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Numbered Foreheads Concluded


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#1 ThunderCloud

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Posted 20 February 2013 - 04:03 AM

Although I believe the logic to answer the original problem was present, it was distributed among several postings... It did not seem right to mark any single post as the best answer. Therefore, here is the bonus round. :)

 

If approached correctly, this version is not much harder than the original.

 

The puzzle:

 

Three perfect logicians had stickers placed on their foreheads so that none could see their own sticker but each could see one another's. They were told that each sticker has a single positive integer written on it (i.e.1, 2, 3, ...), and that the sum of the integers on all three stickers is either 1002 or 1003. They were then asked, in turn, to identify the number on their own sticker. Upon being asked, each logician would name their number if they were sure that they knew it, give up if they were sure that they would never know it, or otherwise 'pass' (or say "I don't know"). The question was repeated, again in turn, until EACH of the three logicians had either named their number or given up. All three stickers actually had the same number written on them. Who among the three logicians was able to deduce his number, and who among them gave up? (Furthermore, how did each answer?)


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#2 ThunderCloud

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Posted 10 March 2013 - 11:42 PM

Hint:

This is not a tedious problem... not much harder than the original. There is a simple way to solve it; a pattern to be observed. An exhaustive description of how each logician answered in his turn would not be very long... ;)


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#3 ThunderCloud

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Posted 13 November 2013 - 05:36 AM

Still unanswered...

 

Can you find the complete sequence of responses? It is considerably shorter than one might initially think...


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#4 harey

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Posted 13 November 2013 - 07:37 PM

Spoiler for

 

I know a better and harder version, if I find it, I'll post it. Does it matter if it is in French?


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#5 antel0pe

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Posted 13 November 2013 - 09:42 PM

Could they communicate beforehand so that they can deduce a plan?


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#6 Rainman

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Posted 13 November 2013 - 10:07 PM

Spoiler for


Edited by Rainman, 13 November 2013 - 10:08 PM.

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#7 ThunderCloud

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Posted 15 November 2013 - 12:41 AM

Spoiler for

 

I know a better and harder version, if I find it, I'll post it. Does it matter if it is in French?

I am not sure I follow your reasoning here -- care to elaborate further?


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#8 ThunderCloud

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Posted 15 November 2013 - 12:42 AM

Could they communicate beforehand so that they can deduce a plan?

 

No. The logicians cannot agree upon a strategy in advance. However, you may assume each of them to be "perfect" in that they will deduce all that they logically can. You may further assume that each logician will assume the others to behave this way as well (i.e., that it is generally known that all three logicians are "perfect").


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#9 harey

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Posted 15 November 2013 - 12:43 AM

Spoiler for 2nd try

Edited by harey, 15 November 2013 - 12:50 AM.

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#10 ThunderCloud

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Posted 15 November 2013 - 12:44 AM

Spoiler for

 

I think you are on the right track, and very close. ;)


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