### Switch Tables

I wanted to talk about a certain quirky algorithm for using two (or more?) dice to index a table. You’re probably familiar with d66, where you roll two six-sided dice: one acting as the tens column, and the other acting as the ones column. It’s similar to how you roll d100 or d% by rolling what are otherwise two ten-sided dice. Anyway, result of d66 maps to a table with 36 entries, each one indexed to numbers like 11, 23, 54, or 66. This method relies upon you deciding which die acts as the tens digit and which one acts as the ones digit. What if it didn’t?

**d66 (Regular)**

```
1 2 3 4 5 6
d66 ----------------
1 | 11 12 13 14 15 16
2 | 21 22 23 24 25 26
3 | 31 32 33 34 35 36
4 | 41 42 43 44 45 46
5 | 51 52 53 54 55 56
6 | 61 62 63 64 65 66
```

## Switch Tables

Instead, what if the tens digit is the lower of the two rolls? This means that if you roll 1 & 6 or 6 & 1, the result is always 16 because 1 is less than 6. This reduces the total number of entries on the table from 36 to 21, where there are 6 double pairs (11, 22, 33, 55, 55, 66) and 15 non-double pairs. Since there are two ways of arriving at any non-double, each one is twice as likely as any double. Hence the total distribution of the table adds up to 15 × 2 + 6 = 36 still.

**d66 (Ascending)**

```
1 2 3 4 5 6
d66 ----------------
1 | 11 12 13 14 15 16
2 | 12 22 23 24 25 26
3 | 13 23 33 34 35 36
4 | 14 24 34 44 45 46
5 | 15 25 35 45 55 56
6 | 16 26 36 46 56 66
```

This is where it gets interesting: using this technique alongside the original, you can roll on the same d66 table three different ways. Let’s list them:

**Arbitrary:**One die is the tens digit, and one die is the ones digit. 1 & 6 become 16, whereas 6 & 1 become 61.**Ascending:**The lower die is the tens digit, and the higher one is the ones digit. Both 1 & 6 and 6 & 1 become 16.**Descending:**The higher die is the tens digit, and the lower one is the ones digit. Both 1 & 6 and 6 & 1 become 61.

In the arbitrary method, every entry is possible and each has an equal likelihood. In the ascending method, about one side of the table is “rerouted” to their opposite pair (e.g. 6 & 1 becomes 16, so 61 is not possible). In the descending method, the opposite side of the table is rerouted. Only entries made of doubles appear in all three tables, and only in the arbitrary method are they just as likely to occur as every other entry.

**d66 (Descending)**

```
1 2 3 4 5 6
d66 ----------------
1 | 11 21 31 41 51 61
2 | 21 22 32 42 52 62
3 | 31 32 33 43 53 63
4 | 41 42 43 44 54 64
5 | 51 52 53 54 55 65
6 | 61 62 63 64 65 66
```

This is why I’m referring to them as **switch tables**. You can switch how you roll on them to represent different circumstances, by selecting for different distributions. Even better, you’re always rolling the same dice. The only thing that changes is how you read them.

To be clear, I gave it a fancy name just to refer to it throughout the post.

## An Example

We’ve just talked about d66, but this works for any pair of same-sized dice. Let’s take d33 as a simple example, and suppose that we want to rewrite the reincarnation spell to use a switch table. Our table has, overall, nine entries, but the method we use will determined which entries are possible. In this case, we will use the ascending method for lawful characters, the descending method for chaotic characters, and the arbitrary method for neutral characters.

```
1 2 3
d33 -----------------------
1 | Human Troll Goblin
2 | Elf Halfling Troll
3 | Dwarf Gnome Werekin
--------------------------
In Order:
11: Human
12: Elf
13: Dwarf
21: Troll
22: Halfling
23: Gnome
31: Goblin
32: Troll
33: Werekin
```

Only neutral characters have an equal chance of becoming any other creature on the chart, namely one-ninth. Meanwhile, lawful characters can not become chaotic, so they only have six creatures to roll from: they also only have a 3-in-9 chance of becoming a neutral creature, while each lawful entry has its own a 2-in-9 chance. Chaotic characters are the opposite of lawful characters, but their math turns out the same way.

Suppose that a lawful character rolls 3 & 1 while a chaotic character rolls 1 & 3. If the lawful character were neutral (or chaotic), their result would be 31 and so they would become a goblin; however, their result is actually a 13 because they are lawful, so they become a dwarf. The chaotic character is the opposite: they would have become a dwarf if they were lawful or neutral, but because they are chaotic, they read their result as 31 and so become a goblin.

