The inhabitants of Ganymede are all infamous for being either knights, liars or knaves. At the spaceport, four flight schedulers with Jovian Galactic Corp. were up-dating their manager on which two inter-lunar shuttles had the best and worst ‘on-time’ departure records the previous month (about 7 Earth days).
From the following discussion, what type of Ganymedan are each of the four, and which two of the fleets' 21 shuttles (each assigned a different two-digit prime number) scored the best and worst departure ratings?
1) B. "The two numbers are less than and greater than 50 respectively."
2) C. "B. is a liar."
3) D. "The difference between the two numbers is a square."
4) A. "The difference between the two numbers is a two digit palindrome."
5) C. "The difference between the two numbers is an odd number."
6) A. "The sum of the two numbers is greater than 100."
7) B. "The sums of the two digits in each number are both primes."
8) D. "The average of the two numbers is even."
9) A. "D. is a liar."
10) C. "The sums of the two digits in each number are both even."
11) D. "C. is a knave."
12) B. "The sum of the two numbers is less than 100."
13) C. "The two numbers are both above 50."
14) D. "The two numbers are both below 50."
15) B. "The average of the two numbers is odd."
16) A. "The shuttle with the best ‘on-time’ departure rating has the higher number."
1) Determine types (assume there is at least one knight):
a.) D is either a liar or a knave because the average of the two numbers cannot be even if their difference is a square.
b.) C is either a liar or a knave because the difference between the two numbers cannot be an odd number.
c.) A and B are internally consistent, but they disagree with each other, so one must be a knave or a liar and the other a knight.
d.) If A was a knight:
-D would be a liar (9).
-C would be either a liar or a truthteller.
-C cannot be a truthteller, so C would be a liar.
-the numbers, then, MUST be 61 and 83, because the sums of the two digits in each number could not be even (10). however, their average would then be even, which cannot happen, because D says it is, and D is a liar.
-therefore, A is not a knight.
-therefore, B is a knight.
e.) C is a liar, because all of C's statements are false if all of B's statements are true.
f.) D is also a liar, by the same reason. (for (3), if the sums of the digits of the two numbers are both primes (7), the difference between them cannot be a square because one or both of those pairs must be comprised of two odd digits, which would add up to an even number, which would not be prime.)
g.) A is a knave, because A is not a knight (1-d.), and not a liar (9).
To summarize: A is a knave, B is a knight, C is a liar, D is a liar.
2) Find the numbers:
a.) B's criteria mean there can be one of only two pairs of numbers: 23 and 67, or 61 and 29.
b.) A is a knave, so some of what A says must be true. The difference between the first two numbers, 23 and 67, is a two-digit palindrome (44), so that is true and those are our numbers.
c.) If (16) were true, then A would have more true statements than false statements, so (16) must be false and 23 has the best rating and 67 has the worst.
Posted by Maria
on 2011-05-27 22:49:09