April 13, 2015
https://www.facebook.com/mehtarahulc/posts/10152715119216922
https://www.facebook.com/mehtarahulc/posts/10152715119216922
Three important LOGIC \ MATHS problems which have immense application in activism / politics . None are trick questions. The explain the concept of Common Knowledge in Logic. Seehttp://en.wikipedia.org/wiki/Common_knowledge_(logic) and it is DIFFERENT from http://en.wikipedia.org/wiki/Common_knowledge .
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Problem-1 :
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(1A) There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
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(1B) The king said to all --- "Exactly one of you have black cap. Those who have black cap should raise their hands at the stroke of the bell". .
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(1C) The king rang bell once. The king rang bell again. None raised his hands. Then king rang bell 3rd time and still no one raised hands. And then king rang bell 4th time, and all and only those who had black caps raised their hands.
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So questions are --- (q1a) how did those with black caps concluded that that they have black caps (q1b) How did those with white caps decided that they have white caps? (q1c) why did king had to ask question 4 times, and not 3 or 5 times. Please note --- if your replace numbers 10 and 4 my N and m, then also problem and solution word.
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Problem-2
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(2A) Same as (1A). There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
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(2B). HERE , THE PROBLEM DIFFERS FROM (1A). The king said to all --- " I wont tell you how many of you have black caps and may be none of you have black caps. And those who have black cap should raise their hands at the stroke of the bell". But each mathematician did see that at least one has black cap.
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(2C) The king rang bell once. The king rang bell again. None raised his hands. Then king rang bell 3rd time and still no one raised hands. And then king rang bell 4th time, and 5th time and 10th time and 100th time. But none raised their hands. So questions are --- (q2a) why didnt mathematicians raised hands?
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Problem-3
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(3A) Same as (1A). There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
.
(3B) HERE , THE PROBLEM DIFFERS FROM (1A). The king asked his guard to take one mathematician away into a closed room. Please note --- one mathematician was taken away from all. And then the king said to the nine mathematician that --- " At least one of have black cap. And all those who have black cap should raise their hands at the stroke of the bell". .And then the 10th mathematician was brought back in the circle,
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(3C) But 10th had not heard the statement.
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(3D) The king kept ranging bell, but none raised the hand. Why NOT? Please note
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Problem-4
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(4A) Same as (3A). There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
.
(4B) Same as (3B) HERE , The king asked his guard to take one mathematician away into a closed room. Please note --- one mathematician was taken away from all. And then the king said to the nine mathematician that --- " At least one of have black cap. And all those who have black cap should raise their hands at the stroke of the bell". And then the 10th mathematician was brought back in the circle,
.
(4C) DIFFERENT FROM 3C. The 10th mathematician had also heard the king's statement when he was in room. Some the 9 mathematicians think that 10th one didnt hear the king's statement.
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(4D) The king started ringing bell,
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(4E.1) Say 10th mathematician had black cap. And he decided to take advantage of the fact that other 9 dont know that he had heard the statement when he was taken away in the room. Then will the 4 mathematicians with black caps raise hands? Then will the remaining 3 mathematicians with black caps raise hands?
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(4E.2) Say 10th mathematician had white cap. And he decided to take advantage of the fact that other 9 dont know that he had heard the statement when he was taken away in the room. Then will the 4 mathematicians with black caps raise hands?
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The problems will explains why "everyone knows" is FAR FAR LESS POWERFUL than "everyone knows that everyone knows".
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And "everyone knows that everyone knows that everyone knows" is more powerful than "everyone knows that everyone knows" .
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And "everyone knows that everyone knows .... infinite times " with N persons more powerful than "everyone knows that everyone knows .... M times " for any large finite M .
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How does this apply in politics / activism?
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Many times , there are certain actions that will happen only if "everyone knows that everyone knows" and those actions will NOT happen even if "everyone knows".
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Example .
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1. say everyone knows everyone knows that everyone knows .... that "everyone thinks that party-A is corrupt"
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2. say everyone thinks that "party-B is only slightly less corrupt". But everyone doesnt know that everyone knows that "party-B is only slightly less corrupt".
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In such case, if paidmedia favors party-B , then paidmedia may manage to convince everyone that "everyone else thinks that party-B is non-corrupt" !!!
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And if paidmedia opposes party-B , then paidmedia may manage to convince everyone that "everyone thinks that party-B is non-corrupt" !!!
.
IOW, paidmedia cant write against common knowledge. But paidmedia can create illusion that "everyone else thinks A is true" even when "everyone thinks that A is false".
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In addition, the concept of common knowledge has many many applications.
.