## As Percentile Dice

Everything above also applies to ten-sided dice, but ten-sided dice have a special property compared to the others: when you roll two together, you can read them as a number from 1 to 100. This is why you can use them as percentile dice, to see if an event with some percent likelihood actually occurs. We can use the switch table as a form of “advantage” or “disadvantage”, producing a number from 1 to 100 but with a bias toward one end or the other.

**d00 (Regular/Arbitrary)**

```
0 1 2 3 4 5 6 7 8 9
d00 ----------------------------
0 | 00 01 02 03 04 05 06 07 08 09
1 | 10 11 12 13 14 15 16 17 18 19
2 | 20 21 22 23 24 25 26 27 28 29
3 | 30 31 32 33 34 35 36 37 38 39
4 | 40 41 42 43 44 45 46 47 48 49
5 | 50 51 52 53 54 55 56 57 58 59
6 | 60 61 62 63 64 65 66 67 68 69
7 | 70 71 72 73 74 75 76 77 78 79
8 | 80 81 82 83 84 85 86 87 88 89
9 | 90 91 92 93 94 95 96 97 98 99
```

Let’s consider the ascending method. We are always going to treat the lower result as the tens digit, and the higher result as the one digit. The result is an average of 35.66, and a standard deviation of 25.33. This indicates a bias towards the left side of the original distribution. A character who rolls less than or equal to a target score of 50 would have approximately an 75% chance of success, whereas usually they would have a 50% chance. This is just like rolling with advantage.

**d00 (Ascending)**

```
0 1 2 3 4 5 6 7 8 9
d00 ----------------------------
0 | 00 01 02 03 04 05 06 07 08 09
1 | 01 11 12 13 14 15 16 17 18 19
2 | 02 12 22 23 24 25 26 27 28 29
3 | 03 13 23 33 34 35 36 37 38 39
4 | 04 14 24 34 44 45 46 47 48 49
5 | 05 15 25 35 45 55 56 57 58 59
6 | 06 16 26 36 46 56 66 67 68 69
7 | 07 17 27 37 47 57 67 77 78 79
8 | 08 18 28 38 48 58 68 78 88 89
9 | 09 19 29 39 49 59 69 79 89 99
```

Using the descending method, we are going to treat the higher result as the tens digit and the lower result as the ones digit. The resultant average is 65.34, and the standard deviation remains 25.33. Now the same character, with their target score of 50, has a 24% chance of succeeding at their task. This is just like rolling with disadvantage.

**d00 (Descending)**

```
0 1 2 3 4 5 6 7 8 9
d00 ----------------------------
0 | 00 10 20 30 40 50 60 70 80 90
1 | 10 11 21 31 41 51 61 71 81 91
2 | 20 21 22 32 42 52 62 72 82 92
3 | 30 31 32 33 43 53 63 73 83 93
4 | 40 41 42 43 44 54 64 74 84 94
5 | 50 51 52 53 54 55 65 75 85 95
6 | 60 61 62 63 64 65 66 76 86 96
7 | 70 71 72 73 74 75 76 77 87 97
8 | 80 81 82 83 84 85 86 87 88 98
9 | 90 91 92 93 94 95 96 97 98 99
```

My friends John B. from *The Retired Adventurer* and Jenx from *Gorgon Bones* told me that the role-playing game *Unknown Armies* does something like this to represent character skills. I could see it being used more broadly as advantage or disadvantage, as described above. I could even see d00 descending used on its own, as a way to subtly 'buff' player-characters. Who knows?

## Conclusion

I was inspired by two things. First, I was thinking about how although the video game series *Fire Emblem* tells you that you have some percent chance of success at something, under the hood it “rolls” two numbers and takes the average of the two (e.g. 40 and 60 become 50). I wondered if there was a way of similarly messing with dice rolls in order to introduce a bias in the distribution, without rolling twice or doing anything weird like that. The method itself I remembered reading in an old book, specifically the d66 ascending method (IIRC, *Lich van Winkle* showed it to me—calling upon his aid!). Only after doing the math did I realize that you could switch between different ways of rolling on the same table with the same dice. I hope it comes in handy for someone!

Thank you to my friend Nova at *Playful Void* for helping me figure out potential applications of this! She has a really cool d88 switch table she is working on, but I’ll let her talk about it (and link it here when she’s done).

This is really cool! You could use a d66 table for random wilderness encounters and roll ascending for daytime and descending for night. 6 entries can be encountered day or night.

ReplyDeleteDaytime creatures are above the diagonal of doubles and night creatures are below it. Thanks for posting and laying this out so clearly!

Deletethat would be a really handy use case!! thank you as well :D

DeleteIt's too early in the morning for me to wrap my head around the math / probability, but I really like this idea, thanks!

ReplyDeletethank you too!! :)

Delete