Student Exercises
Risk!
Risk is a strategy board game produced by Parker Brothers. It was invented by French film director Albert Lamorisse and originally released in 1957 as La Conquête du Monde (“The Conquest of the World”) in France.
It was later bought by Parker Brothers and released in 1959 with some modifications to the rules as Risk: The Continental Game, then as Risk: The Game of Global Domination.
Risk is a turn-based game for two to six players. The standard version is played on a board depicting a political map of the Earth, divided into forty-two territories, which are grouped into six continents. The object of the game is to occupy every territory on the board and in doing so, eliminate the other players.
Players control armies with which they attempt to capture territories from other players, with results determined by dice rolls.
there are a lot of details to the game. We will investigate only one step, that of one country attacking another. This step is done by the owner of the attacking country and the owner of the defending country throwing dice.
Note that i some versions of the game the defender has at most two dice. In this project we we will study the case of three dice for both the attacker and the defender.
Say the attacking country has n armies and the defending one has m. Then if \(n \ge 4\) and \(m \ge3\) both throw 3 dice. These are ordered from largest to smallest and matched. For each match were the defending country is at least equal to the attacking country the attacking country looses one army, otherwise the defending army does.
Example A: 3 5 5 D: 2 5 6
Sort: A: 5 5 3 D 6 5 2
5 < 6 A looses one army
5 = 5 A looses one army
3 > 2 D looses one army
so A looses 2 armies and D looses 1.
finally neither side can through more dice than they have available for fighting, and A always has to keep 1 army to occupy the country, so for example if A has 3 armies and D has one, A throws 2 dice and D throws 1.
Example A 4 2 D 4, \(4 \ge 4\), so A looses 1 army.
If D has lost all his armies he looses the country and A takes it over. If A has only one army left he can no longer attack. A can decide to stop attacking at any time.
Here are two examples how the attack might go:
Case 1: A has 10 armies and D has 6:
Rolls A: 3 6 1 D: 3 4 2
Matched dice and winner: 6>4(A), 3=3(D) 1<2(D)
A looses 1 army, D looses 2
A has 9 armies left, D has 4
Rolls A: 1 6 5 D: 2 6 6
Matched dice and winner: 6=6(D), 5<6(D) 1<2(D)
A looses 3 army, D looses 0
A has 6 armies left, D has 4
Rolls A: 5 6 2 D: 2 5 1
Matched dice and winner: 6>5(A), 5=5(D) 2>1(A)
A looses 1 army, D looses 2
A has 4 armies left, D has 2
Now that D has only 2 armies, he can roll only two dice:
Rolls A: 3 4 1 D: 5 2
Matched dice and winner: 4<5(D), 3>2(D)
A looses 1 army, D looses 1
A has 3 armies left, D has 1
Now A has only 3 armies left, one of which he cannot use to attack, so he can roll 2 dice. D has only 1 army, so he can roll only one die:
Rolls A: 4 2 D: 3
Matched dice and winner: 4>3(D)
A looses 0 army, D looses 1
A has 3 armies left, D has 0
D has lost all his armies, so A won.
Case 2: A has 2 armies and D has 1, so both can roll only one die because A needs to keep one army back.
Rolls A: 3 D: 3
Matched dice and winner: 3=3(D)
A looses 1 army, D looses 0 A has 1 army left, D has 1 A has no more armies to use for attack, so the game ends and D wins.
Now here is our problem: we want to find a “simple” rule that we can use during an actual game that tells us the odds of winning if the Attacker has n armies for the attack and the defender has m. By “simple” I mean something one can remember and apply during a game, without using a computer. n and m should be in the range of 1 to 50.