## Dice

Tue 20 Nov 2001 by mskala Tags used: math, nov01To make best use of this posting, you should have a quantity of identical standard dice. Attempt to solve the questions for yourself before reading the answers. It is more fun to use real dice, but on the other hand, solving them in your head by visualizing the dice will almost certainly qualify you for membership in some pretentious organization.

### Questions

1. Can you build a stack of eight dice, in a cube, so the numbers match on each of the 12 pairs of faces where the dice touch?

2. Can you do the same thing with 27 dice in a 3x3x3 cube?

3. Can you fill the entire Universe with dice in such a way that they satisfy this constraint?

4. What are the answers to questions 1. to 3. above if instead of the numbers being equal on each pair of touching faces, the numbers must add up to seven? (For instance, three could touch four, but five could not touch one.)

5. What if the numbers on a pair of touching faces must add up to something else instead of seven; for instance, ten?

6. How can you use dice to verify that you are in this Universe and not some other one?

7. Bet on a number from one to six. I roll three dice. You win if your number comes up on any of them. Each die has a one in six chance of coming up with your number, and you have three chances to win, so you win half the time. Right?

8. Explain why the game in 7. above is so popular in those parts of the world where it is legal.

9. Loaded dice are dice with internal weights that bias them so that they always, or at least more often than a naive observer would expect, roll particular values. Loaded dice are useful when cheating in various games. There are also such things as blank dice. What are those good for?

10. Dice sold to consumers are all females, but it is quite easy to catch males in the wild if you go hunting near a casino or gaming convention. Use peanut butter for bait. Spread some on the ground, stand a few yards away, wait for the dice to arrive, and catch them in a butterfly net. Then you can start your own breeding outfit and supplement your income by selling baby dice to your friends and neighbours. What will you call your business?

### Answers

You get a bonus mark if you started by attempting 3. and then did 1. and 2. as special cases.

3. Place two dice together with the one faces touching. Then add two more touching these at the six faces, and two more at the end of the line touching at the one faces, and so on. Keep this up and you'll have an infinitely long string of dice joined by their one and six faces. Rotate the dice in the string so that all their two sides line up, keeping them in contact with each other. Now your string has four sides: one with all twos, two with alternating threes and fours, and one with all fives. Take more strings like this and join them at the alternating three/four sides (which happen to be opposite each other) until you have a flat blanket of dice with all twos on one side and all fives on the other side. Join these blankets to each other, face-to-face, and you can fill space until you run out of dice or they collapse under their own gravity, sucking you into the time dilation of the newly formed black hole's event horizon, to fall for subjective eternity as punishment for your foolish behaviour.

1. and 2. First fill the entire Universe with dice as in 3. above. Then remove everything but a 2x2x2 or 3x3x3 cube, as appropriate. Be sure to get an adult to help you when you empty the Universe. Stars can cause fires if disposed of carelessly.

4. This is actually easier. Just put all the dice in the same orientation (let's say one up, two to the left, and that defines where all the other faces go). Then stack them and they will automatically satisfy the constraint.

5. The 2x2x2 case is possible for some sums because you can move the inconveniently large or small faces to the outside of the cube where they don't get paired. For instance, sums of four and ten are both achievable. You can have fun determining whether any other sums are possible. The 3x3x3 and infinite cases are not possible for any sum except seven, because they each contain at least one die where all six sides touch other dice. Then the one and six sides of that die must both be touching other dice. The one side cannot be part of any sum larger than seven, and the six side cannot be part of any sum smaller than seven. So if all sums are the same, they must all be seven.

6. There are many possible answers to this question. Here are some I like:

- Count the sides on a die. If there are not exactly six, and assuming it's not one of those non-cubical RPG dice, then you're in an alternate Universe for sure. You can get a rough estimate of the number of spatial dimensions in your universe by dividing the number of sides on a die by two.
- Dice in this Universe are labelled with the numbers on opposite faces adding up to seven and with the numbers one, two, and three going in clockwise order around one corner. If your dice go counterclockwise, you may be in a mirror-image Universe.
- Roll a die. If it comes up seven, zero, or four and a half, then you may be in an alternate Universe.
- If the die explodes on contact with your skin, with the force of a large fusion bomb, then you are in a Universe where antimatter predominates.
- If the die rolls you, that's a worrisome sign.

7. No, because when your number comes up on one die, quite often it'll also come up on one or both of the others, thus wasting part of your "three in six". The three one-in-six chances overlap, so the overall chance is less than three in six. Just consider what would happen if there were six dice. By the logic of the question, you should then win every time ("six chances at one in six!") no matter which number you bet on. But that would imply that six dice rolled together will always come up with a perfect straight. Do they? Try it.

8. Because you'll be hassled by the law if you play it for money in a jurisdiction where it's banned.

9. Blank dice are like blank bullets - they sound like real dice, but are unlikely to cause injury. Blank dice are used in movies when it is necessary to depict characters rolling dice. The replica dice cups sold to collectors will only roll blank dice.

10. I'm calling mine "Matt's Nice Dice, Inc." My doe's first litter is due any day now. Call me to reserve a dicimal for yourself, or one for the special child in your life. Pay in advance and I'll give you the chance to pick your own as soon as they're born.

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