Next set of problems will be based on "majority knows that majority knows that majority knows .... " vs "majority knows but "majotity doesnt know that majorirty knows"
.
.
=====
.
Problem-1 :
.
(1A) There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
.
(1B) The king said to all --- "Exactly one of you have black cap. Those who have black cap should raise their hands at the stroke of the bell". .
.
(1C) The king rang bell once. The king rang bell again. None raised his hands. Then king rang bell 3rd time and still no one raised hands. And then king rang bell 4th time, and all and only those who had black caps raised their hands.
.
So questions are --- (q1a) how did those with black caps concluded that that they have black caps (q1b) How did those with white caps decided that they have white caps? (q1c) why did king had to ask question 4 times, and not 3 or 5 times. Please note --- if your replace numbers 10 and 4 my N and m, then also problem and solution word.
.
=========
.
Problem-2
.
(2A) Same as (1A). There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
.
(2B). HERE , THE PROBLEM DIFFERS FROM (1A). The king said to all --- " I wont tell you how many of you have black caps and may be none of you have black caps. And those who have black cap should raise their hands at the stroke of the bell". But each mathematician did see that at least one has black cap.
.
(2C) The king rang bell once. The king rang bell again. None raised his hands. Then king rang bell 3rd time and still no one raised hands. And then king rang bell 4th time, and 5th time and 10th time and 100th time. But none raised their hands. So questions are --- (q2a) why didnt mathematicians raised hands?
.
==============
.
Problem-3
.
(3A) Same as (1A). There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
.
(3B) HERE , THE PROBLEM DIFFERS FROM (1A). The king asked his guard to take one mathematician away into a closed room. Please note --- one mathematician was taken away from all. And then the king said to the nine mathematician that --- " At least one of have black cap. And all those who have black cap should raise their hands at the stroke of the bell". .And then the 10th mathematician was brought back in the circle,
.
(3C) But 10th had not heard the statement.
.
(3D) The king kept ranging bell, but none raised the hand. Why NOT? Please note
.
=================
.
Problem-4
.
(4A) Same as (3A). There were 10 mathematicians. A king placed white or black cap on their head. Each can see color of others' caps but not his own. Exactly 4 of them had black cap and 6 had white cap.
.
(4B) Same as (3B) HERE , The king asked his guard to take one mathematician away into a closed room. Please note --- one mathematician was taken away from all. And then the king said to the nine mathematician that --- " At least one of have black cap. And all those who have black cap should raise their hands at the stroke of the bell". And then the 10th mathematician was brought back in the circle,
.
(4C) DIFFERENT FROM 3C. The 10th mathematician had also heard the king's statement when he was in room. Some the 9 mathematicians think that 10th one didnt hear the king's statement.
..
(4D) The king started ringing bell,
.
(4E.1) Say 10th mathematician had black cap. And he decided to take advantage of the fact that other 9 dont know that he had heard the statement when he was taken away in the room. Then will the 4 mathematicians with black caps raise hands? Then will the remaining 3 mathematicians with black caps raise hands?
.
(4E.2) Say 10th mathematician had white cap. And he decided to take advantage of the fact that other 9 dont know that he had heard the statement when he was taken away in the room. Then will the 4 mathematicians with black caps raise hands?
.
==============
.
The problems will explains why "everyone knows" is FAR FAR LESS POWERFUL than "everyone knows that everyone knows".
.
And "everyone knows that everyone knows that everyone knows" is more powerful than "everyone knows that everyone knows" .
.
And "everyone knows that everyone knows .... infinite times " with N persons more powerful than "everyone knows that everyone knows .... M times " for any large finite M .
.
=============
.
How does this apply in politics / activism?
.
Many times , there are certain actions that will happen only if "everyone knows that everyone knows" and those actions will NOT happen even if "everyone knows".
.
Example .
.
1. say everyone knows everyone knows that everyone knows .... that "everyone thinks that party-A is corrupt"
.
2. say everyone thinks that "party-B is only slightly less corrupt". But everyone doesnt know that everyone knows that "party-B is only slightly less corrupt".
.
In such case, if paidmedia favors party-B , then paidmedia may manage to convince everyone that "everyone else thinks that party-B is non-corrupt" !!!
.
And if paidmedia opposes party-B , then paidmedia may manage to convince everyone that "everyone thinks that party-B is non-corrupt" !!!
.
IOW, paidmedia cant write against common knowledge. But paidmedia can create illusion that "everyone else thinks A is true" even when "everyone thinks that A is false".
.
In addition, the concept of common knowledge has many many applications.
.
Next set of problems will be based on "majority knows that majority knows that majority knows .... " vs "majority knows but "majotity doesnt know that majorirty knows"
.
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