December 2012 Brain Teaser Solution

Q: Paula and Georgia play a game with dice that have colors instead of numbers.  Some faces are pink and some are green.  Paula wins when the two top faces are the same color.  Georgia wins when the colors are different. Their chances are even.
The first die has 5 pink faces and 1 green face. On the second die, how many faces are pink and how many are green? 

A: 3 pink faces and 3 green faces

At first it may seem as if the second die should have the reverse of the first (1 pink and 5 green) but that would make Georgia win too often as the chance of different colors would be too high.
This can be done with a guess and check. 
5 pink and 1 green matched with 2 pink and 4 green would give
P           P
P              P
P              G
P              G
P              G
G           G

PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PG, PG, PG, GG, PP, PP, PG, PG, PG, GG,PP, PP, PG, PG, PG, GG, GP, GP, GG, GG, GG, GG

still too many that match giving Paula an advantage



This can be done with a guess and check. 
5 pink and 1 green matched with 3 pink and 3 green would give
P           P
P              P
P              P
P              G
P              G
G           G

(keep the second column steady while changing the first die)
PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP,
PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)

If you want to keep the first column steady and change the second:
PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG,, PP, PP, PP, PG, PG, PG, GP, GP, GP, GG, GG, GG

PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)



Solution 2:
Bill Adams said that he solved it using probability fractions:

Pink 5/6  x Pink 3/6 = 15/36 Pink Pink

Pink 5/6  x Green 3/6

Green1/6 x Pink 3/6
Green 1/6 x Green3/6 = 3/36 Green Green

Add those together and 15/36 + 3/36 = 18/36 = 1/2!