A390860 a(n) is the minimum number of armies required for an attacker to have a 50% or better chance of beating a defender with n armies in the game Risk.

3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 18, 19, 20, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 41, 42, 43, 44, 45, 46, 47, 47, 48, 49, 50, 51, 52, 53, 53, 54, 55, 56, 57, 58, 59, 59

Offset

1,1

Comments

In the game Risk, battles are settled by dice roll. Attackers get one red die per army (up to 3 dice) and defenders get one white die per army (up to 2 dice). The highest attacker die is paired with the highest defender die, and 2nd highest with 2nd highest. The attacker must have at least one more army than the number of dice rolled, since one army must be left behind occupying the original attacker space. Ties are won by the defender, which gives defenders a significant advantage. Losing a dice roll corresponds to losing an army, which can change the number of dice that can be rolled.

The defender has a significant advantage when smaller numbers of armies are involved. However, as the number of armies increases, the 3:2 attacker to defender dice ratio increases the likelihood of the attacker winning. Thus, a(n) < k for k >= 19.

Links

Table of n, a(n) for n=1..67.

Sharon Blatt, RISKy Business: An In-Depth Look at the Game RISK, Rose-Hulman Undergraduate Mathematics Journal, Vol. 3: Iss. 2, Article 3.

David Consiglio, Jr., Graph of Attackers vs. Defenders (1 - 100)

David Consiglio, Jr., Python Markov-chain exact calculation

Daniel Morris and Megan Vance, Risk and Markov Chains.

Bariş Tan, Markov Chains and the RISK Board Game.

Formula

a(n) = 1 + min{k : A371949(k, mN) / 7776^(n+k-1) >= 1/2}. - Sean A. Irvine, Dec 12 2025

Example

For example, suppose the attacker has 4 armies and is attacking a defender with just 1. The attacker can roll 3 dice against 1 for the defender. Here is a sample battle:
          Attacker          Defender
Roll 1:    5 3 2                 6    --> Defender wins and attacker loses one army
Roll 2:    5 1                   4    --> Attacker wins and occupies defender's space
If an attacker with 2 armies attacks a defender with just 1, the attacker loses more than 50% of the time. But with 3 armies, the odds favor the attacker, so a(1) = 3.
Similarly, if a defender has 4 armies then an attacker must have at least 6 armies to have a 50% chance of winning an attack, so a(4) = 6.

Crossrefs

Cf. A167179, A371949.

Sequence in context: A217031 A104136 A198466 * A212642 A159624 A036288

Adjacent sequences: A390857 A390858 A390859 * A390861 A390862 A390863

Keyword

nonn

Author

David Consiglio, Jr., Nov 21 2025

Extensions

More terms from Sean A. Irvine, Dec 12 2025

Status

approved