Dicey

The premier solution in total paradigm shift for resolving dicey problems of tomorrow, today, used by industry-leading professionals around the world!

In seriousness, this program produces total frequency (probability) distributions of all possible dice rolls for a given set of dice. Dice in such a set can be different or even have arbitrary numbers on the sides.

Installation

The best way is probably to clone this repo to receive future updates:

git clone https://github.com/trinistr/dicey.git

Alternatively, just download dicey (and gnuplot-for-dicey if you need it). That's it.

Requirements

Dicey is developed on Ruby 3.1, but should work fine on 2.7, 3.0, 3.1, 3.2 and 3.3. There are no dependencies aside from default gems and common usage will not even load them.

No installation

1. Copy the contents of dicey.
2. Head over to https://www.onlinegdb.com/online_ruby_interpreter.
3. Delete example script and paste previosuly copied code.
4. Type what you need in the "Command line arguments:" box.
5. Click "Run".

This should also work with other online interpreters, but be aware that many of them run severely outdated version of Ruby. You can check before trying to run Dicey by executing puts RUBY_VERSION (see Requirements above).

Usage

Following examples assume that dicey (or gnuplot-for-dicey) is executable and is in $PATH. You can also just run everything with ruby dicey instead.

💡 Run dicey --help to get a list of all possible options.

Example 1

Let's start with something simple. Imagine that your Bard character has Vicious Mockery cantrip with 2d4 damage, and you would like to know the distribution of possible damage rolls. Run Dicey with two 4s as arguments:

dicey 4 4

It should output the following:

# ⚃;⚃
2 => 1
3 => 2
4 => 3
5 => 4
6 => 3
7 => 2
8 => 1

First line is a comment telling you that calculation ran for two D4s. Every line after that has the form roll sum => frequency, where frequency is the number of different rolls which result in this sum. As can be seen, 5 is the most common result with 4 possible different rolls.

If probability is preferred, there is an option for that:

dicey 4 4 --result probabilities # or -r p for short
# ⚃;⚃
2 => 0.0625
3 => 0.125
4 => 0.1875
5 => 0.25
6 => 0.1875
7 => 0.125
8 => 0.0625

This shows that 5 will probably be rolled a quarter of the time.

Example 2

During your quest to end all ends you find a cool Burning Sword which deals 1d8 slashing damage and 2d4 fire damage on attack. Run Dicey with these dice:

dicey 8 4 4
# [8];⚃;⚃
3 => 1
4 => 3
5 => 6
6 => 10
7 => 13
8 => 15
9 => 16
10 => 16
11 => 15
12 => 13
13 => 10
14 => 6
15 => 3
16 => 1

Results show that while the total range is 3–16, it is much more likely to roll numbers in the 6–13 range. That's pretty fire, huh?

If you downloaded gnuplot-for-dicey and have gnuplot installed, it is possible to turn these results into a graph with a somewhat clunky command:

dicey 8 4 4 --format gnuplot | gnuplot-for-dicey
# --format gnuplot can be abbreviated to -f g

This will create a PNG image named [8];⚃;⚃.png:

💡 It is possible to output JSON or YAML with --format json and --format yaml respectively.

Example 3

While walking home from work you decide to take a shortcut through a dark alleyway. Suddenly, you notice a die lying on the ground. Looking closer, it turns out to be a D4, but its 3 side was erased from reality. You just have to learn what impact this has on a roll together with a normal D4. Thankfully, you know just the program for the job.

Having ran to a computer as fast as you can, you sic Dicey on the problem:

dicey 1,2,4 4
# (1,2,4);⚃
2 => 1
3 => 2
4 => 2
5 => 3
6 => 2
7 => 1
8 => 1

Hmm, this looks normal, doesn't it? But wait, why are there two 2s in a row? Turns out that not having one of the sides just causes the roll frequencies to slightly dip in the middle. Good to know.

💡 A single integer argument N practically is a shorthand for listing every side from 1 to N.

Example 4

You have a sudden urge to roll dice while only having boring integer dice at home. Where to find the cool dice though?

Look no further than roll mode introduced in Dicey 0.12:

dicey 0.5,1.5,2.5 4 --mode roll # As always, can be abbreviated to -m r
# (0.5e0,0.15e1,0.25e1);⚃
roll => 0.35e1 # You probably will get a different value here.

💡 Roll mode is compatible with --format, but not --result.

Diving deeper

For a further discussion of calculations, it is important to understand which classes of dice exist.

• Regular die — a die with N sides with sequential integers from 1 to N, like a classic cubic D6, D20, or even a coin if you assume that it rolls 1 and 2. These are dice used for many tabletop games, including role-playing games. Most probably, you will only ever need these and not anything beyond.

💡 If you only need to roll regular dice, this section will not contain anything important.

• Natural die has sides with only positive integers or 0. For example, (1,2,3,4,5,6), (5,1,6,5), (1,10000), (1,1,1,1,1,1,1,0).
• Arithmetic die's sides form an arithmetic sequence. For example, (1,2,3,4,5,6), (1,0,-1), (2.6,2.1,1.6,1.1).
• Numeric die is limited by having sides confined to ℝ (or ℂ if you are feeling particularly adventurous).
• Abstract die is not limited by anything other than not having partial sides (and how would that work anyway?).

💡 If your die starts with a negative number or only has a single natural side, brackets can be employed to force treating it as a sides list, e.g. dicey '(-1)' (quotation is required due to shell processing).

Dicey is in principle able to handle any numeric dice and some abstract dice with well-defined summation (tested on strings), though not every possibility is exposed through command-line interface: that is limited to floating-point values.

Currently, three algorithms are implemented, with different possibilities and trade-offs.

💡 Complexity is listed for n dice with at most m sides and has not been rigorously proven.

Kronecker substitution

An algorithm based on fast polynomial multiplication. This is the default algorithm, used for most reasonable dice.

• Limitations: only natural dice are allowed, including regular dice.
• Example: dicey 5 3,4,1 '(0)'
• Complexity: O(m⋅n)

Multinomial coefficients

This one is based on raising a univariate polynomial to a power and using the coefficients of the result, though certain restrictions are lifted as they don't actually matter for the calculation.

• Limitations: only equal arithmetic dice are allowed.
• Example: dicey 1.5,3,4.5,6 1.5,3,4.5,6 1.5,3,4.5,6
• Complexity: O(m⋅n²)

Brute force

This algorithm goes through every possible dice roll and adds results together. While quickly growing terrible in performace, it has the largest input space, allowing to work with completely nonsensical dice, including aforementioned dice with strings or complex numbers.

• Limitations: objects on dice sides must implement + and all be addable together.
• Example: dicey 5 1,0.1,2 1,-1,1,-1,0
• Complexity: O(mⁿ)