$
    – Nuclear Hoagie
    Commented May 15, 2023 at 13:21
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A different approach:

I think the best approach is to think of the two dice as a single die. Then, enumerate all the possible merged dice and the number of unique ways to put the merged die on a table.

There are $24$ ways to orient a cube, so by putting one die on the left and the other on the right, there are $24^2$ possible merged dice. However, this has a fair amount of duplicate counting.

Specifically, there are two sources of duplication. The first is that the dice are indistinguishable, so the count must be divided by two. Secondly, this method of generating merged dice also generates four orientations of each merged die (imagine rolling the merged die about its long axis).

Therefore the actual number of possible merged dice is $\frac{24^2}{8}=72$.

The number of ways to orient a rectangular prism is the same as a cube - $24$. However, the question specifies that the viewing angle is ignored, which divides the orientations by four. Another way to think of this is that there are $6$ sides that can face up, and spinning the die without changing the top face generates four orientations per side.

Altogether, this approach generates $72*6$ possibilities, or $432$.

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