Risk dice roller and battle calculator

How to use the Risk dice roller

Above, choose how many dice each side rolls. The attacker uses 1 to 3 red dice and the defender 1 or 2. If you play the Italian RisiKo! or Brazilian War, switch on the option for 3 defence dice. When you roll, the tool sorts each side's dice from highest to lowest, pairs the attacker's highest with the defender's highest, then the second-highest, and marks which army falls in each pair. The result appears underneath: attacker losses and defender losses.

To settle a whole battle in one go, use attack to the end. Enter how many armies are attacking and how many are defending, and the tool keeps rolling with the maximum dice allowed until the territory falls or the attacker has no armies left to continue. It saves you rolling twenty times in a row when the board is crowded, and it lets you check before your turn whether an attack is worth making.

Only count the armies that actually attack. The one left guarding the home territory takes no part: if you have 8 armies in a territory, you attack with 7.

Risk dice rules

The attacker

You need at least 2 armies in the territory you attack from, because one must always stay behind. You roll one red die for each attacking army, up to a maximum of 3 per roll however many you have. In practice, 2 armies in the territory give you 1 die, 3 armies give you 2 dice and 4 or more give you 3 dice.

The defender

In Hasbro's current rules the defender chooses to defend with 1 or 2 armies and rolls one die for each. Never more than 2, however many armies sit in the territory. With a single army, only 1 die can be rolled. Unlike the attacker, the defender can lose their last army, and that is how territories are captured.

How the dice are compared

1. Each player puts their dice in order from highest to lowest.
2. The attacker's highest die is compared with the defender's highest. The lower score loses one army.
3. If both players rolled at least 2 dice, the second-highest dice are compared as well.
4. Any extra dice on the side that rolled more are ignored.

Ties always go to the defender. Scores are never added up: a 6 and a 1 are not worth 7, they are a 6 matched against the opponent's highest die and a 1 matched against their second. That is why, in Hasbro's Risk, a single roll costs at most 2 armies in total, whether both come from the attacker, both from the defender or one each.

A worked example

The attacker rolls 6, 3 and 2; the defender rolls 5 and 3. First pair: 6 against 5, the defender loses an army. Second pair: 3 against 3, a tie, so the attacker loses an army. The attacker's 2 is not compared with anything. Result: one loss each.

After a conquest

When the defender loses their last army, the attacker moves in with the armies that survived the final roll and may bring more across from the home territory, always leaving at least one behind. You can attack as often as you like during your turn as long as you hold territories with 2 or more armies.

Risk, RisiKo! and War: the defence dice change from country to country

The way dice are compared is the same in every version. What changes is how many dice the defender may roll.

That third defence die changes the whole game. In Hasbro's Risk, 3 dice against 2 gives the attacker a slight edge. In RisiKo! and War, 3 against 3 hands the advantage firmly to the defender, and you need a much bigger army to break through. If players at the table learnt the game in different countries, agree on the rules before dealing out the territories.

Risk dice odds for every roll

These figures come from listing every possible combination of the dice: 36 for a 1 v 1 battle, 7,776 for 3 v 2 and 46,656 for 3 v 3. They are not simulations but exact values rounded to one decimal place.

Hasbro's Risk

RisiKo! and War only (defender with 3 dice)

"Losses shared" means each side loses at least one army on the same roll. What the numbers show:

• In a 1 v 1 the attacker wins only 41.7% of the time. There are 6 ties among the 36 combinations and all of them go to the defender.
• 3 v 2 is the usual Risk roll in which the attacker comes out ahead on average: the attacker loses 0.92 armies per roll and the defender 1.08. The defender loses both armies on 37.2% of rolls and the attacker loses both on 29.3%.
• 2 v 2 is the other way round: the attacker loses both armies 44.8% of the time and takes both of the defender's only 22.8% of the time.
• In RisiKo!'s 3 v 3 the exact split is: the defender loses all 3 armies on 13.8% of rolls, the defender loses 2 and the attacker 1 on 21.5%, the attacker loses 2 and the defender 1 on 26.5%, and the attacker loses all 3 on 38.3%. On average the attacker loses 1.89 armies per roll and the defender 1.11.

Risk battle calculator: odds of taking a territory

A single roll tells you little. What matters is whether a full attack will succeed, and that is worked out with a Markov chain: each state is a pair (attackers, defenders), each state leads to others with the probabilities in the table above, and you add up every path that ends with the defender on zero. We assume both sides always roll the most dice they are allowed.

