# Imploding Dice

|Because I can't just be satisfied...

Long story short, I was playing around with random tables and I wanted a way to re-roll a particular die, but exploding die weren't working. So what about *imploding* dice?

## The Gist

Roll a die. The result is your "marker." Continue rolling until you "implode" (i.e., you hit your marker or more). Until implosion, add the result of each roll to your total.

## Why Bother?

Though I like the concept of exploding dice, the mechanic skips values. If I were exploding a d6, there's no way to get a result of "6": a roll of 1-5 means I end up with a result of 1-5. But if I roll a 6, then my *minimum* result is actually 7. [1] On the re-roll, there's no way to get a "12" because another 6 would explode. So my result sets are based on the number of explosions: 1 roll gives me 1-5; 2 rolls gives me 7-11; 3 rolls give me 13-17, etc.

I mean, I get the idea, and it's all sorts of fun to roll exploding damage. But my OCD cries foul, and it *will* be heard.

## Why Implosion?

First, it addresses the problem of skipping values. Second, the marker's value suggests (but does not make certain) the final outcome: regardless of the die used, a low marker suggests greater chance of implosion and therefore an overall lower total; a high marker trends toward more re-rolls and therefore a higher value.

With that in mind, GMs can better intuit which die they might want to implode in a given situation, because a smaller die has a higher chance of a low marker, and therefore greater odds of implosion. [2] Consider, for any given marker, the chance of re-rolling based on the die type used:

## Final Words

OK, again, that was more math than I really wanted to deal with. But I expect people to show their work, so there you are.

Big question: Anyone see this in other games, or have I just invented the best new dice mechanic of 2012? (Roger3, I'm looking at you.)

_______________

- Or, put another way, there's an 83.3% chance of getting 1-5 and a 16.6% chance of getting a 7 or more. Goose-egg percent chance of getting a 6, which is total crap.
- I.e., the chance of rolling the marker or higher.
- Yeah, no freak dice, though the percentage for such could easily be calculated:

**Chance of Implosion = 1 - ((M-1)/S)**

Where M equals the marker and S equals the die's number of sides.

I think I might give this a try next time I’m starting a campaign. Just to see how it works, and ideally in a system that already has rules for exploding dice. I will let you know how it goes.

This die mechanic is definitely new to me. However, I believe I’ve seen implosion or imploding used to refer to the reverse of exploding dice: if you roll a 1 you roll again and subtract. A Google search confirms this. So using implosion might cause some confusion with those familiar with the other die mechanic.

You can also solve the value skipping issue by having rolls after the first be 1dX-1. See Hackmaster and others. Example: I roll a d6 and get a 6. I then roll a d6 and get a 4…my total would be 9 (6 + (4-1)). If the second roll had come up 6, I’d roll a third time. If the 3rd roll was a 1, my total would be 11 (6 + (6-1) + (1-1)).

Doing more math, it has a few issues. 1) you can’t get less than the marker. 2) while there are no numbers (above the marker) that don’t get hit, it’s definitely a wavy plot. 3) as the marker approaches the die size you end up having to roll a lot of dice. And since order matters, you can’t just roll a handful easily.

Roll a d6. 1 to 5 = add 1 to 5 to total. 6 = add 0 to total and roll again. No skipped values.

@shortymonster : I’m now using it to determine number of dungeon levels in a ruin (imploding d6) and number of “rooms” on each level (imploding d12). I’m definitely interested in hearing how you use it.

@Granger44 : Yeah, I think a new name might be in order. I hadn’t heard of (or considered) the dX-1 on the re-roll, most likely because of my prejudice against subtraction on die rolls, but your example is pretty clear.

@Philo Pharynx : Agreed on all counts. Not getting less than the marker is intentional, and order does matter, but I thought that was OK – at least on par with exploding dice. But that brings up an idea:

You could specify the number of dice to throw, counting only those less than the marker. E.g., “Roll number of dungeon levels with imploding 4d6.” Meaning, roll 1d6 to find the marker, then roll 4d6 and add only those dice less than the marker. Something to play with, maybe…

And yeah – The plot on this must look ridiculous.

@Required : So pretty much the same as dX-1 on the reroll. Seems obvious now that it’s been explained – maybe the subhead of this post should have been “Because I can’t do things easy…”

Will someone graph this up? I’d really like to see the probability curves!

Just found this today – I wonder if it could be used as a variant method for rolling stats? It could work pretty well with Tunnels and Trolls.

Actually, the probability distribution isn’t that funky. Here’s the one for imploding d6 (This is actually an approximation; I drew 100,000 samples from the actual distribution and graphed that):

1 =========================================================== 20.0%

2 =============================================== 15.8%

3 ================================================ 16.2%

4 ==================================== 12.0%

5 ================================== 11.4%

6 ======================== 8.0%

7 ================== 6.3%

8 =========== 3.9%

9 ======== 3.0%

10 ==== 1.6%

11 == 0.8%

12 = 0.6%

13 0.2%

14 0.1%

If you think about it, this makes sense. 1..6 was originally uniformly distributed, but the imploding mechanism redistributes mass from each according to how likely it is to get an implosion. So 1 keeps it all, 2 distributes a small amount (and receives some from the imploding 1s), 3 distributes more (and receives more, since you can hit 3 on imploding 1 or 2), and so on. So the graph is actually very regular.

An interesting thing if instead of adding if you *always* add the roll to your total, even when you fail to implode (and thus stop the process right after adding the roll).

Again, for d6:

1 0.0%

2 =========== 3.9%

3 =========== 3.9%

4 ========================== 8.9%

5 ============================ 9.5%

6 =========================================== 14.5%

7 ==================================== 12.1%

8 ======================================== 13.4%

9 =========================== 9.3%

10 ============================ 9.6%

11 =============== 5.2%

12 ========== 3.5%

13 ======== 2.8%

14 ===== 1.8%

15 == 0.8%

16 = 0.5%

17 0.2%

18 0.1%

It’s not quite Gaussian or binomial; that valley at 7 is consistent across runs, for one thing, but it’s very similar. I’m not going to try analyzing the reasons for that formally; it doubtless involves deeply nested binomial distributions or something, but I found it interesting that such a minor change in the rules resulted in such a different distribution, while both versions yield relatively smooth distributions.

Larger dice yield similar distributions.

If the commend system removes newlines, those plots will be unreadable, in which case copy/paste them into a text file and insert a line break after each percent sign to recover the plot.

Also, those both assume that the marker is always set to the value of your latest roll. If the marker is *only* set on the first roll and remains at that value, the plot gains a much longer tail, but is still fairly smooth.

Still d6:

1 ============================================================ 20.0%

2 =============================================== 16.0%

3 ============================================= 15.3%

4 ================================ 10.8%

5 ========================= 8.6%

6 =========== 3.7%

7 ============ 4.1%

8 ======== 2.7%

9 ======== 2.7%

10 ===== 1.9%

11 ==== 1.6%

12 ==== 1.5%

13 === 1.3%

14 === 1.0%

15 == 0.9%

16 == 0.8%

17 == 0.7%

18 = 0.6%

19 = 0.6%

20 = 0.5%

21 = 0.4%

22 = 0.4%

23 = 0.4%

24 0.3%

25 0.3%

26 0.3%

27 0.2%

28 0.2%

29 0.2%

30 0.2%

31 0.2%

32 0.2%

33 0.1%

34 0.1%

35 0.1%

36 0.1%

37 0.1%

38 0.1%

39 0.1%

40 0.1%

41 0.1%

42 0.1%

43 0.1%