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Advanced Math/Dice - Cube Shape Design.


QUESTION: Dear Prof Randy

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

ANSWER: The reason for 6-sided dice is probably because it has desirable features such as 1) it is a regular polyhedron so each side has equal chance of coming up (for properly made dice), 2) it is easy to roll, 3) its stable when landing on a side (doesn't easily tip over), 4) is easy to manufacture and 5) doesn't have too many numbers so in a board game you don't travel too far per roll.

Odd numbered dice could be used in games but the rules would need to be changed to accomodate the changing probablities of particular numbers coming up. 7-sided dice do exist, but they are not regular, meaning an equal probablitiy of any particular side coming up. Games like Yatzee and craps, which award points (or returns) based on the likihood of combinations would have to be modified.

You could probably design dice of particular (non-regular) shape that would have well defined probabiities of a particular sides (numbers) coming up for various number. You'd have to do a little physics along with probability theory to accomplish this. It would then be fun/interesting to design a game utilizing these "designer" probabilities.


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QUESTION: Dear Randy

Thank you.

Do you feel adding negative and positive numbers in a Cube, Heptagon, decagon etc Dice design figures can provide some game rules changes such as Snakes and Ladders, Monopoly, Ludo, Craps etc ?

For example : We always use positive numbers 1,2,3,4,5,6.

i.e. Any Game designed with Move Forward i.e. positive, but in this case as examples -2 indicates move two steps backwards , -4 indicates move four step backwards in the Board by the Player.

Cube Dice : 1,-2,3,-4,5,-6 OR -1,2,-3,4,-5,6
Heptagon Dice : 1,-2,3,-4,5,-6,7 OR -1,2,-3,4,-5,6,-7
Decagon Dice : 1,-2,3,-4,5,-6,7,-8,9,-10 OR -1,2,-3,4,-5,6,-7,8,-9,10.

By Adding Positive and Negative Numbers in Series or Sequence within a Dice may also increase the Game Duration Length ?

Also if we introduce Negative numbers within Dice what is the best way to show in terms of dots i.e. for example negative sign and then dot ?

Business Rule : There is a starting point in the Game where all players keep their color marked pointers. In this case if the Player has a negative number say -2, or -4 or -6 etc obviously he/she will remain at the same starting point.

Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

Prashant, Interesting concepts. I think using negative numbers would definitely be OK. In some board games there are spaces that have instructions like "move back 2 spaces", etc., so negative numbers are not unknown in board games.

In regard to having positive and negative values on dice, my first thought was that the mean value of the numbers (the sum of the values of the numbers on the faces divided by the number of faces; aka expected value) would be very important. In your example for the 6-faced die, the first sequence has a mean of -12+9 = -3, which would mean, on the average, you'd be going backwards! The mean can be adjusted of course (it is +3 for the 2nd sequence). The mean would be related to the duration of the game, but not necessarily in a straightforward way (depending on the game). You could also consider having just one number be negative.

As far as designating negative numbers, consider having the negative dots form a different design from the positive. For instance, a negative five could look like a cross instead of a square with a dot in the middle. Different shapes would have more of an aesthetic appeal, I think. More fun to design, too.

New rules would have to be introduced to accomodate negative numbers, but these shouldn't be too complicated (eg. your example).

Using negative numbers and different probabilities for the numbers on multi-faceted dice should open up new and interesting modes of play. Old, familiar games, such as Parcheesi, where you have multiple pieces you can pick from to move and/or which have consequences for landing on somebody elses space, could support different strategies than are currently used. Games like Risk, where you "battle" someone elses dice throw could modify decisions made in that game. It seems like a rich area to explore. There is also the opportunity to devise a whole new type of game. Good luck!


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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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