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Logical Consistency Puzzles
In my view, some of the most mind-boggling logic puzzles are those that present statements, some of which are true and some false, with the objective being to figure out where the truth of the matter lies. As far as I can tell, this genre was invented by British puzzle-maker Hubert Phillips in the 1930s. In a version of this type, we are given three boxes on which labels are attached. The objective is to figure out in which box something is, such as a coin or a ring, on the basis of the logical consistency of the labels alone, which can be true or false.
Here’s an example:
There is an American quarter in one of three closed boxes, A, B, and C. The boxes have the following labels:
- Box A: The quarter is not in here.
- Box B: The quarter is not in here.
- Box C: The quarter is not in A.
If told that two of the labels are true and one false, could you tell where the quarter is?
The labels on A and C say the same thing—namely that the quarter is not in A. So, the two statements are either both true or both false. They cannot be false, since we are told that there was only one false statement. So, they must both be true. From this, we can deduce that the false statement is B (by elimination). So, contrary to what it says, the coin is in B.
This type of puzzle shows concretely what the logician's notion of logical consistency involves—matching statements in terms of their truth or falsity in order to assess if they are logically compatible or non-contradictory. As Aristotle, the master logician, aptly put it: “A thing cannot at the same time be and not be.”
Puzzles
1. A precious gold ring is in one of three closed boxes, A, B, and C. The labels on the boxes are shown below. Can you locate where the ring is, if all the statements are true?
- A. The ring is not in here.
- B. The ring is not in C.
- C. The ring is not in A.
2. An expensive diamond stone is in one of three closed boxes, A, B, and C. Can you locate it, if all the labels are false?
- A. The diamond is in here.
- B. The diamond is in here.
- C. The diamond is in B.
3. An expensive bracelet is in one of the three closed boxes, A, B, and C. Can you locate it, if two of the labels are true and one is false?
- A. The bracelet is in B.
- B. The bracelet is in A.
- C. The bracelet is not in A.
4. A gold coin is one of the three closed boxes, A, B, and C. Can you locate it, if two of the labels are false and one is true?
- A. The coin is not in here.
- B. The coin is in A.
- C. The coin is not in here.
5. A radiant ruby is in one of three closed boxes, A, B, and C. Can you locate it, given that there is only one true label (and two false ones) and that the ruby is in the box with the true label?
- A. The ruby is in here.
- B. The ruby is not in A.
- C. The ruby is in A.
Answers below...
1. B. Since the labels are true, it is easy to examine their consistency.
- A: "The ring is not in here." This is true, so it is not in A.
- B: "The ring is not in C." This is also true, so it is not in C.
This leaves B as the only possibility, since we have eliminated A and C. The statement on C ("The ring is not in A") is true, of course, but it changes nothing.
2. C. All three statements are false. So, the labels on A and B eliminate them as the diamond holders (since they are false statements). This means that the diamond is in C. C’s label is false, but changes nothing.
3. B. The labels on B and C contradict each other. So, one is true and the other is false. We have thus located where the false label is—either on B or C, but not A. This means that A’s label is true (since there are two true labels). It says that the bracelet is in B. And that’s where it is.
4. C. The labels on A and B contradict each other. So, one is true and the other is false. We have now located where the single true statement is—either on A or B, but not C, whose label is therefore false. It says that the coin is not in C. But this is false. So, contrary to what it says, the coin is in C.
5. B. The labels on A and C say the same thing—namely that the ruby is in A. So, they are either both true or both false. They cannot be both true, since we are told that there was only one true statement. So, they are both false. This means that B’s label is the only true one and, therefore, that B that contains the ruby. Its label simply says (truthfully) that the ruby is not in A.
