### Table of Contents

- Crafting the sparkiest D20 you will ever see
- The mathematics of rolling with advantage
- How to win every bar bet
- Have you ever heard of a D1
- An explnation of why dice sets are platonic solids
- Watch Vi Hart get a one-roll Yathzee using a D4
- Ohter shapes for a D12
- The only numbers that roll the same probablity as 2D6
- Impress your friend with a dice trick
- A dice-y way of choosing who goes first

Tabletop roleplayers live and die by their dice. It is no wonder that they obsess over them. In a sense, it is part of a ritual that initiates you into the hobby. One’s first dice set often has a personal meaning.

Dimension 20 GM Brennan Lee Mulligan has said,

From time immemorial, dawn of civilization, human beings have loved shiny little doodads that they can get their little mitts on.

- Brennan Lee Mulligan

The importance of dice in the hobby elevates what one might consider an everyday object. This is why I have gathered these ten videos revealing the design and mathematics of dice.

## 1. Crafting the sparkliest D20 you will ever see

This video comes from a channel that I recently discovered on YouTube. Called Hedron Rocks. The rock carver is Doug. He states on his website that he got into the hobby several years ago.

“I started playing dnd over 3 years ago with some friends and I immediately took to how cool the dice are.”

Doug’s words ring true for many of us in this hobby. Though, few of us would learn how to cut gems to create our own set.

## 2. The mathematics of rolling with advantage

In this video, Matt Parker answers the question, “If you are rolling two dice, picking the best so you have an advantage…what is the expected value from rolling identical dice at the same time?” This is the average value that you will most likely roll when throwing down identical dice.

## 3. How to win every bar bet

These are special dice that are non-transitive. The dice look like your standard six-sided die. You will notice right away what is different. The numbers on these dice are not the same. What is important is that each die has different probabilities when rolling.

In this game, you want to roll the highest number. Your friend picks a die first. You then pick your own knowing that yours has the probability of rolling higher. What makes them unique is that there is no best die. No matter what your friend picks there will be one that rolls higher on average!

## 4. Have you ever heard of a D1

Michael Stevens of VSauce presents these unique dice they invented, Denary Dice. Their geometries allow you to roll for any possible choice from one to ten. There is a D1, D2, D3 and so on.

I missed out on getting these myself. They were part of a special subscription box. If I ever find them, they will be the pride of my dice collection.

## 5. An explanation for why dice sets are platonic solids

There are five platonic solids: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. In our hobby, these represent our dice sets. You know them better as the d4, d6, d8, d12 and d20. It turns out their symmetric geometries make them fair dice.

This video also discusses dice of many different sides. You may have heard of a die like the D30 mentioned in this video. You may have seen a D100 once or twice. But are these dice also fair? Meaning that each side has the same possibility of coming up.

## 6. Watch Vi Hart get a one-roll Yahtzee using a D4

On a six-sided die, there is only a 1 in 1,296 chance that you will roll a Yahtzee on your first turn. Mathematician Vi Hart tackled this task by using 6 d4 instead, a 1 in a 1024 chance. Yes. When rolling six d4 it is more likely to happen than rolling five d6.

She humorously tells us facts about Tetrahedral dice as she goes. It is shocking the amount of rolls it takes. Do you have a guess on what that number is?

## 7. Other shapes for a D12

This twelve-sided die’s look is super funny and weird. It is like someone squished it. You may not want to use this die. It has a special property that might change your mind. This die rolls fairly like a dodecahedron.

These dice are by The Dice Lab. They have full polyhedral sets of skew dice. Which might be nice to add to your collection. This company also makes a D120 for purchasing.

## 8. The only numbers that roll the same probability as 2D6

This is a fact that I find very interesting. This video from Combo Class is about the odd distribution of the sums of numbers on two six-sided dice. This is a very common thing in games and gambling.

He shows the probabilities of getting any sum from rolling 2d6. Then asks if there are non-zero integers, less than ten, that create the same distribution. Or in other words, if we could change the numbers on the dice what should they be? The answer is that there is only one solution. These are Sicherman dice.

## 9. Impress your friend with a dice trick

Martin Gardner brought recreational mathematics to the public in Scientific American. His column influenced many people who entered mathematics and physics. He was also fond of magic and card tricks.

This is one example of a trick that involves three six-sided dice. Make sure you pause before four minutes so you have a chance to figure things out. Once you know, try this one with your friends and family.

## 10. A dice-y way of choosing who goes first

This is a simply stated problem that does not have a simple solution.

You are trying to decide who goes first in a game by rolling four four-sided dice. You will choose your order by who rolls from highest to lowest. There is some probability that two people will roll the same and that could go on forever. Is there any dice that would allow you to decide the order with no-rerolls. And each person has an equal chance of being 1st, 2nd, 3rd, and 4th.

We love these sorts of problems in mathematics. Not for their solutions, but the process of finding the answer. After finding a solution, there are more interesting problems to tackle. You end up learning a lot and yes, this is fun for some people.

From the common D6 to the D20 there is a lot to learn about these objects. I have a dragon’s hoard of a dice collection. Collected from many years attached with many memories. I hope you take the time to think about your own and where those shiny doodads came from.