While wandering around the world, two logicians make the grave error of trespass
ID: 3119957 • Letter: W
Question
While wandering around the world, two logicians make the grave error of trespassing on the Mad Duke's territory. They are captured and brought before the Duke.
"MWAHAHAHAHA!" cackles the Mad Duke, "I will give you a chance to earn your freedom."
The logicians are imprisoned in two cells on opposite sides of a great tower, which sits in the exact geographical midpoint of the Mad Duke's dukedom.
"In my dukedom, there are either 10 or 13 obelisks,” proclaims the Mad Duke, “which are distributed unevenly across my lands.
"From your two cells, each of you can only see one half of the dukedom.
"Your cell walls are too thick for you to communicate with each other.
“Each day at 5pm, I will visit each of your cells and ask you if you know how many obelisks are in my dukedom.
"If you tell me the correct number, you will both be freed.
"If you tell me the wrong number, you will both be killed.
"If you choose not to answer, you will have another opportunity to answer on the next day at 5pm.”
At the end of the first day, neither logician believed that she had enough information to determine how many obelisks were in the entire dukedom.
"Well," chuckled each logician to herself, "looks like it'll be a little while before we can get back on the road."
What is the longest possible number of days that the two logicians could spend in the tower before they will be able to determine whether there are 10 or 13 obelisks in the dukedom?
Explanation / Answer
X= number of obelisks visible to 1st logician
Y= number of obelisks visible to 2nd logician
Since no logician could answer on the first day, we can say that both X and Y were less than 11, else the person who is seeing 11 or more obelsisk would have confirmed the total to be =13
Thus X+Y=10 or 13 and X,Y <11
X= {0,1,2,3,4,5,6,7,8,9,10}
Y= {0,1,2,3,4,5,6,7,8,9,10}
Since X said he doesn't know that means he has anywhere from 0 to 10 obelisks
Now Y also said he doesn't know that means, If Y had 0,1 or 2 obelisks visible then he could have assumed that X had 10 but a total of 11 or 12 wouldn't be possible. But since Y said he doesn't know on first day, he doesnt have 1 or 2 obelisks visible. As in, if Y had 0, 1 or 2 obelisks visible, that would mean X has either 10-0 =10 or 13-0 = 13 (for Y=0) 10-1 =9 or 13-1=12 obelisks visible (for Y=1) and X has either 10-2=8 or 13-2=11 obelisks visible. But X having 11 or 12 isn't possible so Y cannot be = 0, 1 or 2
Thus after first day the followign possibilties stay:
X= {0,1,2,3,4,5,6,7,8,9,10}
Y= {3,4,5,6,7,8,9,10}
DAY 2:
Since the game has to conitue for longest duration, we have to look at cases where X isn't able to comment on second day too. X knows that Y has atleast 3 obelisks visible. So if X has 10 or 9 or 8 visible, he can easily conclude that there are total of 13 obelsisk since he can see at least 8 and there are at least 3 on the other side so total will have to be =13. Hence for X to not be able to answer on day 2 he cannnot see 8 or 9 or 10 obelisks.
Thus X= {0,1,2,3,4,5,6,7}
Now Y knows the available possibilities of X. For Y to not be able to answer, ,let's consider this, he knows that X has a minimum of 0 obelisks. If Y can see <6 obelisks that would mean that there are a total of 10 obelsisk as the maximum that X can have is =7. So if Y has <6 then total cannot be =13 so it would be =10 and Y would be able to answer. Hence Y cannot have value <6
Y= {7,8,9,10}
Thus after end of day 2 the possibilities are:
X= {0,1,2,3,4,5,6,7}
Y= {7,8,9,10}
DAY 3: Now X knows the possible values of Y are 7 or 8 or 9 or 10. Now if he can see 0,1 or 2 obelisks that would mean that Y can haev amaximum of 10 so a total of 12 is max which means 13 isn't possible. So he can say that the answer is 10 but since we are to take this longer he caanot say this. So X cannot be 0 or 1 or 2
X= {3,4,5,6,7}
Y= {7,8,9,10}
Now Y knows the possible values of X. He knows that X is at least =3. If Y is 8 or 9 or 10 that would mean that a total of 10 isn't possible since 8+3 is >10 so it can be said with certainty that the total is 13. But since we shouldnt be able to say that Y cannot be 8 or 9 or 10. That leave Y =7 at the end of day 3.
Thus, by end of day 3:
X= {3,4,5,6,7}
Y= {7}
DAY 4: Now X knows that Y can see 7 obelisks. So he can comment with certainty total how many obelsis are possible since he will be able to see either 10-7 =3 or 13-7 =6 obelisks. Thus if X see 3 obelsisk he will comment the total obelsisk =10 and both will be set free. Similarly if X can see 6 obelsisks he can comment the total number to be =13 and both will be set free.
Thus the longest possible time the two logicians could spend in the tower is = 4 days and will be set free latest by 5pm on day 4.
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