Introduction: Making Tangloids!
Tangloids is a puzzle game developed by the Danish scientist, mathematician, inventor, designer, author and poet Piet Hein. The game was conceived while he was attending to a meeting at the Institute for Theoretical Physics of the University of Copenhagen (now the Niels Bohr Institute).
According to Martin Gardner's book "New Mathematical Diversions", the Austrian and Dutch physicist Paul Ehrenfest was discussing the connection of a problem in quantum theory to the braid theory developed by the Austrian mathematician Emil Artin. Piet Hein and other attendants worked out a demonstration that later became as the original idea for the Tangloids game.
In Tangloids we have two plaques attached by three strands of the same length. The game is played in turns by two players. Each player hold one of the plaques. In each turn one of the players make some braid by rotating his plaque either sidewise and/or between the strands as many times as he wants. Then the other player has to untangle the braid using only the plaque he holds. In the next turn the players reverse their roles, and the one who is fastest untangling the braid wins.
The game can also be played as a solitaire puzzle. In this case one of the plaques is fixed, and the player manipulates the other. Some braid is proposed as the start of the puzzle, and the player hast to figure out how to untangle it by weaving the plaque in and out through the strands keeping the plaque horizontal, that is, using no rotations at all.
Now let's make some Tangloids!
Step 1: Materials & Tools
- Two plaques. They can be of any rigid material:
- Wood. I'm going to use two popsicle sticks for making one Tangloids set.
- Plastic. I'm going to use part of an old DVD case for making another Tangloids set.
- Cardboard. I'm going to use pressed cardboard for a third Tangloids set. Keep in mind that wood and plastic are preferable over cardboard.
The plaques can be of any solid material. In any case wood and plastic are preferable over cardboard.
- A pencil to mark the position of the holes in the plaques for the string.
- A ruler to measure the plaques and place the holes.
- A utility knife to cut the plastic and the cardboard.
- A file to remove and clean the holes in the popsicle sticks after drilling them.
- A tool for making the holes:
- For the wood popsicle sticks a drill is the best choice. A gimlet can be also used, but in my experience it can easily break them.
- For the plastic and the cardboard a single-hole puncher is enough.
Step 2: Prepare the Plaques
Each plaque has to be the same size, and require three holes. The description below gives the details for each of the three materials used.
Wood (popsicle sticks)
The size of the popsicle sticks is good for the plaques of the Tangloids, so there is no need for cutting them. You can start directly by marking the position of the three holes in both sticks.
The first mark must be at the center of the the popsicle stick. As a typical stick has a length of 4.5 inches (approximately 11.43 centimeters), the center will be at 2.25 inches (approximately at 5.7 centimeters).
The other two marks must be at the center between each end of the popsicle stick and the first mark. So in a 4.5 inches stick this two marks must be at 1.125 and 3.375 inches (approximately 2.85 and 8.55 centimeters).
Drill the holes at the marked positions in both sticks.
Cut two rectangular pieces of about 2x5 centimenters. Bigger sizes are also posible. I'm using this small dimensions to allow the final Tangloids plastic set to be carried in a small canister.
As with the popsicle sticks, the mark for the first hole must be at the center of each plaque, in this case at 2.5 centimeters. The other two must be also at the center between both ends of the plaque and the first mark, so they must be at 1.25 and 3.75 centimeters.
Finally make the holes at the marked positions using the hole puncher.
Cut two rectangular pieces of about 2.5x10 centimeters. Again the size depends on how big do you want your Tangloids and how hard is the cardboard you are using.
As with the popsicle sticks and the plastic pieces, the mark for the first hole must be at the center of each plaque, this time at 5 centimeters. And again the other two must be also at the center between both ends of the plaque and the first mark, so they must be at 2.5 and 7.5 centimeters.
Finally make the holes at the marked positions using the hole puncher.
Step 3: Add the String
Run the string through the first hole of the first plaque, knotting the end to keep it form sliding out of the hole. Then run it through the first hole of the second plaque, then across the plaque and through the middle hole. Continue through the middle hole of the first plaque, and again across the plaque and through the third hole. And finally through the third hole of the second plaque, knotting the end to keep the string in position.
And that's it! You have finished your Tangloids.
You can add some finishing touches, like painting each strand to differentiate the braids, and decorating the plaques. As you can see in the pictures I have painted the strands of the popsicle sticks Tangloids set so they are now blue, red and white. This way it's easier to define a set of puzzle problems to be solved in a solitarie game.
Step 4: Solitaire Puzzles Based on Tangloids
As explained in the introduction, Tangloids can also be played as a solitaire puzzle game. Here you have a couple of examples. In the examples the untangled Tangloids set has the blue strand on top and the white on bottom.
In the first example, maybe the most known one, you have a braid made by two sidewise rotations of the right plaque from top to bottom. How can it be untangled just by weaving the right plaque in and out through the strands?
To untangle this braid you just need to weave the plaque, from down to up, under all the strands.
In the second example the braid is made by a forward rotation of the right plaque through the red and white strands, and then followed by a forward rotation between the blue and red strands. Again, how can it be untangled just by weaving the right plaque in and out through the strands?
This is a bit more complicated, but still easy. You just need to pass the plaque below the white strand from down to up close to the other plaque (on the other side of the braid), and then pass the plaque below the blue strand from up to down again close to the other plaque.
Step 5: The Challenge
There are many braids that can be made using the posible rotations of the plaques that can be solved just by weaving one of the plaques in and out through the strands, but some cannot be untangled that way.
So in the spirit of the contest, I challenge you, estimated maker, to solve the three braids shown in this pictures just by weaving the right plaque.
- The first braid is formed by rotating the plaque forward twice through the blue and red strands.
- The second braid is formed by rotating the plaque forward through the blue and red strands, and then backward through the red and white strands.
- The third braid is formed by rotating the plaque backward through the red and white strands, then forward through the red and blue strands, then sidewise from up to down, and then forward though the red and blue strands.
Make a braid by rotating the plaque backward twice through the red and white strands, and then forward through the red and blue strands. Can it be solved just by weaving the right plaque? Why or why not?
The solutions will be published after the end of the Puzzles Challenge. In the meantime you can vote for this instructable if you liked it.
Step 6: The Solutions
Watch the video to see how to solve the three challenges.
The answer to the bonus challenge is that it is not possible to untangle the proposed braid by just weaving the plaque. According to the braid theory, to untangle a braid weaving one plaque requires that the braid is the result of an even number of rotations, and the bonus challenge used only three. If you add any of the six possible rotations then you will be able to untangle the braid according to the rules.
Participated in the