jnu question

[priyal]

I) The house numbers of 3 individuals: A,B and C are,;1)distinct from each other, 2)lie between 1 and 99.
both facts are known to all 3 individuals. B asks A'when C is not present,2 questions: 1)is your house number a perfect square? 2)is your house number greater than 50? Assuming A's answers to be correct, B is able to infer house number of A from answers hiven by A.A's house number, however is not the same as infered by B because A answered only second question truthfully.C asks A when B is not present, 2 questions:1)is your house number a perfect cube?2) is your house number greater than 25? assuming A's answer to be correct, C is able to infer house number of A from answer to two questions given by A.A's house number, however is not same as infered by C because A answered only second question truthfully.deduce the house numbers of 3 individuals given that:1)A's house number is less than house number of both B and C. 2)sum of house numbers of 3 individuals is perfect square multiplied by two.


II) four persons A,B,c and D have to share rs.4 among themselves in units of one rupee.first A proposes a distribution and all of them, including A ,vote on it. if atleast 50%of those voting agree with A,the proposal is accepted. if not,A loses her voting rights and B gets to propose a distribution and all except A vote on it. once again,B's proposal is accepted if atleast 50% of those eligible to vote agree on it. if not,B also loses her voting rights and C gets to propose ansd so on D.
assume that each person prefers more mony to less, and will always vote against a distributionin which she gets zero.what distribution should A propose in the beginning?

[Amit Goyal]

Problem 1
A's House Number is 55
B's House Number is 81
C's House Number is 64

Problem 2
A offers 3 to himself, 1 to C and 0 to B and D.

[priyal]

sir,
for first ques i was able to infer B's n C's house number but how you arrived at A's number?

and please give me steps of second question.

[Amit Goyal]

Hi Priyal,

For getting house number of A, we will first use the following information:
Sum of house numbers of 3 individuals is perfect square multiplied by two.
Given that B's House Number is 81, C's House Number is 64, let A's house number be x. Also we know that 100 > x > 50. Now, x+81+64 is a perfect square multiplied by 2. This implies that (x+81+64)/2 = (x + 145)/2 is a perfect square. Since 100 > x > 50 this implies that
245/2 > (x + 145)/2 > 195/2. Now we look for a perfect square in the interval (195/2, 245/2)
=(97.5, 122.5)
Two Possibilities are 100, 121
Solving,  (x + 145)/2  = 100 give us x = 55
and (x + 145)/2  = 121 give us x = 97
Now using the condition that A's house number is less than house number of both B and C we rule out x = 97
Hence, A's house number is 55.

[Amit Goyal]

Steps for second question:
Let's do the backward induction argument,
When only C and D are left, C will offer 4 to himself and 0 to D and since C's vote carry 50% weight, hence the offer will be agreed.
When B, C and D are left, C will reject any offer that give him less than 4 and D will accept any offer greater than 0 since this is what they are going to get if B is eliminated. So, B will offer 3 to himself and 1 to D and the offer will be accepted by 2/3rd majority. Thus, C will get 0 if B, C and D are left.
So, when its A's turn to make an offer, C will accept any offer greater than 0. So, A will make an offer equal to 3 to himself and 1 to C and the offer will be accepted by 50% votes.

[sohini]

sir how do i go about for calculating B and C's house no.?

[Amit Goyal]

Given that, B asks A when C is not present, 2 questions:
1) Is your house number a perfect square?
2) Is your house number greater than 50?
Assuming A's answers to be correct, B is able to infer house number of A from answers given by A.
Since B is able to infer house number of A from answers given by A the only possibility is that A has answered in 'yes' to both the questions. So that there are two possibilities: 64 and 81. And He can infer the house number of A only if his own house number is one of these two numbers.

Given that, C asks A when B is not present, 2 questions:
1) Is your house number a perfect cube?
2) Is your house number greater than 25?
Assuming A's answers to be correct, C is able to infer house number of A from answers given by A.
Since C is able to infer house number of A from answers given by A the only possibility is that A has answered in 'yes' to both the questions(since A answered 'yes' in reply to B's second question, so he must answer 'yes' to C's second question). So that there are two possibilities: 27 and 64. And He can infer the house number of A only if his own house number is one of these two numbers.

Given that A's house number is less than B and C's house number and A's house number is greater than 50, this implies that C's house number is 64 and B's house number is 81.

For getting house number of A, we will first use the following information:
Sum of house numbers of 3 individuals is perfect square multiplied by two.
Given that B's House Number is 81, C's House Number is 64, let A's house number be x. Also we know that 100 > x > 50. Now, x+81+64 is a perfect square multiplied by 2. This implies that (x+81+64)/2 = (x + 145)/2 is a perfect square. Since 100 > x > 50 this implies that
245/2 > (x + 145)/2 > 195/2. Now we look for a perfect square in the interval (195/2, 245/2)
=(97.5, 122.5)
Two Possibilities are 100, 121
Solving,  (x + 145)/2  = 100 give us x = 55
and (x + 145)/2  = 121 give us x = 97
Now using the condition that A's house number is less than house number of both B and C we rule out x = 97
Hence, A's house number is 55.