What Makes a Good Puzzle?
Eight classic examples.
Posted Jan 04, 2019
Puzzles are experiments in complex and varied thinking, providing satisfaction and enjoyment in their own peculiar ways. Henry E. Dudeney, one of the greatest puzzle-makers of all time, put it as follows: “Puzzle-solving, like virtue, is its own reward.”
But not all puzzles are the same—some seem to be more appealing and popular than others. Sudoku, for example, has broad appeal, perhaps because its rules are easy to understand while still providing considerable challenge. Achieving a completed grid with the numbers in their appropriate cells tends to produce a sense of satisfaction or, as Dudeney phrased it, “its own reward.”
So, what makes a good puzzle—a puzzle that is self-rewarding or satisfying in itself? Like musical tastes, particular types of puzzle appeal to different people. Nevertheless, some puzzles, like some kinds of music, seem to have broader appeal than others. Like music or the other arts, the best kinds of puzzles can be said to have a certain aesthetic allure. The more puzzles produce what psychologists call the “Aha effect,” the more aesthetically-pleasurable they seem to be. As the British puzzle-maker Hubert Phillips put it in his 1937 book "Question Time," solving some puzzles provides an intellectual “kick,” which results from discovering the pattern, trap, or trick they conceal. Interestingly, a phrase similar to “Aha” (in Egyptian) is found in the "Ahmes Papyrus," one of the first collections of math puzzles in history, dating from 1650 BCE.
I have chosen eight classic puzzles that—in my opinion—produce the Aha or aesthetic effect. Solutions are non-obvious and require a blend of logic, imagination, and (in some cases) lateral thinking. As mentioned in virtually all previous blogs, this type of mental engagement is very likely to reap benefits for the brain.
1. Let’s start with one of Dudeney’s classic inventions, which he introduced in the July 1924 issue of Strand Magazine. It has come to be known as an alphametic. You are presented with an arithmetical operation concealed by words. The goal is to reconstruct the original operation by determining what numbers the letters stand for logically. Below is Dudeney’s puzzle:
SEND + MORE = MONEY
2. For my second choice, I have gone with a famous lateral thinking puzzle. I am not sure who invented it. I remember seeing it in a wonderful puzzle collection put together by Paul Sloane, titled "Lateral Thinking Puzzlers," published in 1991:
A person walks into a bar and asks for a glass of water. The bartender reaches under the counter, takes out a gun, and aims it at the man. The person says thank you and leaves. What happened?
3. Here is another classic lateral thinking puzzle, apparently devised by Albert Einstein. It goes as follows:
A group of nature aficionados, having pitched camp, set forth to go and photograph bears. They walk 15 miles due south, then 15 miles due east, where they sight a bear. They return to camp by traveling 15 miles due north. What was the color of the bear?
4. The following puzzle is found in many puzzle collections, but I am not sure who was its inventor:
A bottle and a cork together cost 55 cents. The bottle costs 50 cents more than the cork. How much does the cork cost?
5. Trickery is one of the ingredients of a good puzzle. Below is a well-known trick puzzle that causes consternation in many who come across it for the first time:
Lucia has seven daughters. Each daughter has one brother. How many children does Lucia have?
6. Below is another of Dudeney’s mind bogglers, which he published in the Strand Magazine (volume 77, 1929):
Arrange all the 10 digits in three arithmetical sums, employing three of the four operations of addition, subtraction, multiplication, and division, and using no signs except the ordinary ones implying those operations.
7. The next puzzle type, invented by the late Martin Gardner, involves deducing what number of draws required to make a match. I have discussed this genre in previous blogs:
In a box there are 10 balls, five white and five black. With a blindfold on, what is the least number you must draw in order to get a pair of balls that match in color (two white or two black)?
8. One of the most famous of all arithmetical puzzles comes from the pen of Italian mathematician Niccolò Tartaglia (1499-1557):
A man dies, leaving 17 camels to be divided among his heirs, in the proportions 1/2, 1/3, 1/9. How can this be done?
1. The answer is: 9567 + 1085 = 10652
2. The person had the hiccups, requesting a glass of water to help get rid of them. The bartender took out the gun out, instead, to scare the person’s hiccups away. It obviously worked.
3. How can the group members travel as stipulated and end up back at the camp? On a two-dimensional surface this is, of course, nonsensical. But the earth’s surface is spherical, not planar. The camp is pitched at the North Pole, and the travel directions described by the puzzle will lead the group back to the camp, no matter how far east they go. Hence, the bear is a polar bear, which is white.
4. If the puzzle is read in a cursory or unreflective way, one might come to the erroneous conclusion that the cork costs five cents. If such were the case, then the bottle (costing 50 cents more) would cost 55 cents, and the total cost would be 60 cents, not 55. But that is not what the puzzle states. The appropriate solution can be shown by setting up an equation. Let x be the price of the cork. This means that (x + 50) is the price of the bottle (which stands for “50 cents more than the price of the cork”). The two prices together add up to 55 cents. The relevant equation is, therefore: x + (x + 50) = 55. Solving it produces: x = 2 ½. The cork thus costs 2 ½ cents. This means that the bottle, being 50 cents more, cost 52 ½. Together, the prices add up to 55: that is, 2 ½ + 52 ½ = 55.
5. She has eight children, namely, seven daughters and one son. The son is, of course, a brother to each and every daughter.
6. Dudeney’s solution is the following one. Note that all the digits are used, including 0 (in the number 20):
7 + 1 = 8
9 - 6 = 3
4 × 5 = 20
7. The answer is three. First-time solvers of this puzzle type (as discussed in previous blogs) might be duped into thinking erroneously because of the way the puzzle is presented. So, it is worth going through the solution in detail. Suppose the first ball we draw is white. If we are lucky the next ball will also be white, and it’s game over. The same reasoning applies to drawing two black balls in a row. But we cannot assume this lucky outcome, called a best case scenario, because the puzzle tells that we must get a matching pair, luck notwithstanding. So, we must, on the contrary, assume the worst case scenario—that is, that the first two draws produce two balls of different color. Let’s assume we draw a white ball out first. Then, under this scenario, we will draw a black ball next. Thus, after two draws, we will have taken out one white and one black ball from the box. Obviously, we could have drawn a black ball first and a white one second, under the same scenario. The end result would have been the same: one white and one black ball. Therefore, no matter what color the third ball we draw is, it will match the color of one of the two we had already pulled out. If it is white, we will have two white balls; if it is black, we will have two black balls. So, the least number of balls we will need to draw from the box in order to get a pair of balls that matches is three.
8. Dividing up the camels in the manner decreed by the father would entail having to split up a camel. This would, of course, kill it. So, Tartaglia suggested “borrowing an extra camel,” for the sake of argument, not to mention humane purposes. With 18 camels, he arrived at a practical solution: one heir was given 1/2 (of 18), or 9, another 1/3 (of 18), or 6, and the last one 1/9 (of 18), or 2. The 9 + 6 + 2 camels given out in this way, add up to the original seventeen. The extra camel could then be returned to its owner. Is this really a solution? I leave that decision up to the reader.