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K Sengupta
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Council and Assignment Puzzle
« on: Dec 1st, 2009, 10:02am » |
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On the island of Numeria each of the natives is one of two types: Truth-Tellers who always tell the truth, or Liars who never tell the truth. The island is governed by a Council of Elders who will only answer questions that have numerical answers. In fact the only answers they give are whole numbers, either zero or positive. Furthermore, they will never give an answer greater than the current number of council members. This number can vary daily, but is never less than 4 or more than 40. Also, the Council will only answer questions whose correct answer is independent of who is asked (e.g., no questions such as "How old are you?").
One day three native students, Ann, Bob, and Cal, were given an assignment by their teacher to question the council. They each asked a question, which was answered by every council member. Afterward they reported to their teacher and made the following statements:
(1) Ann: I asked the council how many of them were Truth-Tellers.
(2) Bob: I asked the council how many of them were Liars.
(3) Cal: Those statements are not both true!
(4) Ann: All of the answers I received were different.
(5) Bob: All of the answers I received were different.
(6) Cal: At least two of my answers were different.
(7) Ann: The sum of my answers is a palindrome.
(8) Bob: The sum of my answers is a palindrome.
(9) Cal: The square root of the sum of my answers is not less than the number of council members.
What was the number of council members on that day?
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ThudnBlunder
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Re: Council and Assignment Puzzle
« Reply #1 on: Dec 1st, 2009, 1:51pm » |
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Does 3) claim that 1) and/or 2) is false? Actually, both Cal's statements and his answers seem to be pretty useless.
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| « Last Edit: Dec 1st, 2009, 3:24pm by ThudnBlunder » |
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ThudnBlunder
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Re: Council and Assignment Puzzle
« Reply #2 on: Dec 1st, 2009, 4:21pm » |
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1 truth teller and 9 liars seems to work, giving a palindrome of 55.
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| « Last Edit: Dec 2nd, 2009, 9:43am by ThudnBlunder » |
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Vondell
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Re: Council and Assignment Puzzle
« Reply #3 on: Dec 1st, 2009, 9:36pm » |
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on Dec 1st, 2009, 1:51pm, ThudanBlunder wrote:| Does 3) claim that 1) and/or 2) is false? Actually, both Cal's statements and his answers seem to be pretty useless. |
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Don't forget that they are natives and Cal's answers determine that he is a Liar since they contradict each other which means that Ann and Bob are Truth-Tellers. So, I agree that it must be 1 Truth-Teller and 9 Liars (55) since that is the only way to get all different answers.
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Why is it that people ask for my two cents worth, and then only offer me a penny for my thoughts?
That deal sucks!
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ThudnBlunder
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Re: Council and Assignment Puzzle
« Reply #4 on: Dec 2nd, 2009, 1:44am » |
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But what about the fact that they never answer less than 4?
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| « Last Edit: Dec 2nd, 2009, 4:30am by ThudnBlunder » |
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towr
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Re: Council and Assignment Puzzle
« Reply #5 on: Dec 2nd, 2009, 3:09am » |
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on Dec 2nd, 2009, 1:44am, ThudanBlunder wrote:| But what about the fact hat they never answer less than 4? |
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They can answer less than 4. The number of council members can't be less than four, and they won't answer a number greater than the number of council members, but less is allowed.
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Wikipedia, Google, Mathworld, Integer sequence DB
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SMQ
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Re: Council and Assignment Puzzle
« Reply #6 on: Dec 2nd, 2009, 9:54am » |
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Let's take this from the top:
- (6) and (9) cannot both be true. Therefore Cal is a Liar, and (3), (6) and (9) must all be false.
- (3) false implies (1) and (2) are both true, therefore Ann and Bob are both Truth-Tellers and (4), (5), (7) and (8) are all true.
- (6) false implies Cal's answers were all the same. Therefore the Council was composed of either all Truth-Tellers or all Liars.
- (4) and/or (5) true implies the council was not composed of all Truth-Tellers. Therefore the Council was composed of all Liars.
- A Liar could not answer 0 to Ann's question nor N (the number of council members) to Bob's question. Therefore Ann's answers must have been {1, 2, ..., N-1, N} while Bob's answers must have been {0, 1, ..., N-2, N-1}.
- For N between 4 and 40, the only palindromic sums are:
N = 10: sum to N = 55
N = 11: sum to N = 66
N = 34: sum to N = 595
N = 36: sum to N = 666
- The only consecutive palindromes are 55 and 66. Therefore there were 11 council members, all Liars.
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SMQ
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Re: Council and Assignment Puzzle
« Reply #7 on: Dec 2nd, 2009, 11:56am » |
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on Dec 2nd, 2009, 11:28am, Vondell wrote:| For statements 6 and 9, one of them MUST be true. |
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How do you figure? If the Council is homogeneous (all Truth-Tellers or all Liars) they could all have given the same answer, thus (6) can be false. Only if that answer is equal to the number of council members would (9) be true, but there's nothing which forces that to be the case. They could have all answered "zero" for all we're told.
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Vondell
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Re: Council and Assignment Puzzle
« Reply #8 on: Dec 2nd, 2009, 1:14pm » |
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A misfiring of the synapses. I was still going off of the previous answer of a mix of Lie and Truth(rookie mistake  ) I deleted my other posts to avoid confusion.
But, with the council being all Liars...couldn't the answer be 1 greater than any of your palindromic totals? (0 -> n-1)
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| « Last Edit: Dec 2nd, 2009, 1:17pm by Vondell » |
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Why is it that people ask for my two cents worth, and then only offer me a penny for my thoughts?
That deal sucks!
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ThudnBlunder
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Re: Council and Assignment Puzzle
« Reply #9 on: Dec 3rd, 2009, 5:48am » |
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on Dec 1st, 2009, 9:36pm, Vondell wrote:
Don't forget that they are natives and Cal's answers determine that he is a Liar since they contradict each other which means that Ann and Bob are Truth-Tellers. So, I agree that it must be 1 Truth-Teller and 9 Liars (55) since that is the only way to get all different answers. |
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Yeah, at the time I hadn't realized that Ann, Bob, and Cal are also potential Liars. I should have read the problem more carefully.
Anyway, an excellent problem.
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| « Last Edit: Dec 3rd, 2009, 5:50am by ThudnBlunder » |
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SMQ
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Re: Council and Assignment Puzzle
« Reply #10 on: Dec 3rd, 2009, 6:02am » |
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on Dec 2nd, 2009, 1:14pm, Vondell wrote:| But, with the council being all Liars...couldn't the answer be 1 greater than any of your palindromic totals? (0 -> n-1) |
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The true answer to Ann's question, "how many of you are Truth-Tellers," is zero, so none of the All-Liar Council could answer zero. And since the answers she received were all different they myst have been 1, 2, ..., N. At the same time, the true answer to Bob's question, "how many of you are Liars," is N, so none of the All-Liar Council could answer N. And since the answers he received were all different, they must have been 0, 1, ..., N-1. Both of these totals have to be palindromes, and the only possibility for that is N = 11, giving Ann a total of 66 and Bob a total of 55.
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Vondell
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Re: Council and Assignment Puzzle
« Reply #11 on: Dec 3rd, 2009, 9:08am » |
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Forgot about that. Yet another oversight on my part. Good work, though. 
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Why is it that people ask for my two cents worth, and then only offer me a penny for my thoughts?
That deal sucks!
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