## September 2011 riddle

The following is an oldie but a goodie. Answer either or both parts. Separate lists will be kept for the two parts.

#### Part 1:

Two teams of gladiators are in a competition. Each team is composed of a certain number of gladiators (not necessarily equal to both teams), and each gladiator has a level of armament, which is given as a positive real number.

The competition is a series of combats. In the beginning of each combat, the trainer of each team picks one of her gladiators to compete. As soon as the competitors are chosen, they duel to the death. If the strength of gladiator A is a and the strength of gladiator B is b, the probability that A will overcome B is a/(a+b).

At the end of the fight, the winning gladiator takes his dead opponent's armament. This makes his armament level be the sum of his original armament level and his opponent's original armament level. (In the example, it is a+b.)

The winning team is the team to still have live gladiators after the other team's gladiators have all perished. The strategy of the trainer is the function by which she decides which gladiator steps into the arena at each combat.

The question: prove that the strategies of the trainers do not influence the winning probabilities of either team.

#### Part 2:

Part 2 asks the same question as Part 1. However, this time gladiators are not allowed to take their dead opponents' weapons. Each gladiator remains at his original armament level (or "strength") throughout all his matches.

You are, again, to prove that the strategies of the trainers do not influence the winning probabilities of either team.

### List of solvers:

#### Part 1:

Jan Fricke (1 September 18:00)
Djinn Lu (1 September 19:26)
Lu Wang (2 September 05:43)
Phil Muhm (2 September 06:00)
Joseph DeVincentis (3 September 03:32)
Yan Wang (3 September 08:47)
Pei Wu (6 September 15:43)
Ashish Chiplunkar (8 September 03:30)
Liubing Yu (9 September 13:30)
Sylvain Becker (10 September 01:49)
Gaoyuan Chen (12 September 16:46)
Ganesh Lakshminarayana (13 September 21:50)
Jim Boyce (29 September 09:58)

#### Part 2:

Lu Wang (6 September 06:14)
Ganesh Lakshminarayana (14 September 21:56)
Gaoyuan Chen (16 September 15:55)
Liubing Yu (18 September 04:54)
Jim Boyce (30 September 12:16)

Elegant and original solutions can be submitted to the puzzlemaster at riddlesbrand.scso.com. Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.

Enjoy!