What Is a Logic Puzzle?

10 classic puzzles and what they reveal about the logical mind.

Posted Sep 06, 2020

The twentieth-century philosopher Ludwig Wittgenstein once wrote that: “Logic takes care of itself; all we have to do is to look and see how it does it.” These words bring out the crux of the problem regarding the meaning of logic—namely, it is impossible to define, so all we can do is to observe how it works as we use it. This actually became a major principle in the philosophical movement of twentieth-century pragmatism. 

What makes it difficult to pin down is that logic is not a single construct; it comes in various forms and styles of thinking. Following Wittgenstein, the best way to grasp what it is may be to illustrate it with puzzles that are built on some aspect of logic. One of the first to understand this premise was Lewis Carroll. He deconstructed syllogistic Aristotelian logic in two marvelous books—Symbolic Logic (1896) and The Game of Logic (1886).

For example: If all humans are mortal, and I am a human, then I must conclude that I too am mortal. This conclusion is clear-cut and inevitable and, in this case, matches truth and reality But always the trickster, Carroll also made fun of this kind of thinking, making up syllogisms that were patently silly, even though they were consistent with the rules of syllogistic logic. He called these sillygisms. Examples are the following:

  1. A prudent man shuns hyenas
  2. No banker is imprudent
  3. Conclusion: No banker fails to shun hyenas
  1. No bald creature needs a hairbrush
  2. No lizards have hair
  3. Conclusion: No lizard needs a hairbrush

In one of his classic puzzles, Carroll showed that logic can actually contradict what we might think is common sense. Here is a paraphrase of his puzzle:

You have two clocks. One doesn’t go at all, and the other loses a minute a day. Which should you prefer?

We might pick the clock that loses a minute a day since we would have, at the very least, an approximate time with this clock, while the stopped clock does not work at all. But Carroll chose the latter. 

To grasp his logic, let’s label the clocks as follows: (1) A = the clock that doesn’t go at all; (2) B = the clock that loses a minute a day. Clock B will have to lose 12 hours in order to become synchronized once again with Clock A. Since it takes 60 days to lose one hour, it will need 60 x 12 = 720 days to become synchronized again. That is just under two years’ time. So, it will take almost two years for the clock that loses one minute a day to show the correct time again. The stopped clock, on the other hand, shows the correct time twice a day. The stopped clock, Carroll concluded, is logically the better one. 

So, what is a logic puzzle? Like logic itself, it cannot be so easily defined. It might require us to draw a specific conclusion from a series of statements or facts; it might ask us to figure out certain relations present in given facts; it might require us to identify which element in a set does not belong in it; and so on. 

However, there are two general characteristics that apply to all logic puzzles. First, no specific background information or knowledge is required to solve them. Second, exact thinking is required, whereby facts are matched and evaluated for consistency. One of the great makers of logic puzzles in the twentieth century, Henry E. Dudeney, went so far as to claim that: “The history of puzzles entails nothing short of the actual story of the beginnings and development of exact thinking in man.” The puzzles here are versions of classic logic puzzles, designed to cast light on various aspects of logical thinking.

The French philosopher René Descartes refused to accept any belief, even the belief in his own existence unless he could “prove” it to be logically true. Descartes maintained that logic was the only effective way to solve all human problems, most of which were caused by the emotions and the passions. Like Mr. Spock in the Star Trek saga, he believed that logic could be used to great advantage for the betterment of the human condition because “errors” in thinking could be reduced to errors in logic and thus easily fixed. However, as Spock constantly found out, if only human life were so simple!

Puzzles

You may already be familiar with some of these classic nuts since they have been recycled over and over. But, even if one is familiar with them, the different versions might still produce some enjoyment.

1. Last week, three young men, Alex, Raj, and Francesco, and three young women, Jill, Caterina, and Shamila decided to go out dancing together. One of the women was dressed in red, one in green, and one in blue. The men also wore outfits in the same three colors. At one point, Alex, who was dressed in red, and who was dancing with Shamila, turned and said to Jill, who was dressed in green and dancing with Raj: “Not one of us is dancing with a partner dressed in the same color.” Can you tell what colors they wore?

2. On a table there five billiard balls, one of which weighs less than the other four. Otherwise, they are identical. What is the least number of weighings you will need to make using a balance scale to be sure that you can identify the culprit ball?

3. Five people were brought in for questioning yesterday. One was suspected of having robbed a bank. Here’s what each one said:

Virginia:         I didn’t do it.

Sonya:            I didn’t do it.

Doreen:         Sonya is innocent.

Mitch:             I didn’t do it.

Lisa:                Virginia is innocent.

Four of the suspects told the truth; one lied. Can you identify the robber?

4. A rare gold coin is placed in one of three boxes, which are then locked. Each box has an inscription written on it: BOX A: “The coin is in here”; BOX B: “The coin is in here”; BOX C: “The coin is in A”. Can you tell in which box the coin is if only one of the inscriptions is true?

