I haven’t done one of these for a while, and I was bored, so I let my math nerd out to play. During the past few weeks, I’ve run across various conversations about math-related topics and I guess they’ve been percolating in the back of my head since then. Here’s the result.

### Is 1d4+1d6+1d8 the same as 3d6?

I’ve seen variations on this, such as 1d14+1d4. Obviously, that would be quite different from 3d6. But 1d4+1d6+1d8 is less obvious. Clearly, it would not be identical (4*6*8=192, 6*6*6=216). However, I couldn’t seem to visualize anything beyond that so I really had no idea how similar the two would be. The only way to find out was to test both functions.

To do this, we will need to enter every possible combination of numbers, add up each set, and count how many of each result we come up with. I did this, with both methods, and here are the findings.

Click on anything that is too small to read. : )

#1d4+1d6+1d83d6
310.52%10.46%
431.56%31.39%
563.13%62.78%
6105.21%104.63%
7147.29%156.94%
8189.38%219.72%
92110.94%2511.57%
102311.98%2712.5%
112311.98%2712.5%
122110.94%2511.57%
13189.38%219.72%
14147.29%156.94%
15105.21%104.63%
1663.13%62.78%
1731.56%31.39%
1810.52%10.46%
192100%216100%

So the two methods are indeed quite similar, yet clearly not quite the same. The alternate method favors median values. The chart is actually a bit misleading though. It appears that the extremes are identical, but that is not actually the case. A truer representation would be to chart distribution percentages instead of raw data. This would squish down the second bar graph, showing that the extremes are actually reduced.

But I lost interest and didn’t go any further. This already gave us our answer: 1d4+1d6+1d8 is indeed not the same as 3d6.

Edit: I modified the table to include the percentage distribution and added a second chart using that data. As you can see, the first table was extremely misleading. The second is a much more accurate portrayal. So much so that it actually reversed my prediction. The alternate method (1d4+1d6+1d8) slightly reduced the median values and slightly favors the extremes. I’m quite surprised at just how similar the two methods are. They are so close in fact that I think that the alternate method could be used, just for fun, with no real consequence whatsoever.

### Dice Progressions

Another conversation I observed was regarding dice progressions (1d4 < 1d6 < 1d8, etc.). It was very straightforward when only including single dice, but when multiple dice were used, people seemed to have some confusion about the order (and magnitude) of the progression. In a couple of instances, they recommended a progression that had occasional large steps mixed in and, in at least one case, a progression that got smaller before growing larger again.

It may not be as interesting as the first case, but I was in the mood to chart things, so I included it. Here is a graphical representation of various dice combinations. There is no progression implied, simply a comparison in order to visually see the differences.

Edit: I added a second chart with columns ordered by average result. I think it more clearly displays the various rolls.

Hopefully, that was enough to get all the math out of my system for a while. It also served as a distraction to help me step back from another problem I’ve been working on (with little success). Maybe tomorrow I can look at that other problem with fresh eyes and see what I’ve been missing.