3 logicians - 8 stamps

anna_gg · 2012-01-22 11:13:57

Here is another:
3 logicians and one moderator have decided to play a game. The moderator has a set of 8 stamps, of which 4 are red and 4 are black. He affixes two stamps to the forehead of each logician, so that each of them can see all the other stamps except those two in the moderator's pocket and the two on his or her own head. He then asks them in turn if they know the colors of their own stamps. Target of the game is to have at least one of the logicians guess the correct colors of his own dots (and of course explain his reasoning). If the 1st player does not know the answer, the 2nd is asked and then the 3rd, then 1st again etc (1-2-3-1-2-3 etc).
Before the beginning of the game, the first logician is asked to which of the 3 positions he wants to sit. What should be his choice and why?

John E. Franklin · 2012-01-23 13:00:47

Let's start the thinking off at least.
K=blacK R=Red (K is an industry standard for black, since B is blue)

caseKK: Hidden:  KK   Exposed: KKRRRR
caseRR: Hidden:  RR   Exposed: RRKKKK
caseKR: Hidden:  KR   Exposed: KRKRKR

John E. Franklin · 2012-01-23 13:07:41

Next we should break caseKR into the 3 forehead (front=[francais for forehead]) combinations:
caseKR1: KR KR KR
caseKR2: KK KR RR
Next we should break caseRR into the 3 foreheads:
caseRR1: RR KK KK
caseRR2: KR KR KK
Next we should break caseKK up just like caseRR's symbiotic relationship:
caseKK1: KK RR RR
caseKK2: KR KR RR

John E. Franklin · 2012-01-23 13:16:05

Next we'll look at some possibilities.
Pretend you have KR on your forehead, list what you see on the others 2 gals foreheads?
KR KR
KK RR
KR KK
KR RR
Now pretent you have RR on your forehead, list what you see on the other 2 gals foreheads?
KK KR
KK KK
KK RR
KR KR
Now convert the latter list for RR on your forehead to KK on your forehead with substitutional logic.
RR RK
RR RR
RR KK
RK RK

John E. Franklin · 2012-01-23 13:24:46

Next we'll letter the 12 combinations you might see AND what MIGHT be on your forehead, maybe:
A.) KR KR => KR
B.) KK RR => KR
C.) KR KK => KR
D.) KR RR => KR
E.) KK KR => RR
F.) KK KK => RR
G.) KK RR => RR
H.) KR KR => RR
I.) RR KR => KK
J.) RR RR => KK
K.) RR KK => KK
L.) KR KR => KK
Now we must find the ambiguous ones and true ones looking for duplicates on left sides of equations.

John E. Franklin · 2012-01-23 13:29:34

A = H = L --> any of 3 possibilities on your forehead

B = G = K --> any of 3 possibilities on your forehead

C = E ----> either KR or RR on your forehead

D = I ----> either KR or KK on your forehead

F ----> you absolutely have RR on your forehead

J -----> you absolutely have KK on your forehead

Those are the 12 thingy-do's

John E. Franklin · 2012-01-23 13:46:36

So, looking at all the posts so far, we see early on that there are either 4 and 0 shown or 3 and 1 shown or 2 and 2 shown to you by looking at the two other gals.
If 4 and 0 is shown, you know what you have. (what you've got).
If 3 and 1 are shown to you, then the missing 1 could be on you or in the pocket.
If the 1 is on you then the second person will see bicolors on you and doubles or bicolors on the 3rd guy.
The third guy will see what the second guy didn't see (backwards).
However, if 2 and 2 are shown to you, then you could have either alll red or all black or both.
The second person would see either 2 and 2 or 3 and 1 or 4 and 0, so this is very hard.
I have no idea what each person would guess and I give up for now.

Math Is Fun Forum

#1 2012-01-22 11:13:57

3 logicians - 8 stamps

#2 2012-01-23 13:00:47

Re: 3 logicians - 8 stamps

#3 2012-01-23 13:07:41

Re: 3 logicians - 8 stamps

#4 2012-01-23 13:16:05

Re: 3 logicians - 8 stamps

#5 2012-01-23 13:24:46

Re: 3 logicians - 8 stamps

#6 2012-01-23 13:29:34

Re: 3 logicians - 8 stamps

#7 2012-01-23 13:46:36

Re: 3 logicians - 8 stamps

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