Logic Puzzle Update
Based on a great commenter suggestion, I've made each box clickable -- thus if you correctly solve the puzzle, you will be rewarded with an actual picture of what could very well have been the ring of King Wingbipeekaboo. Or else you get death.
I'll follow
steeldraco 's nomenclature for the boxes:
123
456
789
As noted by more than one commenter, the statement on #9 is tricky. The best strategy is to just ignore it until the end and let the other boxes dictate whether it should be true or false. Then at the end, check its validity.
Let's start with box #1, and let's assume that it is true. This automatically makes #3 true as well. But we know that exactly one of (#4,#5) plus at least one of (#7,#8) must be true no matter what. This adds up to a total of four which must then be true, violating the assumption that exactly 3 are true. Thus #1 must be false.
F23
456
789
Now let's look at #2 and #3 together. If #2 is true, then #3 must also be true. But if #2 is false, #3 must also be false, and here's why: In order for #3 to be true when #2 is false, there must be 4 true boxes or less. We know from the above discussion of box #1 that at least two of (#4,#5,#7,#8) must be true. Thus if #3 is also true, this makes 3 boxes that are true, violating what we've already established -- that #1 is false. If two of (#4,#5,#7,#8) AND #3 AND some other box are true, this violates the assumption that #2 is false and is thus invalid. Now we've eliminated all the possibilities for #2 false and #3 true, and thus have shown that #2 and #3 must be the same no matter what.
FFF FTT
456 or 456
789 789
OK, now we've got two cases to work with. For the first, we are limited to 4 total false boxes and we've already used three. Plus we know that either #4 or #5 MUST be false. Therefore the rest must be true, and we have two possibilities:
FFF FFF
FTT or TFT
TTT TTT
I've highlighted the possible locations of the rings red. In both cases the ring must be in the first column. In the first case, the ring is in a box labeled false, but we do not know which one -- either #1 or #4. In the second case, the ring is in a box labeled true, and we know that since #7 is true, it cannot be located there, and the only other possibility is #4.
OK. Let's go revisit the case where #2 and #3 are both true:
FTT
456
789
in this case, exactly four statements are true. However, two are already listed, and we know that there must be two trues in (#4,#5,#7,#8), so all the rest must be false. Not only that, but we know that exactly one each of (#4,#5) and (#7,#8) must be true.
FTT
45F
78F
By evaluating #6, we determine that the ring is not in the first column, so #7 must be true. Therefore #8 is false and is the location of the ring. #4 and #5 then follow and we get a solution of
FTT
FTF
TFF
OK, so our three possible solutions are:
FFF FFF FTT
FTT or TFT or FTF
TTT TTT TFF
Now we can evaluate the truthfulness behind box #9. If we look at the third possible solution (where #9 is false), we get the following logical statement: "You DO need to assume that the puzzle is solvable in order to solve it. However, as long as #9 is false, we get a singular solution WITHOUT having to make this assumption. So let's set this one aside and move on to the first two possibilities.
If we assume that #9 is true, then we have two possibilities. However, one of them leads to a degenerate case -- we can't tell which box has the ring. If we assume that the puzzle is solvable, we can rule out this case and claim victory with case 2. BUT, we had to make the assumption that the puzzle was solvable! This invalidates #9 being true!
It might seem that none of these three cases are possible, but in fact case 3 turns out to be completely valid. Why? Although we did not use the assumption that the puzzle is solvable to come up with the location of the ring (in box #8), we DID use it to rule out the other two cases, leaving a single solution:
FTT
FTF
TFF
The ring is in box #8.
I'll follow
steeldraco 's nomenclature for the boxes:
123
456
789
As noted by more than one commenter, the statement on #9 is tricky. The best strategy is to just ignore it until the end and let the other boxes dictate whether it should be true or false. Then at the end, check its validity.
Let's start with box #1, and let's assume that it is true. This automatically makes #3 true as well. But we know that exactly one of (#4,#5) plus at least one of (#7,#8) must be true no matter what. This adds up to a total of four which must then be true, violating the assumption that exactly 3 are true. Thus #1 must be false.
F23
456
789
Now let's look at #2 and #3 together. If #2 is true, then #3 must also be true. But if #2 is false, #3 must also be false, and here's why: In order for #3 to be true when #2 is false, there must be 4 true boxes or less. We know from the above discussion of box #1 that at least two of (#4,#5,#7,#8) must be true. Thus if #3 is also true, this makes 3 boxes that are true, violating what we've already established -- that #1 is false. If two of (#4,#5,#7,#8) AND #3 AND some other box are true, this violates the assumption that #2 is false and is thus invalid. Now we've eliminated all the possibilities for #2 false and #3 true, and thus have shown that #2 and #3 must be the same no matter what.
FFF FTT
456 or 456
789 789
OK, now we've got two cases to work with. For the first, we are limited to 4 total false boxes and we've already used three. Plus we know that either #4 or #5 MUST be false. Therefore the rest must be true, and we have two possibilities:
FFF FFF
FTT or TFT
TTT TTT
I've highlighted the possible locations of the rings red. In both cases the ring must be in the first column. In the first case, the ring is in a box labeled false, but we do not know which one -- either #1 or #4. In the second case, the ring is in a box labeled true, and we know that since #7 is true, it cannot be located there, and the only other possibility is #4.
OK. Let's go revisit the case where #2 and #3 are both true:
FTT
456
789
in this case, exactly four statements are true. However, two are already listed, and we know that there must be two trues in (#4,#5,#7,#8), so all the rest must be false. Not only that, but we know that exactly one each of (#4,#5) and (#7,#8) must be true.
FTT
45F
78F
By evaluating #6, we determine that the ring is not in the first column, so #7 must be true. Therefore #8 is false and is the location of the ring. #4 and #5 then follow and we get a solution of
FTT
FTF
TFF
OK, so our three possible solutions are:
FFF FFF FTT
FTT or TFT or FTF
TTT TTT TFF
Now we can evaluate the truthfulness behind box #9. If we look at the third possible solution (where #9 is false), we get the following logical statement: "You DO need to assume that the puzzle is solvable in order to solve it. However, as long as #9 is false, we get a singular solution WITHOUT having to make this assumption. So let's set this one aside and move on to the first two possibilities.
If we assume that #9 is true, then we have two possibilities. However, one of them leads to a degenerate case -- we can't tell which box has the ring. If we assume that the puzzle is solvable, we can rule out this case and claim victory with case 2. BUT, we had to make the assumption that the puzzle was solvable! This invalidates #9 being true!
It might seem that none of these three cases are possible, but in fact case 3 turns out to be completely valid. Why? Although we did not use the assumption that the puzzle is solvable to come up with the location of the ring (in box #8), we DID use it to rule out the other two cases, leaving a single solution:
FTT
FTF
TFF
The ring is in box #8.