A is the number of attacking armies, not counting the one left in the home territory. D is the number of armies in the defending territory. Figures are percentages.

Hasbro's Risk (defender rolls at most 2 dice)

Italian RisiKo! and War (defender rolls up to 3 dice)

In Risk, attacking with equal numbers starts to pay off from 5 v 5 (50.6%) and improves slowly: 10 v 10 gives 56.8%. With small armies the defender's tie advantage weighs more and 3 v 3 stays at 47.0%. Under the Italian rules the picture is very different: 10 v 10 succeeds only 19.0% of the time, and even 10 v 5 reaches just 68.1%.

How many armies you need to attack with an advantage

The table gives the minimum number of attacking armies (not counting the one left behind) needed to take a territory with D defenders at odds of 50%, 75% and 90%.

In Hasbro's Risk, matching the defender army for army puts you around 50%, and about 3 more attackers than defenders takes you past 75%. In RisiKo! and War you need far more: against 5 defenders, 8 attackers give you an even chance and 16 are needed for 90%.

Practical tips

• Attack with 3 dice whenever you can. You need 4 armies in the home territory. With 2 dice or 1 the odds drop sharply.
• As the defender in Hasbro's Risk, roll 2 dice whenever you have 2 armies. Against 3 attack dice, defending with 2 costs you 1.08 armies per roll and removes 0.92; with 1 die you lose 0.66 and remove only 0.34. In every situation, 2 dice remove more attackers for each army you lose.
• In RisiKo! and War, the defender should also roll the maximum. With 3 dice against 3, the attacker loses 1.89 armies per roll on average and you lose 1.11.
• Watch the end of the attack. Once you are down to 2 or 1 attackers you roll fewer dice and the odds fall away. An attack that starts well can leave you holding the new territory with a single army, wide open to a counter-attack.
• Check the table before you attack. If you are below 50%, it may be better to wait a turn and reinforce.
• In RisiKo!, territories with 1 or 2 defenders are the cheap targets. Against 1 or 2 defenders the odds are the same as in Risk, because the defender has no army left to roll a third die.

History: from La Conquête du Monde to Risk, RisiKo! and War

The game was invented by the French film director Albert Lamorisse, maker of the short film Le Ballon rouge (The Red Balloon, 1956), which won the Palme d'Or for best short film at Cannes. His game went on sale in France in 1957 as La Conquête du Monde (The Conquest of the World), published by Miro. In that version the defender could roll up to 3 dice.

Parker Brothers bought the North American rights and launched it in 1959 as Risk. One of their changes was to limit the defender to 2 dice so that battles would not drag on. The game cost around $7.50 at a time when most board games sold for about $2, because of its large board and many pieces, yet it sold more than 100,000 copies in its first year. It was later renamed Risk: The Game of Global Domination, and since Hasbro bought Parker Brothers in 1991 it has been published by Hasbro.

The game reached Italy in 1968 through Giochiclub, a Milan publisher, and kept the original French rule of 3 defence dice. In 1977 it passed to Editrice Giochi, which added secret objective cards. Since 2016 the brand has belonged to Spin Master. RisiKo! has gone its own way and has tournaments with their own rules, such as a dedicated deck of objective cards.

In Brazil the story is that of Grow, a company founded in São Paulo in 1972 by four partners: Gerald Reiss, Roberto Schussel, Oded Grajew and Valdir Rovai. The name comes from their initials. Its first big release was War, a version of Risk with objective cards and 3 defence dice.

Lamorisse did not see his game's long career: he died in 1970 in a helicopter accident in Iran while filming a documentary.

Risk dice facts

• The classic board has 42 territories across 6 continents in Risk, RisiKo! and War alike.
• A well-known maths slip. In 1997 the journal Mathematics Magazine published an analysis of Risk using Markov chains. It assumed the dice in each pair were independent of each other, which they are not, because they are sorted before being compared. In 2003 the statistician Jason Osborne corrected the calculation in the same journal, and the result came out more favourable to the attacker.
• In Hasbro's Risk the attacker can never lose more than 2 armies in one roll, however many dice are thrown. In RisiKo!, with 3 against 3, the attacker can lose 3.
• The colours: attack dice are red in every version; defence dice are blue (white in some editions) in Risk, blue in RisiKo! and yellow in War.