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Exploding d6 Resolution
Every 6 is re-rolled and added, recursively. Computed exactly from the closed form up to a total of 120; residual truncated mass is reported below and is negligible.
A total of exactly 6, 12, 18, or any multiple of 6 is impossible on a single die, because a 6 always explodes. This leaves visible gaps in a single-die distribution that fill in as dice are added.
Summary
| Dice | Mean | Median | No die explodes | Residual mass beyond 120 |
|---|---|---|---|---|
| 1d6! | 4.2 | 3 | 83.3% | 6.7e-016 |
| 2d6! | 8.4 | 7 | 69.4% | 1.4e-014 |
| 3d6! | 12.6 | 11 | 57.9% | 3.0e-013 |
A plain d6 averages 3.5. Exploding raises that to 4.2 per die — a 20% increase — and adds an unbounded right tail.
Chance of rolling at least a given total
| Total | 1d6! | 2d6! | 3d6! |
|---|---|---|---|
| 3+ | 66.7% | 97.2% | 100% |
| 5+ | 33.3% | 83.3% | 98.1% |
| 7+ | 16.7% | 58.3% | 90.7% |
| 9+ | 11.1% | 38% | 75.5% |
| 11+ | 5.6% | 25% | 57.4% |
| 13+ | 2.8% | 16.7% | 42.1% |
| 15+ | 1.9% | 10% | 30.3% |
| 18+ | 0.5% | 5.1% | 17.1% |
| 21+ | 0.3% | 2.3% | 9.3% |
| 25+ | 0.1% | 0.8% | 3.9% |
| 30+ | 0% | 0.2% | 1.2% |
The tail
Exploding dice have no maximum. These are the probabilities that make the tail real rather than theoretical.
| Dice | 2x mean or better | 3x mean or better | Chained explosion (2+ in a row) |
|---|---|---|---|
| 1d6! | 11.1% | 2.78% | 2.8% |
| 2d6! | 6% | 0.64% | 5.5% |
| 3d6! | 3.1% | 0.18% | 8.1% |
Depth tiers — hit location by explosion chain
A single exploding d6 partitions itself into exact tiers of five, separated by the unreachable multiples of 6. Reaching tier N requires N-1 explosions, so deep locations are unreachable without them.
| Tier | Results | Explosions needed | Each result | Tier total |
|---|---|---|---|---|
| 1 | 1-5 | 0 | 16.667% | 83.333% |
| 2 | 7-11 | 1 | 2.778% | 13.889% |
| 3 | 13-17 | 2 | 0.463% | 2.315% |
| 4 | 19-23 | 3 | 0.077% | 0.386% |
| 5 | 25-29 | 4 | 0.013% | 0.064% |
Tier totals form an exact geometric series that sums to 1: 5/6, then 5/36, then 5/216, and so on. The unreachable multiples of 6 are not a defect here — they are precisely the boundaries between layers.
Full distributions
1d6 exploding
| Total | Chance | At least |
|---|---|---|
| 1 | 16.67% | 100% |
| 2 | 16.67% | 83.33% |
| 3 | 16.67% | 66.67% |
| 4 | 16.67% | 50% |
| 5 | 16.67% | 33.33% |
| 6 | impossible | 16.67% |
| 7 | 2.78% | 16.67% |
| 8 | 2.78% | 13.89% |
| 9 | 2.78% | 11.11% |
| 10 | 2.78% | 8.33% |
| 11 | 2.78% | 5.56% |
| 12 | impossible | 2.78% |
| 13 | 0.46% | 2.78% |
| 14 | 0.46% | 2.31% |
| 15 | 0.46% | 1.85% |
| 16 | 0.46% | 1.39% |
| 17 | 0.46% | 0.93% |
| 18 | impossible | 0.46% |
| 19 | 0.08% | 0.46% |
| 20 | 0.08% | 0.39% |
| 21 | 0.08% | 0.31% |
| 22 | 0.08% | 0.23% |
| 23 | 0.08% | 0.15% |
| 24 | impossible | 0.08% |
| 25 | 0.01% | 0.08% |
| 26 | 0.01% | 0.06% |
| 27 | 0.01% | 0.05% |
| 28 | 0.01% | 0.04% |
| 29 | 0.01% | 0.03% |
| 30 | impossible | 0.01% |
2d6 exploding
| Total | Chance | At least |
|---|---|---|
| 2 | 2.78% | 100% |
| 3 | 5.56% | 97.22% |
| 4 | 8.33% | 91.67% |
| 5 | 11.11% | 83.33% |
| 6 | 13.89% | 72.22% |
| 7 | 11.11% | 58.33% |
| 8 | 9.26% | 47.22% |
| 9 | 7.41% | 37.96% |
| 10 | 5.56% | 30.56% |
| 11 | 3.7% | 25% |
| 12 | 4.63% | 21.3% |
| 13 | 3.7% | 16.67% |
| 14 | 3.01% | 12.96% |
| 15 | 2.31% | 9.95% |
| 16 | 1.62% | 7.64% |
| 17 | 0.93% | 6.02% |
| 18 | 1.16% | 5.09% |
| 19 | 0.93% | 3.94% |
| 20 | 0.75% | 3.01% |
| 21 | 0.57% | 2.26% |
| 22 | 0.39% | 1.7% |
| 23 | 0.21% | 1.31% |
| 24 | 0.26% | 1.11% |
| 25 | 0.21% | 0.85% |
| 26 | 0.17% | 0.64% |
| 27 | 0.12% | 0.48% |
| 28 | 0.08% | 0.35% |
| 29 | 0.04% | 0.27% |
| 30 | 0.05% | 0.23% |
3d6 exploding
| Total | Chance | At least |
|---|---|---|
| 3 | 0.46% | 100% |
| 4 | 1.39% | 99.54% |
| 5 | 2.78% | 98.15% |
| 6 | 4.63% | 95.37% |
| 7 | 6.94% | 90.74% |
| 8 | 8.33% | 83.8% |
| 9 | 9.03% | 75.46% |
| 10 | 9.03% | 66.44% |
| 11 | 8.33% | 57.41% |
| 12 | 6.94% | 49.07% |
| 13 | 6.25% | 42.13% |
| 14 | 5.56% | 35.88% |
| 15 | 4.94% | 30.32% |
| 16 | 4.4% | 25.39% |
| 17 | 3.94% | 20.99% |
| 18 | 3.09% | 17.05% |
| 19 | 2.55% | 13.97% |
| 20 | 2.08% | 11.42% |
| 21 | 1.72% | 9.34% |
| 22 | 1.45% | 7.62% |
| 23 | 1.29% | 6.16% |
| 24 | 0.99% | 4.88% |
| 25 | 0.78% | 3.89% |
| 26 | 0.62% | 3.11% |
| 27 | 0.49% | 2.49% |
| 28 | 0.4% | 2% |
| 29 | 0.35% | 1.6% |
| 30 | 0.27% | 1.25% |