5. In a box there are 21 balls, 2 white, 2 black, 14 green, and 3 red. With a blindfold on, what is the least number you must draw in order to be sure that you will get a pair of balls that matches in color (2 white, 2 black, 2 green, 2 red)?

6. This is a puzzle invented by philosopher Max Black in the 1940s; he designed it as an example to show what critical thinking might involve. It has come to be known as the "mutilated chessboard problem." Remove two opposite corner squares of a typical chessboard. A chessboard has 64 squares, alternating black and white. You are given a large number of dominoes—each one the size of two chessboard squares, with each one equally half black and half white, so that when you put it on the chessboard it will cover a black and white square perfectly. Is it possible to cover the entire chessboard with the dominoes? You cannot place the dominoes on top of each other, and they must lie flat.

7. A boy answers his cellphone. “Who is speaking?” he asks. A man answers: “Guess Your mother’s mother is my mother-in-law. Who is the man?

8. If there are six pairs of black gloves and six pairs of white gloves in a drawer, all mixed up. With a blindfold on, what is the least number of draws that are required in order to guarantee a matching pair of black or white gloves?

9. Six friends—Amber, Jill, Eddy, Kristina, Sierra, and Louie—got together for a friendly foot race. Amber beat four others, but came in behind Jill. Eddy beat Kristina. Sierra beat Louie, but not Kristina. Who came in third?

10. Consider the following “odd-one-out” puzzle. In the set of digits, one does not belong. Which one? This is rather tricky. Note that the question uses the word digits, not numbers.

1234, 8011, 7102, 8211, 5212, 1117, 2332

Answers

1. Alex is dressed in red, so Shamila cannot be dressed in the same color. She is also not dressed in green, because Jill is. So, by elimination, Shamila is dressed in blue. Now, since Jill is dressed in green and Shamila in blue, this means that Caterina is the one dressed in red. Now, Raj is dancing with Shamila who is in blue; so he is not dressed in that color, nor is he dressed in red, because Alex is. So, he is dressed in green. This leaves Francesco as the young man dressed in blue. Summary: Alex—red; Raj—green; Francesco—blue; Jill—green; Shamila—blue; Caterina—red.

2. We will need two weighings.  We put one ball to the side. This leaves four balls to weigh. So, we put two on one pan of the scale and two on the other. If the pans balance out, it means that all the balls on the scale weigh the same and, thus, that the culprit ball is the one we had put aside. But we cannot assume that this will happen, since that is luck, not logic. We are told to be sure, not lucky. So, we consider the other possibility—one of the two pans rises and the other falls. This means that the one that rises holds the suspect ball, since it weighs less. We now discard the two good balls, and put each of the two other balls, one of which is the culprit ball, on separate pans. Again, the pan that falls will contain the culprit ball. Summary: It will take two weighings to be sure that we will identify the culprit ball.

3. Virginia and Lisa say the same thing in different ways—namely that Virginia is innocent. So, their statements are either both true or both false. They cannot be false since we are told that there was only one false statement in the set. So, they are both true. Similarly, Sonya and Doreen say the same thing—namely that Sonya is innocent. So, their statements are either both true or both false. They cannot be false since there was only one false statement in the set. So, they are both true. To summarize, the four truth-tellers are Virginia, Sonya, Doreen, and Lisa. This means that Mitch is the liar and, contrary to what he says, he’s the robber.

4. If the inscription on A is true, then so is the one on C—since both indicate that the coin is in A. This means that they are either both true or both false. They cannot be true since we are told that only one of the inscriptions was true. So, they are both false. This means that the inscription on B was the true one. As it says, the coin is inside B.

5. We cannot assume luck, since we must be sure to draw out a matching pair. So, let's assume a worst-case scenario—that is, we draw a white, black, green, and red ball in some order. These are the colors of the four balls that are now outside the box. Now, the next draw from the box  will produce a ball in one of these colors—hence a match with one of the balls already outside. So, five draws will be needed in total to be sure. 

6. It is not possible. The opposite-corner squares that were removed are of the same color. That is, with two white, or two black, squares taken from the board, there will not be an equal number of black and white squares left behind. And each domino must cover one black and one white square.

7. He is the boy’s father. The mother of the boy’s mother, who is his grandmother, is the mother-in-law of his father.

8. We might draw all 12 left-handed gloves, of both colors, as a worst-case scenario. However, the thirteenth glove will be a right-handed one and also match one of the previous twelve already outside in color. Thus,  13 gloves have to be drawn out in order to ensure that we get a pair of matching gloves.

9. There are six places: first place, second place, third place, fourth place, fifth place and sixth place. From the fact that Amber (A) came in before four others but after Jill (J), we conclude that Amber came in second and Jill first: J—A. Among the four others, we are told that Eddy (E) beat Kristina (K) and that Sierra (S) beat Louie (L) but not Kristina: E—K— S—L. We join this part of the formula to the other part above to get: J—A—E—K—S—L. It shows that Eddy came in third.

10. Except for 8311, the digits together add up to 10.