Go First Dice

In many board games, players typically roll dice to establish the starting order. For instance, the highest roll usually leads off, with turns proceeding clockwise around the table. However, this method presents two main issues: first, if two players tie, it necessitates another roll, wasting precious time. Secondly, the subsequent player placements are not random.

I aimed to design a set of dice that would fairly determine the order for 2 to 4 or more players, ensuring each player's chance was equal. My preferred choice was the five platonic solids, though I was open to alternatives. Importantly, ties were a definite no-go—only one roll per game!

When there are just two players, the process is relatively straightforward. If the player with the lowest number starts and we use coins, it's a simple matter of labeling them accordingly:

Coin 1: 1,4

Coin 2: 2,3

Ultimately, it boils down to whether coin 1 shows a number higher or lower than the two consecutive numbers on coin 2. To apply the same logic to platonic solids, we can simply replicate the faces. For instance, with cubes, we could propose:

Cube 1: 1,1,1,3,3,3

Cube 2: 2,2,2,2,2,2

If we want all the numbers to be unique—which I prefer—we could design them in the following way:

Cube 1: 1,2,3,10,11,12

Cube 2: 4,5,6,7,8,9

Things start to complicate when three players are involved. I must admit that I attempted to solve this using a blend of algebraic methods and trial and error in Excel but struggled to come up with a viable solution. Eventually, I resorted to some clever programming and created a simulation that randomly assigned numbers to the faces of three dice from 1 to 18 until I stumbled upon a valid configuration. Remarkably, it found a solution in minutes, which looked like this:

Cube 1: 3,4,9,10,13,18

Cube 2: 2,5,7,12,15,16

Cube 3: 1,6,8,11,14,17

There are 6³There are 216 different combinations for rolling three dice. With six possible arrangements for these three players, you can believe me when I say that out of 216 potential outcomes, each arrangement appeared exactly 36 times (216/6).

Since I had already developed a simulator for this purpose, I decided to extend it for the case of four players. This simulation ran for hours, investigating trillions of combinations but to no avail. Consequently, I returned to solving the problem mathematically, expanding on the three-dice solution with the following considerations:

Cube 1	4	5	10	15	18	23
Cube 2	3	6	8	17	20	21
Cube 3	2	7	9	16	19	22
Cube 4	1	11	12	13	14	24

My conclusion was that the player using Cube 4 should have a¼probability of taking the first or last turn. To evaluate the chance of going first, if he rolled a 1, he would automatically go first no matter what the other three dice showed, since a roll of 1 is the minimal value. This gives a probability of 1/6. If Cube 4 rolled between 11 and 14, the other three players would need to roll sums of 15 or higher for Cube 4 to remain the smallest. Each had three rolls greater than 14, resulting in a calculated probability of Cube 4 being the lowest: (1/4) + (4/6)*(3/6)^3 = 1/6. This approach allowed me to allocate a fair chance to each player.¼chance of going first.

However, if Cube 4 was indeed the smallest, the ranking of the remaining players was not equally likely. For instance, if Cubes 1 to 3 all rolled 15 or higher, the likelihood of Cube 1 being the lowest would be 1/3. However, the actual computation reveals it as prob(cube 1 = 15) + prob(cube 1 = 18)*prob(cube 2 = 20 or 21)*prob(cube 3 = 19 or 22), resulting in 1/3 + (1/3)*(2/3)*(2/3) which simplifies to 13/27.

Thus, I thought to transform Cubes 1 to 3 into dodecahedrons (12-sided dice), by duplicating the original six faces, while also incorporating 24 additional faces. The new configuration is as follows:

Cube 1	5	6	11	12	15	20	31	32	37	38	41	46
Cube 2	4	7	9	14	17	18	30	33	35	40	43	44
Cube 3	3	8	10	13	16	19	29	34	36	39	42	45

For Cube 4, I included the two smallest and two largest values: 1, 2, 47, and 48, along with eight numbers ranging from 20 to 29. This configuration maintained the probability for Cube 4 of being either first or last at (2/12) + (8/12)*(6/12)^3 =¼. If Cube 4 lands between 10 to 39, it reverts to the three-dice configuration, which has already been shown to be effective. Hence, the four-dice solution looks like this:

Die 1	5	6	11	12	15	20	31	32	37	38	41	46
Die 2	4	7	9	14	17	18	30	33	35	40	43	44
Die 3	3	8	10	13	16	19	29	34	36	39	42	45
Die 4	1	2	21	22	23	24	25	26	27	28	47	48

You can trust that out of the 12^4 = 1,296 potential rolls of these four dice and the corresponding 4! = 24 different orders, each arrangement has 1296/24 = 54 corresponding combinations.

I didn't stop there, but moved on to consider the five-player scenario. Using similar reasoning applied in the four-player case, the best feasible solution ended up being 840-sided dice. Instead of drawing out an extensive list of numbers for this newsletter, I shared the exact die configurations in my forum on Wizard of Vegas, within the thread. Go First Dice The total number of combinations for rolling five 840-sided dice amounts to 3,485,099,520,000, so I corroborated the outcome through random simulations and confirmed that the dice functioned as intended.

The video that sparked my deep dive into this topic is called Go First Dice, featured on the Numberphile channel (which is one of my favorites!). I must acknowledge that James Grime approached the four-dice scenario via a similar pathway to mine, but I aspire to contribute additional insights to this conversation.

All visuals included in this newsletter were created utilizing Copilot .