петък, 1 февруари 2013 г.

Martin Gardner and topological puzzles

 During the lecture

At the puzzle party

This is the title of my talk presented at the Gardner's G4G event   ( www.g4g-com.org , see also)    on 19.10.2012 at Sofia University and on 14.12.2012 at Veliko Tarnovo University,  Bulgaria. The aim of the talk was to describe a method which can help us in solving some  puzzles, normally called "string and wire puzzles". Following Gardner I call them "topological puzzles", because in their solution one can apply some informal topological reasoning. Topology is a branch of mathematics which studies properties of things invariant under continuous deformations. We, however, will use topology informally.  The method,  is based on the  notion which I call   "topological  equivalence of puzzles". It will be described below by means of two examples. The main idea is the following. Trying to solve some topological puzzle sometimes we see  that if we can make certain deformation of the wire part of the puzzle, then we obtain  a puzzle which is known or easily solvable and in this case we say that the two puzzles are topologically equivalent. But since deformation is forbidden,  we try to overcame  it by some manipulations with the rope part of the puzzle. If this is successful, then we obtain the solution. Practically this can be done by making a new model of the puzzle in which the wire part is made by some soft material adaptive to deformations. Hence  topologically equivalent puzzles may have quite similar solutions and the method is just to see if the new puzzle, which we do not know how to solve, is topologically equivalent to a puzzle with known or obvious solution. After the talk there were a puzzle party in which I presented many topological puzzles, which can be solved by the method of topological equivalence.

Example 1. In Fig. 1,2 and 3 below one can see some  topological puzzles and the goal for all  is to separate the metal part from the rope. The puzzle 1, which is taken from a book by Gardner, is quite popular and easy to solve, the puzzle 2 is a simplified version of 1 and has the same solution:  move the loop along the rope and  go through the second hole,   then go around the  two balls at the ends of the rope, and finally go back - the rope is free. This shows that the solution does not depend on the form of the metal part - we are working only with the two holes.  However, looking at puzzle 3 we see that it is quite different - there is no  loop in the wire like in the puzzles 1 and 2, which has been essential in the solution.

Fig. 1
Fig. 2

Fig. 3
Fig. 4
 To solve  puzzle 3 suppose that the wire part is deformed and straighten (this can be done if we consider a version of 3 in which the wire part is made by some soft material). After this transformation you can see that we obtain the puzzle 2, which we already know how to solve. In this case we say that 3 is "topologically equivalent" to 2 (and to 1). The problem then is to avoid the deformation, which is forbidden, and to replace it by some manipulation with the rope. It is possible to do this and to obtain the configuration of the rope shown in Fig. 4, which is the same  as in the puzzle 2 and 1 and which yields the solution. On Fig.5 and Fig. 6  you can see other versions of  puzzle 4 which can be solved in a similar way. 

Fig. 5
Fig. 6
Example 2.   Look at the puzzles in Fig. 7 and Fig. 8. Puzzle 7 is easy to solve operating with the loop like in puzzles 1 and 2. Puzzle 8, however,  looks quite different - there is no loop in it and it is quite similar to the impossible puzzle shown on Fig. 11. But nevertheless 8 has a solution: deforming  the wire part (together with the rope ) giving it the form of a ring, we see that  we can obtain the same configuration as in Fig. 7. Hence, 7 and 8 are "topologically equivalent" puzzles. Again, reasoning as in Example 1, we are looking for some operations with the rope, avoiding the  deformation and obtaining a configuration of the rope similar to that in puzzle 7. This is possible and such  a configuration is shown on Fig. 9, which yields the solution. On Fig. 10 you can see a generalization of the puzzle 8 which can be solved in a similar way.

Let us note that puzzles 3 and 6 are special cases of the puzzle Spiral-n  (spiral with n turns)  for n=1 and 2  and puzzles 8 and 10 are special cases of the puzzle Fan-2n (fan with 2n feathers)  for n=1 and 2. Both Spiral-n and Fan-2n were presented at IPP Design Competition 2011. One can wonder why we do not consider  puzzle Fan-(2n+1). This is just because Fan-(2n+1) is topologically equivalent to the obviously impossible puzzle shown on Fig.  11.

Fig. 7
Fig. 9
 Fig. 8

Fig. 10   

Fig. 11