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The English word puzzle covers a broad range of meanings, referring to everything from riddles and crosswords to Sudoku and conundrums in advanced mathematics. As a generic label, it is a convenient one for classifying diverse manifestations of what is arguably a singular phenomenon within the mind—the desire to seek answers to challenging questions. The word was first used to describe a game in a forgotten book published around 1595, titled The Voyage of Robert Dudley Afterwards Styled Earl of Warwick & Leicester and Duke of Northumberland. It derives in all likelihood from the Middle English word poselen “to bewilder, confuse,” a meaning that certainly can be applied to solving many puzzles.
So, what is a puzzle? Scott Kim, a brilliant puzzle-maker himself, defines a puzzle insightfully as something that is fun and has a right answer. This does not imply in any way that puzzles are hardly trivial, or just fun-based. The object of recreational mathematics, for instance, is to explore mathematics through puzzles. However, even so, determining what constitutes a puzzle under the “recreational” rubric is unclear. As mathematician Charles W. Trigg remarked in a 1978 issue of Mathematics Magazine: “Recreational tastes are highly individualized, so no classification of particular mathematical topics as recreational or not is likely to gain universal acceptance.”
To tackle the issue of the meaning of puzzles in human life, the current crossword editor of the New York Times, and a brilliant puzzle creator, Will Shortz, came up with the term “enigmatology,” as a technical term for the scientific study of puzzles of any kind—verbal, mathematical, logical, visual, and so on.
One way to differentiate puzzles within enigmatology is to distinguish between what can be called “open” from “closed” puzzles. The latter do not hide the answer, but give it at the start as an “end-state” that must be reached. The fun in this case is how to get to the end-state. Examples of closed puzzles abound. They include Sudoku, the Rubik’s Cube, the Tower of Hanoi, and so on and so forth. In all closed puzzles, we are given a set of rules of how to reach the end-state. The challenge is how to achieve it.
Open puzzles, on the other hand, do not involve end-states or rules. The answer in open puzzles is never obvious; the fun in this case is figuring out what the answer is, not how to get to it. This is why closed puzzles are often described as producing an “Aha” effect.
Below I will present versions of five classic puzzles that fall, in my estimation, into the closed variety. If you are not already familiar with them, there is a strong possibility that they will produce an Aha effect on you, revealing what an open puzzle does to the mind. I have discussed a few of these types in previous blogs. The ones here are new formulations. The intent is to exemplify what is meant by an “open puzzle,” and how truly “enigmatic” it is.
Five Classic Examples
1. The following puzzle is a version of a genre invented by one of the greatest puzzle makers of all time, the Englishman Henry E. Dudeney (1857-1930).
A young girl who is an only child answers her cell and asks, “Who are you?” A woman’s voice responds, “Figure it out. Your father’s father is my father-in-law.” Who is the woman?
2. Here is a version of another of Dudeney’s inventions, which now constitute a puzzle category unto their own, known generally as “Logic Puzzles.”
In a certain company, the positions of director, engineer, and accountant are held by Barry, Jasmine and Shamila, but not necessarily in that order. The accountant is single and intends to remain so to his last breath. Shamila is married to one of Barry’s siblings. Both Barry and Shamila earn more than the engineer. What position does each person fill?
3. As is well known, Lewis Carroll (1832-1898) was not only a writer of children’s books and a mathematician, but also one of the greatest creators of puzzles of all time. Here is one of his classic nuts:
A bag contains one counter, known to be either white or black. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter?
4. The prototype for the following puzzle comes from the fertile mind of the late Martin Gardner (1914-2010), another great puzzle maker..
In a box there are 24 shoes, 12 colored green and 12 colored red; the shoes are made up of matching pairs, of course, both in color (either green or red) and orientation (either left-footed or right-footed). The shoes are all scrambled inside the box. With a blindfold on, what is the least number you must draw out in order to get a pair of shoes that match in color (2 green or 2 red) and orientation (a left-footed and a right-footed pair of the same color)?
5. This puzzle is based on a series invented in the Indian mathematician Dattathreya Ramchandra Kaprekar (1905-1986). It is a brilliant example of the type of puzzle in which we are required to complete a sequence logically:
What number comes next: 28, 38, 49, 62, 70, 77, 91, …?
Answers
1. The woman is the girl’s mother. The father of the girl’s father, who is her grandfather, is her mother’s father-in-law.
2. The accountant is a single person, whereas Shamila is married. So, she cannot be the accountant. She cannot be the engineer either, because we are told that she earns more than the engineer, whoever that might be. So, this leaves the position of director as the only possibility for Shamila. Now, consider Barry. He is not the engineer, since he too earns more than the engineer; he is not the director (Shamila is). So, by elimination he is the accountant. Thus, the only position left for Jasmine to fill is that of engineer.
3. Let B and W1 stand respectively for the black and white counters that might be in the bag at the start, and W2 for the white counter added to the bag. Removing a white counter from the bag entails three possible combinations of two counters, one inside and one outside the bag: (1) Inside: W1—Outside: W2; (2) Inside: W2—Outside: W1; (3) Inside: B—Outside: W2. In (1), the white counter drawn out is the one that was put into the bag (W2), and the white counter inside it (W1) is the counter originally in it. In (2) the white counter drawn out is the one that was originally in the bag (W1), and the white counter inside it (W2) is the counter that was put in. In (3), the white counter drawn out is the one that was put in the bag (W2), since there was no white counter originally in it. The counter that remains in the bag is a black one (B). In two of the three cases, a white counter remains in the bag. So, the chance of drawing a white counter on the second draw is two out of three.
4. In this type of puzzle, you cannot assume that luck is on your side, since it asks us that we “must” be sure of getting a matching pair. So, we must assume the worst-case scenario for the purposes of the puzzle. This means that all left-footed or right-footed shoes might be drawn out first. In the 24 shoes inside the box, twelve shoes are left-footed and twelve are right-footed, of different colors of course. So, we might draw out either all the 12 left-footed shoes or all the 12 right-footed shoes, no matter their color. Either way, the next, or thirteenth draw, will produce a shoe that will match the color of one of the shoes already outside and its opposite orientation. If the shoes drawn are all left-footed, the thirteenth draw will produce a right-footed shoe that will match one of the colors of the shoes outside; vice versa, if the shoes drawn are all right-footed, the thirteenth draw will produce a left-footed shoe that will match one of the colors of the shoes outside. So, you will ned to make 13 draws in total to ensure a match. Of course, you might be lucky and get a match with fewer draws—which is actually more likely in reality.
5. The answer is 101. First, add the digits in 28. The sum of its digits is 2 + 8 = 10. Now, add the result of 10 to 28 and you will get the next number in the sequence, namely 38. Now, add the digits in 38, 3 + 8 = 11. Add this result to 38 (38 + 11) to get the next number in the sequence, namely, 49. And so on.
