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Numbered Foreheads Concluded
Posted 15 November 2013 - 09:20 AM
Edited by harey, 15 November 2013 - 09:29 AM.
Posted 11 February 2014 - 09:48 AM
There can be a solution in one pass disregarding the answers of C. The numbers are a little bit high for practical purposes, so I will illustrate it on a sum of 12 or 13, everyone having 4.
"!n" means "I do not see the number n"
"<-(xy)" means "because otherwise I would know the combination is (a=x b=y)"
Both A and B see c=4, they both know a+b=8 or a+b=9.
They also know that the other one knows that.
By saying "I do not know", they in fact tell:
1 A: !8 <- (18)
B: !8 <- (81)
!1 <- (17); (18) excluded in 1A
2 A: !1 <- (71); (81) excluded in 1B
!7 <- (27); (17) excluded in 1B
B: !7 <- (72); (71) excluded in 2A
!2 <- (26); (27) excluded in 2A
3 A: !2 <- (62); (72) excluded in 2B
!6 <- (36); (26) excluded in 2B
B: !6 <- (63); (62) excluded in 3A
!3 <- (35); (36) excluded in 3A
4 A: !3 <- (53); (63) excluded in 3B
!5 <- (45); (35) excluded in 3B
--> B knows A sees 4.
--> A realises he never will get more information
A already knows that b=4, he can imagine this dialogue (the answers of B will not change whether a=4 or a=5), so his very first answer is:
1 A: I will never be able to tell.
if b=4, this makes sense
if b=5, A would have announced he would be able to tell.
From the answers of A and B, C deduces c=4 (if necessary, I will explain).
The remaining problem is whether the answers of C are redundant or whether they lead to another distribution.
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