*OptiDice TM*

*Standard polyhedral dice optimally designed for fairness!*

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Our designs of the standard polyhedral dice are optimized for fairness by balancing the distribution of numbers, using numerals that are physically balanced, and sizing the dice based on both manufacturing and game play considerations. OptiDice are available in blue, red, green, white, and black. All of our offerings are available online at The Dice Shop at Mathartfun.com.
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**d6**

*Single die: $2.50
Set of five dice: $10.00*

**d8**

*Single die: $2.50
Set of five dice: $10.00*

**d12**

*Single die: $2.50
Set of five dice: $10.00*

**d10 0-9 and 00-90**

*Single die: $2.50
Set of five dice: $10.00
The upper hexagon denotes tens*

**Magic-numbered d20**

*Single die: $2.50
Set of five dice: $10.00*

**OptiDice Design Philosophy**

*1. Balanced distribution of numbers.* To roll fair, dice should be physically balanced, but in practice this is never the case due to small inaccuracies in molds, nonuniform changes introduced during tumbling, and density variations due to defects like voids. In addition, it's possible to affect the roll of dice to a degree by carefully controlling the manner in which they're tossed. For these reasons, dice are more fair if they are as numerically balanced as possible. For example, a void inside a die near a vertex (a point where three or more faces come together) will cause that vertex to preferentially face up when the die is tossed. The effect of such defects can be minimized by arranging the numbers such that the sum of the faces meeting at each vertex is as nearly the same as possible. Our d20 is "magic numbered", meaning the vertex sums are ideally balanced. A magic numbering of the other standard polyhedral dice isn't possible, but ours are as close as the math allows under the convention of largest number being across from smallest, next largest across from next smallest, etc. For the d4 and d6 there is essentially no choice. For the d8, d10 and d12 the numerical balancing of OptiDice is considerably better than other dice on the market.

*2. Physically-balanced numerals.* The removal of material for numbers is another source of physical imbalance. A face numbered 1, e.g., will be heavier than a face numbered 8. To remove this source of imbalance, our copyrighted set of numerals have the same area, as shown below. Each numeral is made up of ten hexagons of the same size, including the decimal point for the 6 and 9. Furthermore, the numerals are positioned on the faces such that the center of mass of the numeral lies at the center of the face. Due to the small size of numbers required for a d20, it isn't practical to use these designs for our d20.

*3. Optimum size.* Dice will be better balanced physically if the polyhedron is accurately shaped. In manufacturing technologies errors, as well as the amount of material removed during tumbling, are generally absolute. E.g., mold dimensions are accurate to within some error such as .1 mm. The larger the polyhedron, then, the more accurate the shape will be. On the other hand, a fair roll depends on having a good sampling of all the available states (faces of the dice). Small dice tumble more than large dice for a typical toss, meaning dice that are too large fail to sample the states well. Coupled with the above argument on shape accuracy, this implies there is an optimum size for fairness. From observing the tumbling of dice of different sizes, our opinion is the dice in standard poly sets are quite a bit smaller than optimum. For this reason, OptiDice are significantly larger than most standard polyhedral dice.

**Video about our OptiDice set:
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