## Wednesday, 6 August 2008

### Logical Games and solutions to the last games

You might know the story of the prisoner, the princess and the tiger. Do you know that we can make a lot of logical games with this story? Here you can find some.

Connaissez-vous l'histoire du prisonnier, de la princesse et du tigre? Il existe de nombreux jeux logiques fondés sur cette histoire. En voici quelques-uns.

A) Logical games:
A king put a lot of his princes in prison, but the prisons are now overloaded. He decides to empty them and to get rid of his innumerable daughters, thanks to a logic game.
The game is simple: the prince must choose a door among many. Behind the door, it may be a tiger or a princess. The only way to find what is behind is logic (and we suppose that you want to find the princess)

Un roi a emprisonné de nombreux princes. Ses prisons sont bien trop pleines. Il décide, pour vider ses prisons et pour se débarrasser des nombreuses filles nées d'un trop grand nombre de concubines, de soumettre les princes emprisonnés à un jeu.
Le jeu est simple: le prince doit choisir une porte parmi plusieurs. Derrière une porte se trouve un tigre ou une princesse. Le seul moyen de déterminer ce qui se trouve derrière la porte est le raisonnement (on suppose que vous voulez trouver la princesse).

*First game:
Door 1: "There is a princess in this room and a tiger in the other one"/ "Il y a une princesse dans cette cellule et un tigre dans l'autre".
Door 2: "There is a princess in one room and there is a tiger in one room"/ "Il y a une princesse dans une cellule et un tigre dans une cellule".
Rule: one message is telling the truth, one is a lie/une affiche dit la vérité et l'autre ment.

*Second game:
Door 1: "Both rooms are hiding a princess"/ "Les deux cellules contiennent une princesse".
Door 2: "Both rooms are hiding a princess"/ "Les deux cellules contiennent une princesse".
Rule: the message on the door 1 is telling the truth when there is a princess and is telling a lie when there is a tiger; the message on the door 2 is telling the truth when there is a tiger and is telling a lie when there is a princess/ l'affiche sur la porte 1 dira la vérité quand il y a dans la cellule une princesse et mentira quand il y aura un tigre, tandis que l'affiche sur la porte 2 mentira quand il y aura une princesse dans la cellule et dira la vérité quand il y aura un tigre.

*Third game:
Door 1: "A room at least is hiding a princess"/ " Une cellule au moins contient une princesse".
Door 2: "The other room is hiding a princess"/ "L'autre cellule contient une princesse".
Rule: same as the second game.

B) Solutions:

*Mathematical game's solution/solution du jeu mathématique:

d: voters from the Conservatives/électeurs de droite
g: voters from the Liberals/électeurs de gauche
A : set of voters in the village/ensemble des électeurs dans le village
d+g =A

First ballot/Premier tour:

$d-1= g+ 1$
$(A-g) -1= g+ 1$
$A-g-1=g+1$
$A-2g=1+1$
$2g=v-A+2$
$g = {A-2\over 2}$

Second ballot/Second tour:

$d+1=2(g-1)$

$d+1=2((A-d)-1)$

$d+1=2(A-d)-2$

$d+1=2A-2d-2$

$d=2A-2d-3$

$3d=2A-3$

$d={2A-3\over 3}$

Voters/Électeurs:

$d+g=A$

$({2A-3\over 3})+({A-2\over 2})=A$

$6({2A-3\over 3})+6({A-2\over 2})=6A$

$2(2A-3)+3(A-2)=6A$

$4A-6+3A-6=6A$

$7A-12=6A$

$7A-6A=12$

$A=12$

*Logical game's solution/solution du jeu logique:

A says he cannot identify a color that is not the one he has on his hat.

B says he cannot identify a color that is not the one he has on his hat.

We have no indication about C, so let's examine his case:
First hypothesis: C is yellow. But A said that there is no peer (if there was a peer, A would have said that he knows at least one color which is not the one on his hat). So B is not yellow. If so, B could have said that he knows a color which is not the one he has on his hat. Consequently, C is not yellow.

Second hypothesis: C is red. But A said there is no peer. So B is not red. If so, B could have said he knows a color which is not the one he has on his hat. Consequently, C is not red.

There is only one remaining possibility: C is green. If so, neither A nor B can tell a color which is not the one they have on their hat. Nothing more can be told about A and B.