Let me first introduce myself and this blog, which is titled Total Brain Workout. I am a professor of anthropology at the University of Toronto, and one of my main areas of cultural research is puzzles. I have also been teaching an undergraduate course on puzzles (their history and cultural meaning) at the University for many years. It is one of the most satisfying courses I have ever taught because, by the end of it, the students not only develop puzzle-solving skills (which they may have thought they didn't have at the start), but also come out of it with a better perspective about the role of puzzles in human life.
This blog is modeled on puzzles that I put together for my recent book, The Total Brain Workout: 450 Puzzles to Sharpen Your Mind, Improve Your Memory & Keep Your Brain Fit, published by Harlequin Books. It will contain one or two puzzles, and (if relevant) the history behind them, to tease your brain and thus keep it fit. As I indicate in the preface to that book, it would seem that such apparently "trivial amusements" foster brain growth, by stimulating logical and creative thinking regions of the brain. Research has come forward to suggest (although not prove beyond a shadow of a doubt) that puzzles sharpen the mind, improve memory, and keep the brain fit throughout life, and especially later life. As a boomer myself, and a puzzle addict since my childhood, I welcome this news. If puzzles are to the brain what physical exercise is to the body, then let's do puzzles-not just for fun, but more importantly for brain fitness. And even if the research is not exactly what it is claimed to be, so what! Doing puzzles cannot hurt.
Puzzles are as old as human history. They are found in all cultures throughout time. One of the first documented puzzle-and still one of the most famous-is the Riddle of the Sphinx. According to myth and legend, when Oedipus approached the city of Thebes he encountered a gigantic sphinx guarding entrance to the city. The menacing beast confronted Oedipus, posing the following riddle to him, and warning him that if he failed to answer it correctly he would die instantly at the Sphinx's hands:
What has four feet in the morning, two at noon, and three at night?
The fearless Oedipus answered (paraphrasing his statement somewhat): "Humans, who crawl on all fours as babies, then walk on two legs as grown-ups, and finally need a cane in old age to get around." Upon hearing the answer, the astonished sphinx killed itself, and Oedipus entered Thebes as a hero for having gotten rid of the terrible monster that had kept the city in captivity for so long. Ironically, by solving the riddle the devastating prophecy, which Oedipus tried to elude-that he would kill his father (which he did unwittingly on the way to Thebes) and marry his mother, the widowed queen of Thebes-came true.
Why are we so intrigued by stories such as this one which revolve around puzzles? The answer might lie in the origin of the English word puzzle itself, which comes from the Middle English word poselen "to bewilder, confuse." And indeed, puzzles generate bewilderment and confusion, because they cannot be solved by applying any formula or method mindlessly. They always require a dose of creative, unconventional thinking, which psychologists call "insight thinking." This is essentially an intuitive grasp of a pattern or twist concealed by the puzzle.
Given their appeal, some puzzles have given origin to commonly-held ideas, such as the one that life is comparable to the three main parts of a day (the Riddle of the Sphinx). Others are the source of everyday expressions. Here is a brainteaser that gave origin to the expression "thinking outside the box." Many readers undoubtedly know it:
Without letting your pencil leave the paper, can you draw four straight lines through the following nine dots?
Those who may not have come across this puzzle before might tend to approach it by joining up the dots as if they were located on the perimeter (boundary) of an imaginary square or flattened box. But this reading of the puzzle does not yield a solution, no matter how many times one tries to draw four straight lines without lifting the pencil. A dot is always left over. It is at this point where creative thinking comes into play: "What would happen if I extend one or more of the four lines beyond the box?" That hunch turns out, in fact, to be the relevant insight. One possible solution is as follows:
Can you find the others? It should now be obvious why this puzzle gave rise to the expression "thinking outside the box," which entered the English language around the middle part of the twentieth century when people in business and education started referring to it as a prototypical example of what creative or "lateral" thinking is all about. It continues to be cited by psychologists as an example of how the mind tends to impose unnecessary limitations upon methods of attacking problems.
Who invented the puzzle? I have looked into several sources and have been able to trace it as far back as 1914, in the first edition of puzzlist Sam Loyd's (1841-1911) Cyclopedia of Puzzles. But the principle it embodies is probably older, as Martin Gardner indicates in his 1960 edition of Loyd's work (titled The Mathematical Puzzles of Sam Loyd).
The Nine-Dot puzzle is a 3 × 3 version of what can be called generally a Dot-Joining puzzle. Can you solve the Sixteen-Dot (4 × 4) and Twenty-Five Dot (5 × 5) versions? Again, you just connect the dots without lifting your pencil. How many lines are required in each of these two cases? Do you detect a correlation between number of dots and number of connecting lines?
As a final word on Dot-Joining puzzles, I should mention that, as with any puzzle genre, once the general principle involved in solving them is deciphered, the genre starts losing its appeal. However, like any good joke, Dot-Joining puzzles can be played on others over and over to great effect.
I await your answers, solutions, discussions, anecdotes, etc. for this particular puzzle, including any general formula for solving any general version (n × n) of the puzzle (if there is one). I also welcome suggestions for future puzzles on this blog. This is going to be fun! (Scroll down for the answers)
Each of the following constitutes only one possible solution.
Six lines are needed for this version of the puzzle. As mentioned other solutions are possible. All involve six lines.
Eight lines are needed for this version of the puzzle. Other solutions are possible.
Can we generalize? By making the Nine-Dot puzzle as complex as we desire (increasing the number of dots to 16, 25, 36, 49, etc.), a pattern seems to emerge through inspection. This pattern can be charted as follows:
Dots Lines Required
3 × 3 (3 + 1) = 4
4 × 4 (4 + 2) = 6
5 × 5 (5 + 3) = 8
6 × 6 (6 + 4) = 10
n × n n + (n - 2) = 2n - 2
I should point out that I have not tested this pattern beyond a 6 × 6 version of the puzzle. As with all inductively-derived formulas, there is no way to be sure that the formula works all the time. It thus can be called, simply, a working formula.