OVERRUN — conquest with the maths shown

The rules, and the five that everyone gets wrong

Each turn has three parts. Draft new armies and place them. Attack across as many borders as you like. Fortify once, then pass. Take every region on the map and you have won.

1. How many armies you get

Your territory count divided by three and rounded down, with a floor of three, plus the bonus for every continent you hold completely. So two territories still pays three, and twelve territories pays four. Holding seven of Khoren's eight regions pays nothing — the bonus is all or nothing, and that is the entire strategic shape of the game.

2. You must leave one army behind

A region holding a single army cannot attack at all. The number of dice you roll is your armies minus one, capped at three — so four armies is the first stack that rolls a full three dice. The defender rolls one die per army, capped at two.

3. The defender wins ties

Both sides sort their dice highest first. The top attacking die is compared with the top defending die, and the second with the second. Equal numbers go to the defender. This is the rule online versions invert more often than any other, and it is not a detail: flipping it moves a three-against-two throw by 24.8 percentage points. Our test suite re-derives the whole table with ties going the wrong way purely to prove how much it is worth.

4. Occupying takes as many armies as you rolled dice

Win a region on three dice and at least three armies must move in. You may move more — everything except the one that stays behind. This stops a conquest being garrisoned by a single token army and it is why over-extending actually costs you something.

5. Fortify moves along a connected path

One move per turn, between any two of your regions joined by a chain of regions you occupy. Both variants are in print — the older wording says the destination must be adjacent — and both are implemented here. Connected-path is the default because adjacent-only punishes exactly the player who has built a deep, secure interior, which is the correct thing to have done, and because a beginner whose sensible-looking move is silently refused concludes the game is broken.

The dice are solved, and the solution is on this page

This is the rare game whose core randomness has an exact, published answer. Enumerate all 6⁵ = 7,776 ways three attacking dice can meet two defending dice and you get exact fractions, not estimates:

Fighting a battle to the death is an absorbing Markov chain, which also solves exactly. That solution is what the panel beside the board is showing you before every attack. The published reference is Jason Osborne's “Markov Chains for the RISK Board Game Revisited” (Mathematics Magazine 76:2, April 2003, 129–135), and our engine reproduces all one hundred cells of his battle table plus both figures he quotes in prose.

The correction matters more than the table. Osborne's paper exists because the previous published solution was wrong: Baris Tan (Mathematics Magazine 70:5, 1997) printed (0.237, 0.504, 0.259) for that same row, having treated the two dice comparisons as independent when the defender's two order statistics are strongly dependent. Two peer-reviewed papers, one right — so our engine derives the table combinatorially and only then compares it to the page of a book. The suite contains a test named for Tan's numbers that fails loudly if we ever emit them.

And a third check, against the licensed product itself

SMG Studio develops the official licensed version of this genre and open-sourced its dice engine. Their README works two examples all the way to the last digit. We reproduce both: their 0.292566872427984 for a standard three-against-two round, and their 0.0222128001707278 for the chance of losing exactly twelve of thirty troops attacking a capital held by fifteen. Their code is C#, ours is JavaScript, and neither was written from the other.

⚠️ That second number took a correction to reach. Read carelessly it looks like a standard battle, where our answer is 0.0717 — nothing like theirs. A capital defends with three dice. Chasing the convention instead of quietly dropping the check is what turned a mismatch into the strongest cross-check we have.

Why we made the dice verifiable rather than just promising they are fair

“The dice are rigged” is the loudest complaint in this entire category, and every operator answers it the same way: our random number generator is cryptographically secure, trust us. That answer is worthless, because seeing the shuffler does not tell you when it gets called.

And on the licensed official game, the players are right. SMG's own repository documents a mode called Balanced Blitz which, in their words, works by “adjusting the true probabilities of every possible outcome in a given Risk battle.” On the worked example above it reports 0.0100 where the true likelihood is 0.0222 — less than half. Players have sensed that for years and had no way to prove it.

