Here is a known riddle: See above a picture of nine dots. Now, connect the dots by drawing four straight, continuous lines that pass through each of the nine dots, and never lifting the pencil from the paper. If you are not familiar with the riddle, you can stop here and try to solve it yourself, before continuing the read. If you already know the solution, or just want to relax and enjoy the answer, it’ll come soon.
Some time ago we published an article called Infinitely Many… Solutions! in which we discussed a common urban legend where a problem is presented in front of an enthusiastic and maybe a bit unusual student. The student, instead of giving an expected solution, gives many alternative solutions, “rebelling the institution”, in his own way. Here we plan to do the same: As we said above, the riddle is well known, and many of you already know the solution, but maybe there are some alternative solutions? Let’s check it out (and for those of you who like riddles, we encourage you to find more solutions yourselves).
To those who are not familiar with the riddle, the task may seem easy. How hard it can be to find four connected lines that pass through each of the nine dots?! Normally, approaches similar to these will be attempted:
All these attempts use at least five lines. Therefore, soon comes the realization that it’s probably impossible to solve the riddle. But there is a solution – how come? The “Nine dots puzzle” is often used to demonstrate the “Thinking outside the box” approach, and encourage creative thinking, unconventional perspective, and fresh ideas.
The puzzle itself is an old one, and one of its first mentions is in Sam Loyd’s 1914 Cyclopedia of Puzzles (the same Sam Loyd who appears in “Alice in Palindrome Land” article). Sam Loyd’s original formulation of the puzzle entitled it as “Christopher Columbus’ egg puzzle.”
The way Sam Loyd presented the puzzle was an allusion to the story of Egg of Columbus that demonstrates how a brilliant idea or discovery often seems to be simple or easy after the fact. In the story, dating from at least the 15th century, Christopher Columbus is told that finding a new trade route was inevitable and no great accomplishment. As a response, Columbus challenges his critics to make an egg stand on its tip. After his challengers give up, Columbus does it himself by tapping the egg on the table to flatten its tip.
OK, OK, now to the solution of our riddle. The known outside-the-box solution does exactly that: Continues the lines beyond the imaginary box. No one said that there is a border or an edge and that you cannot draw beyond it. What previously seemed to be impossible now becomes simple, after you’ve extended your perspective on the topic.
But we promised more than that: If we already think outside the box, and understand that the created boundaries were artificial, what other artificial boundaries we’ve created that limited us to solve the riddle?
Another artificial boundary is created due to terminology. We said “connect the dots by drawing four straight lines”, but we didn’t say that we must connect the dots in the middle. We found an internet page where someone named Lars Hellvig, from Stockholm, Sweden, said that he saw a solution in a book about children’s inventiveness when given different problems and “free hands” to solve them. In the book, only three lines are used, and dots are not connected in the middle. It’s true that in mathematics, dots have no dimension (or have a zero dimension), hence has to be connected “in the middle”, but this fact is not mentioned or enforced in the riddle (to be honest, Sam Loyd did require to connect the eggs through their center, but this requirement was omitted over the years). It may seem like cheating, but in a way, it’s also a possible solution.
And we have more!
Here is another artificial limitation: The riddle requires “never lifting the pencil from the paper”, but it never says that we cannot go through the same line twice! That brings us to another solution:
Now, all these are nice, but there is another solution, where only one straight line is needed to solve the riddle. Don’t believe it? The problem is that you are probably reading this riddle on a screen. Leave the screen and go back to the real, physical, world! Now, try to solve this riddle with paper and a pencil. The solution requires some topological thinking: Connect the page in a spiral, putting one line of dots in front of another line, then a single line can be drawn connecting all nine dots – which would appear as three lines in parallel on the paper when flattened out.
Since we are already working on a paper, we can create an origami solution as well:
And now to the last solution in our article: What if instead of using paper, we will use a transparent sheet? Then, we can fold it in a way that each line of dots is strictly above the previous line, making the whole riddle contain only one line of dots (or even only one dot, if we fold it further). Now solving it with only one line is as simple as possible.
Telling someone to think outside the box isn’t the most useful recommendation most of the time, because it’s hard to follow. However, by practicing it, as we did in this riddle, one can achieve a more divergent way of thinking. Now, looking back at all these solutions, we already can say that at least here, we thought outside the box. Can you find more solutions?
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