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Topological Puzzles by Dimiter Vakarelov

Topological Puzzles: An analysis, Internal Classification and Creation

In this feature and essay dedicated to Martin Gardner, we will explore ‘Topological Puzzles’ also commonly known as string and wire puzzles. Looking at the Slocum classification they fall into the category of disentanglement puzzles. Observing Martin Gardner’s terminology, I will use the name ‘topological puzzles’ because they are related to the subject of topology in a sense that their solution sometimes requires some informal and non-trivial topological reasoning.

What is topology?

Let us mention that topology is a branch of mathematics which studies properties of things preserved by soft deformations. For instance, if we make a knot with a string and glue at the ends of the string, then unless we use scissors, there is no way that we can untangle the string. So, the property of the string to be ‘knotted’ is a topological property. Since this article is addressed to all puzzlers, I will not include in the text unknown formal mathematics, but only some kind of informal topological reasoning consistent with our everyday practice which can help in solving topological puzzles.

For people who are interested in possible applications of topology to (topological) puzzles, I can say that the branch of topology called ‘knot theory’ is the most relevant. I can suggest ‘The Knot Book: An elementary introduction to the mathematical theory of knots‘, by Colin C. Adams. This is a very nice book containing beautiful pictures of knots.

Topological puzzles

Topological puzzles consist of several pieces made of wire, wood, plastics and string (rope), tangled in some way. The goal is to separate some parts of the puzzle or to remove them in another place without deforming the structure or cutting the string. Different elements of the puzzle play different roles: for instance, a ball or a ring attached to a structure prevents this element from going through a certain hole. Very often different puzzles contain details of a typical shape forming closed or open loops of two kinds – rigid loops, made by wire, and soft loops, containing a string; see Fig. 0.1 for closed loops and Fig. 0.2 for open loops.

To solve a puzzle, one must find a series of admissible steps after which the goal is reached. Such puzzles are called solvable. There are, however, unsolvable topological puzzles that are named also ‘impossible puzzles’ (they are different from the puzzles which look like an ‘impossible object’). Sometimes the ‘impossibility’ of a given puzzle is obvious. For instance, if you have a string passing through a ring and two balls with a diameter bigger than the diameter of the ring attached to the ends of the string, then obviously the ring cannot be separated.

The situation for some other impossible puzzles is not so obvious, for instance, such is the well known Steward Coffin’s Eight Puzzle described in the book ‘Creative Puzzles of the World’ by P. Van Delft and J. Botermans. Its impossibility is proved by some heavy mathematical methods from algebraic topology and knot theory. Such mathematical proofs are quite complicated and are understandable only by specialists, so they will not be included in this essay.

Topological puzzles are qualified by some puzzlers as the most difficult. One of the reasons for this is that they try to solve a topological puzzle by the easy method of “trial and error” which works perfectly for some other puzzles. For instance, if you have to put the 12 pentominoes in a box 5×10 there is no other method to solve this puzzle – very often the last piece does not fit in the box and you start a new trial without any special strategy and go on until you find accidentally the solution.

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