Four kinds of puzzles
Doing science—as opposed to learning about science—requires creative problem solving, which we have previously discussed as “night science” . We solve problems when we figure out what experiment or analysis to carry out or when we try to make sense of observations and data. Doing this is akin to solving puzzles created for amusement, such as brain teasers or jigsaw puzzles, putting us in a very similar mental space. Such artificial puzzles form a microcosm of problem-solving. Working on them, we experience a fundamental asymmetry: they seem very difficult or even impossible while we are looking for the solution; but once we know the solution, it seems almost obvious.
Puzzles come in different flavors, each with their own premises that circumscribe the type of the expected solution. In the conceptually simplest class of puzzles, you are presented with all of the pieces and the possible types of connections—all you need to do is to figure out how they fit together. The archetype of this class is the jigsaw puzzle, where your solution efforts are rewarded through the global image that emerges from connecting the pieces locally. As an example of a mathematical jigsaw puzzle, consider the following (also shown in Fig. 1a):
How can you obtain the number 25 by combining all of the numbers 2, 4, 6, and 8 with three different operations out of +, −, *, and /?
It is possible to solve this puzzle by brute force, trying out all possible combinations—though there are many. As the scale of such puzzles increases, so does their complexity.
In the second class of puzzles, the parts are also clearly defined, but a logical leap is required to arrive at the solution. Such “logical puzzles”—often called brain-teasers—pose a well-defined problem, the solution of which frequently involves the use of mathematical tricks. For example, consider the following logical riddle (Fig. 1b):
Imagine you have 12 coins, 11 of which have the same weight. The remaining coin is either heavier or lighter than the rest. Can you find the odd coin with only 4 weighings? You have to use a “unique” digital balance scale that compares two weights. It outputs one of three symbols, corresponding to the following: “left is heavier,” “equal weights,” and “right is heavier.” You can distinguish the three symbols, but you do not know their meaning.
While finding the solution is not easy, its general structure is clear: in the first weighing, you put n coins on the left and n′ other coins on the right, and so forth for the three other weighings. That is all there is—there can be no tricks, such as messaging the scale’s inventor or melting the coins. What is the precise logic of choosing the solution? If we were to provide you with the solution immediately, it might seem straightforward. But trying to solve it, you will appreciate that it is not. What will be required is to shed light on the logical structure of the problem—a simplifying insight that makes the solution possible.
These first two classes of puzzles can be considered closed-world: the constituents of the solution and their possible connections are known in advance; the challenge is to assemble them in a meaningful way. In contrast, other classes of puzzles are open-world. Here, the answer is not inscribed within a closed box—you are missing important information on the components or the structure of the solution. Hence, in the third class of puzzles, you need to make a connection to a realm external to the problem formulation. Consider the following (Fig. 1c):
A man cooks nine meatballs for his ailing father. He gives them to his daughter to bring to her grandfather. To make sure that all meatballs arrive uneaten, he labels the pot containing the meatballs with the roman numerals “IX,” using a permanent marker. On the way, the girl eats three of the meatballs. She has the marker, though, and while she cannot erase her father’s mark, she can add something to it. What should she do so that her grandfather does not suspect that anything is amiss?
You might realize soon that it is impossible to get to a smaller number than 9 when staying within the roman numeral system. Thus, this puzzle requires you to connect the aspects presented in the puzzle description to something external to it. While the connection may be a simple one, finding it is complicated by the vast size of the search space.
Finally, in the fourth class of puzzles, one does not need to connect to a whole other world, but rather look outside the box. You need a deeper insight that requires a mental leap—a trick! Typically, this requires you to drop a constraint on the search space that was not part of the problem description, but that you added implicitly. Consider the puzzle of the nine dots shown in Fig. 1d, which you might have seen before:
Can you connect the dots with an uninterrupted line consisting of 4 straight edges?
Finding the solution using five connected lines is quite simple. If you are like most people faced with this problem for the first time, you may unconsciously restrict your lines to the virtual box circumscribing the outermost dots. We can find the solution only when we drop that implicit assumption and allow ourselves to think outside of this box. It is interesting to note that the solutions to such puzzles often make us laugh, because of the startle we feel when realizing an unexpected, alternative way to see the same set of facts.
These four puzzle archetypes can be arranged on a two-dimensional grid (Table 1). The first dimension concerns the completeness of the problem formulation: closed-world (Classes I and II) versus open-world (Classes III and IV). The second dimension concerns the type of insight required: finding connections (Classes I and III) versus the need to reframe the problem—either through an insight into the problem structure (Class II) or by moving our thinking out of the box (Class IV). If you know where you are on this grid, you know which type of puzzle you are in.