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Emulation and Fairness Techniques

Here I describe how one or more of one type of dice can be used as a substitute for others: for example, how to pick a random day of the week using normal 6-sided dice.

Emulating a Coin Toss

The most common techniques are to use "odd/even" or "high-low" on a single die roll. For example, rolling a d6, the odd-even rule would assign one result to 1, 3, and 5. If your dice are the normal kind with pips (dots), notice that the odd numbers are the ones that have a dot in the centre.

Emulating Rock-Paper-Scissors (or "Who Goes First?")

The standard game of "rock paper scissors" is played in tournaments because people have trouble being fully random, and their opponent can notice this and take advantage of it. In gaming, RPS can be played with dice to make it truly random. Specialty d3 dice sometimes have the letters R, P, S on them for this purpose. Another die with a multiple of three sides (like a d6 or d12) could be used, so long as you divide its outcomes evenly among "rock", "paper" and "scissors".

To make a "rock-paper-scissors" type decision with other larger dice like d20s, just let the larger number win; with a d20 the odds of having to roll again are only 1 in 20.

But the "largest wins" allows cheating with loaded dice. Some fairness can be restored by using this system:

If both players are trying to win and use loaded dice, it will work against them. This requires a re-roll 1/10 of the time, but that could be mitigated by using a similar approach with percentile dice.

Who Goes First with More Than Two Players

A full turn ordering could be determined this way too. If this is desired and there are a lot of players, percentile dice is probably the best because, for example, if six people each roll d20, two of them will roll the same thing more than 50% of the time.

Eric Harshbarger, Robert Ford and James Ernest developed a set of four d12s each bearing 12 numbers from 1 to 48, distributed in such a way that no two numbers occur on two of the dice, and if rolled together each of the dice is equally likely to end up being highest, 2nd, 3rd, or lowest in ranking (and all 24 permutations are equally likely). They're called "Go First Dice" and are sold by Mathartfun and Maths Gear

Odd Dice

Game Masters often need to choose from an odd number of possibilities, e.g. deciding which of 5 PCs (Player Characters) will be the first to be hit in an ambush.

Players occasionally need "odd dice" too, most famously for Goodman Games' Dungeon Crawl Classics series, which in 2005 began using an extended set of D&D dice: d3, d4, d5, d6, d7, d8, d10, d12, d14, d16, d20, d24, d34, and percentile d100.

Emulating N sides with 2N-sided dice

The simplest trick for emulation is if you have a die with twice the needed number of sides:

Emulating 2N sides with N-sided dice

To "double" a die you can use a d6 plus an N-sided die. Here we'll use a d6 and a d12 to "emulate" a d24.

This is sometimes called "using the d6 as a control die". A similar technique can be used in many different ways, some requiring more mental arithmetic than others. For example, 2d6 can be used to emulate d9 — and there are specially-labeled d6s that do just that — but it's not nearly as easy as the next trick:

Emulating N-1 with N-sided Dice

To roll a d7 using a d8, just roll the d8 and if you roll 8, roll again.

The same method is useful in general for any unusual dice rolls — just use the next-larger type of dice. For example, to roll a d17 using a d20, roll the d20 and roll again if you get an 18, 19, or 20.

Emulating N+1 with N-sided dice

To roll a d7, use a d6 and this technique:

It can be done with 2d6 if the dice are distinguishable so you can tell which is the "first" die.

This technique can be used to select a random day of the week:

first roll second roll weekday
1 anything but 1 Monday
2 anything but 2 Tuesday
3 anything but 3 Wednesday
4 anything but 4 Thursday
5 anything but 5 Friday
6 anything but 6 Saturday
1 through 5 same as 1st die Sunday
6 6 start over

Of the 36 combinations of two d6 rolls, each weekday can be chosen 5 different ways; with just one combination (double 6's) left over: 6×6 = 5×7+1 = 36.

The same technique is used with a d4 to make d5 rolls, with a d8 to make d9 rolls, and with a d10 to make d11 rolls. However, in each of those cases the previous trick is easier and takes fewer rolls on average. But this technique is viable for using a d12 to make d13 rolls, and similarly for certain higher numbers.

Emulating Two Dice With One

This is a solution to a problem that does not really exist. If you rolled two N-sided dice but were supposed to roll one, can you (fairly) use the outcome of the two dice to pick a number from 1 to N? Yes! Just add the two together, and if the answer is greater than N, subtract N.

Emulating Two d6 With Three d6

A problem that is even less likely to exist in real life: If you have three indistinguishable d6 dice trapped inside a box, so that you can cannot roll them individually but must roll all at once, can you use this to fairly generate dice-rolls of a single d6, or of 2d6? Again, the answer is yes, and I wrote a separate article about it.


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