The Dice

C.A.T.S. uses six-sided dice only. Every roll explodes: a rolled 6 is re-rolled and added, recursively.

Measured behaviour: analysis/exploding-d6.md.

What this supersedes

This replaces the earlier "one standard seven-piece polyhedral set" constraint entirely. That analysis — comparing 2d10, d20, and d100 roll-under — is void as a decision, though the reasoning is preserved in analysis/resolution-comparison.md and in the OQ-20 history.

Note the practical change: a standard polyhedral set contains only one d6. A d6-only game needs a handful of them, so the component requirement is now "several d6s" rather than "one polyhedral set." That is arguably a better place to land — d6s are the most widely owned die in the world, salvageable from any board game — but it is a different requirement, not the same one.

Why exploding dice behave the way they do

A single exploding d6 has a clean closed form:

Exploding raises the average by 20% and replaces the ceiling with a long right tail.

Scaling by dice count

Mean scales linearly at 4.2 per die. Explosion frequency does not — it climbs toward certainty. At three dice you re-roll on more than two rolls in five, which is a real cost against the speed goal and the PA's roll budget.

The tail is real, not theoretical

Roughly one roll in ten produces double the expected result on a single die. That is frequent enough to shape play rather than being a curiosity.

What this settles

Roll-over, not roll-under — forced

Exploding dice push totals upward. Under roll-under, a 6 exploding would drive a character toward failure, which is self-defeating and reads as a punishment for a lucky roll.

So high must be good. OQ-23 is resolved in favour of roll-over: succeed by meeting or beating the Target Number, with Margin = Roll − Target Number.

This also makes Difficulty a single clean number rather than a subtracted term, which suits the no-opposed-rolls lock and gives the PA one value to compute and display.

An explosion is a continuation, not a second roll

An earlier draft claimed exploding dice were incompatible with one-roll resolution, on the grounds that an explosion is an extra sequential roll. That was wrong and is withdrawn. An explosion continues the same roll toward one result; the situation is still resolved by one roll producing one Margin. There is no second question being asked and no waiting on another party.

Explosion is the critical mechanic

The brief's Critical Edge fired on matching dice. With d6-only that mechanic needs rebuilding anyway — and it does not need replacing, because an explosion already is the critical. It is self-announcing, needs no lookup, occurs at a known 16.7% per die, and its magnitude scales with how far the chain runs.

One mechanic covering both critical results and damage spikes is cheaper than two.

Depth-tiered hit location

The unreachable multiples of 6 partition a single exploding d6 into exact tiers of five — 83.3% / 13.9% / 2.3% — so a critical location cannot be reached without exploding. What looked like an oddity in the distribution becomes the load-bearing feature.

Those percentages belong to a dedicated location die only. If depth is instead read from the attack roll's explosions (Option A in the hit-location document, and currently the leading choice), then with three dice the chance that at least one explodes is 1 − (5/6)³ = 42.1%, not 13.9% — deep hits become three times as common. The two figures cannot both be advertised, and which one is true depends on an unresolved choice.

Likely rule, detailed in 07-hit-location-depth.md.

Margin gains an unbounded tail

Under margin-driven damage and progress, an unbounded Margin means unbounded damage and unbounded progress. This delivers precisely the "typical result with rare dramatic spikes" curve the one-set constraint previously made impossible — the reason exploding dice were promoted in the first place.

It also breaks every bounded band in the brief. The Penetration Margin bands (−4 or lower through 8+) assumed a bounded 2-20 range. They need rebuilding against an unbounded distribution. See OQ-24.

What is now open

On dice count specifically

Dice count and skill entry have to be decided together, because "skill adds dice" and "skill adds a modifier" produce completely different games. Adding dice raises the mean and the explosion rate and the variance simultaneously, so skilled characters would become dramatically swingier — the same convexity problem flagged in OQ-07, amplified.