[plasmid]

Suppose you have eight dice (the regular six-sided kind for you D&D players). As many people know, if you look at any one face of a die, you know that the number on the opposite face will be 7 minus the number you're looking at. For these eight dice, you also confirm that they have the same chirality: that is, if you have face 1 pointing up and face 2 pointing to your right, face 3 will be pointing toward you.

Is it possible to take those eight dice and put them together in a 2x2x2 cube such that all faces that are touching have the same number, and no two dice whose faces are directly touching will have numbers pointing in the same direction? Just contacting along an edge or a vertex doesn't count, an entire face needs to be touching for those rules to apply.

For a bit of clarification, suppose one of the dice has its face with a 1 pointing up. Then you know that the die on top of it must have its face with the 1 pointing down. Although the two faces numbered 1 are touching, they are not pointing in the same direction since one is pointing up and the other is pointing down. However, if both dice had their face with 2 pointing toward you, that would break the rules.

[Guest]

yes?

[Guest]

Suppose you have eight dice (the regular six-sided kind for you D&D players). As many people know, if you look at any one face of a die, you know that the number on the opposite face will be 7 minus the number you're looking at. For these eight dice, you also confirm that they have the same chirality: that is, if you have face 1 pointing up and face 2 pointing to your right, face 3 will be pointing toward you.

Is it possible to take those eight dice and put them together in a 2x2x2 cube such that all faces that are touching have the same number, and no two dice whose faces are directly touching will have numbers pointing in the same direction? Just contacting along an edge or a vertex doesn't count, an entire face needs to be touching for those rules to apply.

For a bit of clarification, suppose one of the dice has its face with a 1 pointing up. Then you know that the die on top of it must have its face with the 1 pointing down. Although the two faces numbered 1 are touching, they are not pointing in the same direction since one is pointing up and the other is pointing down. However, if both dice had their face with 2 pointing toward you, that would break the rules.

I don't understand this part. If I'm looking at a 5, whats on the oppisite side?

[plasmid]

I don't understand this part. If I'm looking at a 5, whats on the oppisite side?

The 2. Generally, dice are made such that the 1 and the 6 are on opposite sides (if the 1 is facing towards you, the 6 is facing away from you). 2 and 5 are on opposite sides, and 3 and 4 are on opposite sides.

[Guest]

yeah opposite sides of a die=7....I just misread that line, it seemed as tho you were taking 7 away from what we were looking at.. I get it now.

[Guest]

This is hard to explain in writing, and I don't know how to get pictures so here it goes...

The bottom four I arranged:

Top-Left die=1 on top, 3 faces you, and 5 faces to the right.

Top-Right die=1 on top, 4 faces you, and 5 faces left.

Bottom-Left die=1 on top, 4 faces you, 3 on the opposite side, and 2 faces right.

Bottom-right die=1 on top, 3 faces you, 4 on opposite of 3 ,2 faces left. All touching sides are the same.

Now the top four dice go:

Top-left die=1 on bottom, 6 on top, 4 faces you, 5 faces right.

Top-right die=1 on bottom, 6 on top, 3 faces you, and 5 faces left.

Bottom-left die=1 on bottom, 6 on top, 3 faces you, 4 on opposite side of 3, and 2 faces right.

Bottom-left die=1 on bottom, 6 on top, 4 faces you, 3 on opposite side of 4, and 2 faces left.

All dice whos faces are touching another, the numbers are the same. Also any two dice that are touching, the number on the outside is facing the opposite direction.

Is that what your looking for?

Edited May 25, 2009 by James8421

[plasmid]

Is that what your looking for?

This is hard to explain in writing, and I don't know how to get pictures so here it goes...

The bottom four I arranged:

Top-Left die=1 on top, 3 faces you, and 5 faces to the right.

Top-Right die=1 on top, 4 faces you, and 5 faces left.

Bottom-Left die=1 on top, 4 faces you, 3 on the opposite side, and 2 faces right.

Bottom-right die=1 on top, 3 faces you, 4 on opposite of 3 ,2 faces left. All touching sides are the same.

Now the top four dice go:

Top-left die=1 on bottom, 6 on top, 4 faces you, 5 faces right.

Top-right die=1 on bottom, 6 on top, 3 faces you, and 5 faces left.

Bottom-left die=1 on bottom, 6 on top, 3 faces you, 4 on opposite side of 3, and 2 faces right.

Bottom-left die=1 on bottom, 6 on top, 4 faces you, 3 on opposite side of 4, and 2 faces left.

All dice whos faces are touching another, the numbers are the same. Also any two dice that are touching, the number on the outside is facing the opposite direction.

If I'm reading that correctly, in the bottom four, the top-left die would have a face touching the top-right die, and they would both have their 1s facing up. That would break the rule by having two dice whose faces are touching and have a number pointing in the same direction.

[Guest]

If I'm reading that correctly, in the bottom four, the top-left die would have a face touching the top-right die, and they would both have their 1s facing up. That would break the rule by having two dice whose faces are touching and have a number pointing in the same direction.

oh thats what you meant by that.

Well in that case I can only get four dice to follow the rules, Three on the bottom, and only one stacked on top on the corner die, after that it is impossible to add more without breaking the rules. (I think).

[CaptainEd]

oh thats what you meant by that.

Well in that case I can only get four dice to follow the rules, Three on the bottom, and only one stacked on top on the corner die, after that it is impossible to add more without breaking the rules. (I think).

it can't be done. Here's my argument:

Let's give a canonical representation of the orientation of one die, in terms of Up,North,East.

123 denotes a die with the 1 facing up, the 2 facing North, and the 3 facing East.

For each face up, there are four possible North faces, and for each one, the East face is predictable. The 24 orientations are:

123 135 154 142

214 231 263 246

312 326 365 351

421 462 415 456

513 541 536 564

632 624 653 645

Our task is to make a square array of dice with another on top, subject to the facing constraints

(1) touching faces must have the same number

(2) neigboring parallel faces must have different numbers

Suppose the bottom layer of dice is laid out:

142 XXX

YYY ZZZ

XXX must not have 1 up or 4 North, and must have 2 to the West (hence 5 E). The only possibilities are 365 and 415

YYY must not have 1 up or 2 east, and must have 3 to the north (to match the south face of 142). The only possibilities are 231 and 536.

For each combination of XXX YYY, let's identify the possible ZZZ.

XXX=365 YYY=231:

ZZZ Must have East 6 because of YYY's East 1, and it must have North 1 because of XXX's North 6. But no die has 1 and 6 on adjacent faces, so this is impossible.

xxx=415 YYY=231

ZZZ must have East 6 because of YYY's East 1, and it must have North 6 because of XXX's North 1, but no die has two 6's, so this is impossible

similar arguments for XXX=536.

So, we can't even fill out the bottom layer

Edited May 28, 2009 by CaptainEd

[plasmid]

James was right and CaptainEd got the proof: a 2x2x2 cube, and for that matter a 1x2x2 square, can't be built with these rules. You could build at most the part of a 2x2x2 cube remaining if you take away two opposite diagonal dice. Which was a little surprising, seeing as how it is possible to build a seemingly more complex 3x3x3 cube with the four corners and one central dice removed:

Top Layer  Middle Layer  Bottom Layer

  []	   [][][]		  []

[][][]	 []  []		[][][]

  []	   [][][]		  []

Likewise, although it's possible to build a 3x3 open ring (like the middle layer of the figure above), it is NOT possible to build a 4x4 open ring. Good job on visualizing what was (at least to me) a pretty complex 3-D puzzle.