Below is fairly well-known puzzle attributed to Albert Einstein. Some claim that 98% of the human population cannot solve it. While this claim is probably unsound, it does point to the severe difficulties people experience when working towards its solution. It is my belief that such difficulties are entirely avoidable. Through an exploration of one illustrative solution, I hope to shed light on the art of learning. Specifically, I intend to show that this puzzle is not inherently difficult; but is interpreted as such because human beings all too often fail to understand the power of abstraction.
Five men of different nationalities and with different jobs live in consecutive houses on a street. These houses are painted different colors. The men have different pets and have different favorite drinks. The following clues are provided:
- The English man lives in a red house
- The Spaniard owns a dog
- The Japanese man is a painter
- The Italian drinks tea
- The Norwegian lives in the first house on the left
- The green house immediately to the right of the white one
- The photographer breeds snails
- The diplomat lives in the yellow house
- Milk is drunk in the middle house
- The owner of the green house drinks coffee
- The Norwegian’s house is next to the blue one
- The violinist drinks orange juice
- The fox is in a house that is next to that of the physician
- The horse is in a house next to that of the diplomat
Determine who owns a zebra and whose favorite drink is mineral water.
1) Extract needless symbolism.
Assigning meaningful variable names not only saves space, but if effectively relieves any symbolic baggage from the problem. Here, numbers differentiate between the five specified characteristic and letters identify individual members of the set.
2) Quantify the problem.
The next phase in our process of abstraction is quantification. Rather than dealing with these extensive logic chains, it is arguably simpler for the brain to visually interpret the data. As we will see, this technique is extremely powerful and reduces this convoluted verbal maze into a straightforward fill-in-the-blank puzzle matrix. So simple, in fact, that I am convinced that anyone determined enough could solve it. Below illustrates one such quantization. As you can see, the house-trait pairs have been represented in a 5×5 matrix, and puzzle pieces have been fabricated, each labelled with the associated clue. The problem is solved by fitting all of these pieces onto the puzzle and identifying the missing squares. The only complication is that two of the puzzle pieces, the two closest to the matrix, can be mirrored about the y-axis. Note that the ribbons above the puzzle pieces represent the associated clue. Blocks 1E, 2E, and 3C have already been filled out based off of clues 5, 11 and 9 respectively.
All we have done up to this point is clarify and illuminate the problem before us. While the previous simplifications and abstractions take time to implement, they are fundamentally trivial constructions. I have concealed the next phase of the solution should you choose to attempt the solution for yourself. The purpose to my writing is not to expound a particular implementation of the solution (although the following is instructive), but to discuss the underlying principles behind effective situations.
3) Solve the problem.
While a brute-force approach would be relatively straightforward, I will illustrate the solution via a more conventional logic tree. Note that the following is not necessarily the most elegant path. I have chosen it for its simplicity and its illustrative strategic methodology. One quick note: we will represent puzzle pieces by (clue number) and columns by [column name]. We will also symbolize placement decisions via the notation Place(clue number, column name). Our working strategy is to find the path of least choice. Here, (8&14) can only fit into only two locations, and is therefore the least complex logical path.
Assume (6&10) is placed in [M&R]. It then becomes apparent that (1) has only one allowed placement:
Here, (8&14) contains the last piece from row 2. Therefore, it must be mirrored about the y-axis and placed. From here, the placement of (4) and then (12) become apparent:
However, we have reached an impasse that can be understood through the Pigeonhole Principle. In the above illustration, (2) and (3) and (7) are mutually exclusive (they all have at least one row in common with another) and may only be placed in [M] and [R]. Just as we know that three pigeons cannot fit into two holes, we can be certain that Path 1 contains no solutions. Since we began our solution by exploring the first possible placement of (6&10), we must know turn to the second.
This path begins with the alternative placement of (6&10) in [R&FR]. As before, (1) and then (8&14) have only one available location:
We have now reached another fork in our journey. There are no mandated placements, so we search for limitations to our choices. One such limitation is that (4) can only be placed in [L] or [R]. Therefore, we once again branch into two paths.
We start by assuming that (4) is in [R]. Given this assumption, we can ascertain the implied locations of (2) and then of (3):
However, in the resultant figure above we simply have no room for (12). We can safely conclude that Path 2a is incorrect and proceed to Path 2b.
Since (4) cannot be in [R], it must be in [L]. This mandates the location of (12) in [R]. This in turn means that (3) must be in [FR].
To our delight, we can now appreciate that Path 2b is the only correct logical journey through our puzzle. The concluding steps are given below, and the desired quantities are shown in yellow:
Finally, our symbolic table can decode this information to provide the desired solution: The Norwegian man drinks the mineral water and the Japanese man owns the zebra.
4) Extract significance.
Hopefully, you will have been able to follow the logical journey above. Please note that I explored every logical journey in order to establish the unique nature of the solution. Given the right setup, however, this process essentially becomes a 5×5 jigsaw puzzle. While there are many legitimate interpretations of our results, perhaps the most useful is the implications of human logical capacity. I would posit that much the human population could easily, albeit not as explicitly, reproduce the logical chain of events required to solve our matrix. The lesson here lies in the reinterpretation of the problem, which transformed the problem from a convoluted mess into a simple puzzle. It is not superhuman ingenuity, but rather this process of chipping away at problems until you uncover their infastructure, that has been the vehicle for the humanity’s greatest accomplishments.