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Number Theory/Dice - Cube Shape Design.


Dear Prof Scott

We use Dice in various indoor games viz Snakes and Ladders, Ludo, Business etc. Dice is cube shape, just curious to know regarding the design i.e. Six sides numbered with each side with 1,2,3,4,5,6.

What could be the reason for designing a Dice with a Cube shape ?
Six is an even number. Can we also have dice manufactured as a Pentagonal shape (i.e. 5 sides numbered with 1,2,3,4,5) or Heptagon (i.e. 7 sides numbered with 1,2,3,4,5,6,7.
i.e. Odd Number Dice.

Some Games are also played with more than One Dice i.e Two Dice.

Can Games be played with One Dice a Cube shape and another Dice with a Heptagon Shape ?

Is it something to do with Probability Theory ?. If we change the Dice design, will there be also probable changes in game business rules viz Snakes and Ladders, Ludo etc ?.

Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

By the way, dice refers to more than one.  When there is only one, it is referred to as a die.
That sounds funky, but that's the way it is.

It is easy to make many dice at once with a criss-cross pattern.  The others aren't quite so easy to do, but with technology going the way it is, they are easy to make as well.

With production going the way it is, they could now make a die with almost any number of sides.
I have seen 4, 6, 8, 10, 12, and 20 sided dice.  To make a die with an odd number of sides,
just make a long cylinder-like shape and cut it into the appropriate number of sides.
For example, to make a 5-sided die, make a long cylinder that looks like a pentagon when looking straight down on it.  It could be tapered off at the ends to a single point so landing on an end would be impossible.  Another way to think of it would be to take a sphere and cut them into the appropriate sided n-gon.

If the die is weighted properly or cut in a non-symmetric pattern, it could be weighted to have one number come up more often.  Many people have done this illegally to make money, quickly passing on to the next town.  Nowadays, however, doing this isn't quite so simple.

Another way to do it would be to take a even balanced six sided die and number three of the sides with a 1, two of the sides with a 2, and one side with a 3.  This would make 1's come up half of the time, 2's come up a third of the time, and 3's only come up one sixth of the time.

Another way to make lower numbers more probable would be to have two evenly balanced dice and multiply the result.  A multiplication sheet would be included if necessary.  For two six sided dice this would give the following:

Result / Number of Ways
1   1
2   2
3   2
4   3
5   2
6   4
8   2
9   1
10   2
12   4
15   2
16   1
18   2
20   2
24   2
25   1
30   2
36   1

For example, a 6 could be gotten by a 1-6,  2-3, 3-2, or 6-1, so there are four ways.
Note that prime number over 6 are not included, as well as some others.  The average is 12.25.
Note that out of 36 possibilities, 23 are less than this and only 13 are greater.

There are even more designs; for example, maybe a (4-sided die#*(12-sided die#, or the sum of 4 8-sided dice, or making each side an increasing power of 2 or 3 or higher.  On that last one, in backgammon, they have what's known as a doubling cube, with the sides labeled 1, 2, 4, 8, 16, and 32 #2^0, 2^1, 2^2, 2^3, 2^4, and 2^5).  Maybe they could make a tripling cube, with 3^0, 3^1, 3^2, 3^3, 3^4, and 3^5 #1, 3, 9, 27, 81, and 243).

As technology moves forward, they may have some shape with a pretty picture in the middle.  The device would be battery operated and would generate any range of values with whatever probability was desired.  It might be made to done at some casino where there were several people betting on the outcome.  It could tally the numbers bet on and give a weighted chance  of getting a number to be higher the fewer people that bet on it.

The possibilities are endless.

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Scott A Wilson


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