3 replies to this topic

[bonanova]

Variation on a familiar theme.

You and two other crack Denizens participate in a competition based purely on logic.
Not surprisingly, you win. But because of the large stakes involved, including but not
limited to the coveted bonanova gold star, a federal investigator has been assigned
to rule out any possible allegation of cheating. Think Slumdog Millionaire. Therefore
You are required to explain precisely how you were able to win.

Here are the facts.

The host, Rookie produced eight colored stamps: four red and four blue.
Two each were affixed to the foreheads of the three contestants.
The remaining two were placed in Rookie's pocket.
Each contestant can see the four stamps on the other two foreheads,
but none can see the colors on his/her forehead, nor the two in Rookie's pocket.

Rookie then asks the contestants in turn if they know their own colors.

A. No, I don't.
B. No, I don't.
C. No, I don't.
A. No, I don't.
B. Yes, I do.

You were contestant B.

How did you convince the investigators that there was no cheating?
For extra credit and the gold star, what were your colors?

[CaptainEd]

Spoiler for notations, proof, and answer

Spoiler for notation

denoting a distribution with
AABBCC: the two colors on A's forehead, followed by those of B, followed by those of C

 

for example:
rbbbrr means A is wearing rb, B is wearing bb, C is wearing rr.

Spoiler for proof

A(1) didn't see ..rrrr, or would have announced bb so bbrrrr is out
A(1) didn't see ..bbbb, or would have announced rr so rrbbbb is out
B(2) didn't see bb..bb or would have announced rr, so bbrrbb is out
B(2) didn't see rr..rr or would have announced bb, so rrbbrr is out
C(3) didn't see rrrr, or would have announce bb, so rrrrbb is out
C(3) didn't see bbbb, or would have announced rr, so bbbbrr is out
C(3) didn't see rrbb, or would have announced rb, so rrbbrb is out
C(3) didn't see bbrr, or would have announced rb, so bbrrrb is out
A(4) didn't see ..rrrb, so rbrrrb is out
A(4) didn't see ..rrbb, so rbbbrr is out
A(4) didn't see ..bbrr, so rbbbrr is out
A(4) didn't see ..bbbr, so brbbbr is out

 

Remaining cases are

 

rr rb rb
rb rb rr
rr rb bb
rb rb rb
bb rb rr
bb rb rb
rb rb bb

Spoiler for answer

They all have contestant B wearing rb

 

[k-man]

Spoiler for pure logic and no cheating

There are a total of 18 possible ways how the stamps may be distributed among 3 players. Each "No" answer eliminates possibilities that have unique combination of colors for the other 2 players and would allow the answering player determine his colors. For example, the first "No" by player A eliminates the possibility that both B and C have 4 stamps of the same color. Each subsequent "No" answer eliminates any remaining possibility with unique combination. After 4 answers 5 possible distributions remain and in all of the player B has one red and one blue stamp. Here is the complete table of possibilities and the answer eliminating that possibility ("P" is for Pocket):

 

A  B  C  P  Answer
BB RR RR BB 1
RR BB BB RR 1
BB RR BB RR 2
RR BB RR BB 2
BB BB RR RR 3
BB RR BR BR 3
RR BB BR BR 3
RR RR BB BB 3
BB BR RR BR 4
BR BB BR RR 4
BR BB RR BR 4
BR RR BB BR 4
BR RR BR BB 4
BB BR BR RR 
BR BR BB RR 
BR BR BR BR 
RR BR BB BR 
RR BR BR BB 

 

 

[bonanova]

You both have it.
CaptainEd got it first.
Congrats both.