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John and Peter participate in a treasure hunting competition. The organizer places a gold medal in one of the lockers. The locker rack contains 49 lockers, each of which has a name indicating its row and column. The organizer gives both of them a list of 10 possible treasure locker location as shown below. The organizer will tell John the treasure locker row (A to G) and tell Peter the treasure locker column (1 to 7).

John: "I have no idea where the treasure locker is, and I do not think Peter will know.".

Peter: "I have no idea just now, but I know the treasure locker location."

John: "I also know where the treasure locker location is".

So, where is the treasure locker location? Please explain your answer.

If John

Doesn’t think Peter will know, he must be sure that the gold medal isn’t in A5 or B3. He therefore mustn’t have been told A or B.

Eliminated lockers:

A2, A5, B3, B6, B7.

The medal is therefore in

rows D or F.

Now, If Peter

was told 1, he still wouldn’t be able to find out the locker after John’s statement, because D1 and F1 are both possible lockers. So the gold medal isn’t there.

More eliminated lockers:

D1, F1

Peter must’ve been told

2, 6, or 7, because he would be able to uniquely determine the locker after John’s statement. There are three possible lockers: F2, D6, and F7.

If John was told

F, then by Peter’s statement, F1 would be eliminated in John’s mind. However, John would not be able to discern between F2 and F7. If John was told D, he would know the locker was D6. This is the only locker where both would know the location given their deductions.

Therefore the gold medal is in

Locker D6.

I have a different answer. It might be a little lateral-thinking, but word choice matters! The treasure is in locker:

F2

The first thing to really pay attention to is when John says:

"I do not think Peter will know".

Why didn't he say I know for a fact Peter does not know? Because he does not know for a fact that Peter does not know. It's like when I go with my friend to play roulette and he puts all his money on 32. I try to talk him out of it by saying "take your money back, I do not think 32 will come up". Why did I say that? Do I know for sure 32 will not come up? Of course not, it's just that it's so unlikely that I do not think it will this time. Same with John, he does not know that Peter for sure does not know where the treasure is, it is just that it is unlikely that he does.

Having determined this, the next step is:

So we cannot eliminate row B, because if John was given row B then Peter could have been given columns 3, 6 or 7. Only if he was given column 3 would he know where the treasure is - that's a 33% probability and unlikely to happen in a sense that an even that occurs 66% of the time is more likely to happen than one that occurs only 33% of the time. Right? That is why John says he does not think Peter will know.

The only row we can eliminate is row A. If John was given row A then Peter would have a 50% chance to either know or not know where the treasure is, so John would be unable to say that he doesn't think Peter will know, because with a 50% chance he cannot take a position of what is more likely to happen - Peter knowing or not knowing.

It's a breeze now:

Now Peter says he didn't know where the treasure was, but now he knows. So Peter was not given columns 3 or 5 because then he would know right away. The only way for him to know where the treasure is after only eliminating row A is if the treasure is in F2. Now for John to know where the treasure is after hearing Peter would have to mean that he was given row F and he also deduced that the treasure is in F2.

And that's that.