Earlier today I asked you to turn this
into this:
The easiest way to solve this puzzle is so straightforward that when you understand it, it is hardly a puzzle at all! All you needed to do was follow the instructions…
I told you that the strands in the braid cross six times. That was a clue.
I’m assuming we all know how to braid. You cross the leftmost strand over the central one, then the rightmost over the central one, then the leftmost, and so on. In the illustration below, 1 crosses 2, then 3 crosses 1, which is now in the centre. The next step would have 2 (on the left) crossing 3 (in the centre), and so on.
For a moment let’s forget that the three strands join at the top end and the bottom end. Start at the top end and braid. Do six crossings since I told you that there are six crossings in the solution. (It is quite fiddly, which is why I recommended you use plastic. Paper may rip.)
If you pinch the sixth crossing between thumb and forefinger, you should have something that looks like the illustration below. (Or watch the video at the top of the post, which makes it much clearer).
Each time we made a nice braid crossing on one side, it was countered by an ugly twist on the other. After six crossings, the side to the left of my thumb is the pattern we are aiming for in the solution - and the side to the right is a crazy mangle of paper/plastic/leather.
What now? Well, we’re in luck. The mangle will disappear. Eventually. Use your other hand to thread the right end through itself a few times and the strands perfectly untangle. Readjust the braid so it flows evenly down the strands.
Voila! The impossible braid is possible after all.
This solution is not particularly elegant, but it works. And it is the easiest to work out for yourself. Sometimes the answer to a question is the simplest response. The question said braid, so braid!
There are fancier ways to make the impossible braid with fewer twists, which are less likely to tear paper. These solutions can be found on the web. My favourite is this one by an American leather store for making dog collars and wristbands.
If this challenge whet your appetite for braiding puzzles, here’s another one, which has historic links to the physicists Niels Bohr and Paul Dirac. You need to get the scissors back out, cut out two pieces of cardboard, and join them together with three pieces of string. The result is topologically equivalent to the strip of paper/plastic with two slits that we braided above.
It is also important to mark the front face of each piece of cardboard with marker pen.
The game that we are going to play is to keep the left piece of cardboard fixed, and rotate the right piece for a full rotation. To make sure we know it is a full rotation, after any rotation the black mark on the cardboard must be face up.
There are six ways to rotate the right cardboard: it can rotate upwards or downwards (as if the middle string is the axis); it can rotate through the top strings from the front, as illustrated below, or from the back; it can rotate through the bottom two strings from the front and back. (If you watch the video above, it makes it much clearer.)
This rotation will end up like this:
The aim of the game is to untangle the string without rotating the cardboard. All you are allowed to do is to move the cardboard between the strings keeping the cardboard flat.
Here’s why the game is interesting. If you make a single rotation of the right piece of cardboard, as above, then it is impossible to untangle the strings without further rotations.
But if you make a second rotation of the right piece, such as below, then it is always possible to untangle the strings. For example, by adding this rotation to the first one:
We will produce this tangle:
This looks like more of a mess! But it isnt. This tangle can be untangled while keeping the cardboard facing you at all times, in other words, using no further rotations.
I’m not going to show you how to do it. But it is really fun and satisfying to do. Once you have solved it, then start with no tangles and make any two rotations with the right cardboard. Whichever two rotations you make the tangle will always be solvable.
We can generalise and state that after an odd number of rotations the model can never be untangled, and after an even number of rotations it can always be untangled. In a certain sense, the second rotation cancels out the first rotation - even though it looks like it is causing more entanglement.
The fascinating property that double rotations can always be untangled has made the string model of interest to physicists exploring how rotations work in space.
The famous British quantum physicist Paul Dirac [1902 - 1984] was known to use the model “to illustrate the fact that the fundamental group of the group of rotations in 3-space has a single generator of the period 2.”
The Danish poet and mathematician Piet Hein frequented Niels Bohr’s Institute for Theoretical Physics in Copenhagen in the 1930s, where he learnt about the string model. Always a playful character, Piet Hein devised a two-player game called Tangloids based on the model.
Player A holds the left piece of cardboard, and player B the right piece. One player then makes two rotations to his piece of cardboard, and his or her opponent must then unweave the tangle without rotating his or her piece of cardboard. Players take turns in tangling and untangling and the fastest untangler is the winner.
Put those leather braids in your hair, people, and get tanglin’!
I post a puzzle here on a Monday every two weeks. If you like this sort of thing check out my other Guardian blog Adventures in Numberland. You can also check me out on Twitter, Facebook, Google+ and my personal website.
If you like colouring for relaxation you might enjoy my latest book, Snowflake Seashell Star: Colouring Adventures in Numberland, which is out now.
And if know of any great puzzles that you would like me to set here, get in touch.