So OVERRUN does not ask for trust. Before your first move we show you a SHA-256 commitment to a secret seed. You supply your own seed — type anything. Every throw is a pure function of (secret seed, your seed, throw number) and nothing else: not the score, not who is winning, not whether you are about to lose a continent. When the game ends we reveal the secret and you can re-derive every roll yourself. The dice cannot react to the board, because they were fixed before there was a board to react to.

The map is original, and it is costed

We had to draw our own world — the specific arrangement of the classic board is its publisher's. Having to build one fresh was an opportunity, because the old board's continent bonuses are not priced against the work of holding them. On that board Australia pays 2 armies for a single border while Africa pays 3 for four: a 2.7× spread between the best and worst deal on the map, which is why one corner is a fortress and another is a tax.

OVERRUN uses a flat rule instead, and the test suite asserts it from the adjacency list on every build so it is a gate rather than a boast:

Every continent pays exactly one army per border you have to hold, and half an army per region you own. And every border is a single named crossing — each of the eighteen border regions has exactly one neighbour outside its own continent, so all nine crossings are genuine chokepoints rather than a smear of coastline. Nivraad is small and cheap to hold, Khoren is large and pays well, and neither is a free lunch.

This game ends, and that is a design decision

The structural complaint about this genre is not the dice — it is that a match takes hours, and that a player knocked out at minute thirty watches for four more. The usual fix is a tight turn timer, and there is good evidence it makes things worse: hurried players take fewer risks and eliminate nobody, so the game drags out further.

So there is no turn timer here. Instead the card sets escalate — 4, 6, 8, 10, 12, 15 and then +5 forever — which makes holding out strictly worse over time, and there is a hard 40-round horizon after which the board is adjudicated on regions held, then armies, then continents. “How long can this take” has a number instead of a shrug.

And here is the real distribution, because a single average would have been misleading. Eighty self-play games per row, all seats on Marshal:

Head to head is the clean, fast game and it is the default. Add seats and it roughly doubles, and somewhere between one in eight and one in six of those games ends on the horizon rather than by total conquest. That is a genuine trade rather than a bug, and it is on the page rather than discovered on your third evening: more players means more diplomacy and a longer grind. Pick your seat count on the start screen — the estimate updates as you do.

How strong the computer actually is

The three levels differ in what they know, not in a difficulty multiplier:

• Recruit attacks whenever it has a bare majority, dribbles its draft one army at a time and spreads itself thin. It is weak by being reckless.
• Captain masses its whole draft in one place and attacks on the rule of thumb that two more armies than the defender is enough. It knows what a continent bonus is worth. It does not compute battle odds.
• Marshal uses the exact published probability for every attack it considers, picks a continent to finish and pushes toward it, scores the position each attack would leave behind, and hunts eliminations for the cards.

The honest limitation: none of them searches past the current turn. There is no opponent modelling, no card counting, and no concept of a temporary alliance — which in a three-or-more-player game is a real part of the skill and we do not simulate it. Marshal is roughly a competent human who has read the odds table and is not thinking two turns ahead. It will rarely make a mathematically bad attack; it can still be outplanned.

Those are measurements, not impressions: over seventy games each with the seats alternated, Marshal beats Captain 66%, Captain beats Recruit 80%, and Marshal beats Recruit 93%. The suite asserts every one of those and fails if two levels ever collapse into each other — which they did, twice, during the build.

What we do not have yet

No tournaments or ladders specific to this game beyond the shared site rating. No map editor and no alternative maps — one map, balanced, is the trade we made. No in-game chat. Live play is two-player only; the three-to-six-seat game is against the computer for now. And live play trusts both clients with the dice the same way every rival does: against the computer the commitment is ours and verifiable end to end, but in a two-human room only the host draws and the guest receives a read-only mirror of the receipt, which means a determined host could in principle restart to reroll. We would rather write that down than let the word “provably” cover a case it does not.