Solution : 7 Coin Puzzle

Pre-analysis :

To solve the puzzle, you might now start a division into cases. You notice, of course, that placing the first coin on 1 is all right, because all vertices play the same role; therefore, with regard to the first coin you do not need to make any division into cases (on 1, on 2, etc.). For the second coin, you have the seven possibilities of placing it on 2, on 3, . . . , on 8; it is irrelevant along which free line you move (when the vertex can still be occupied along two free lines, which is not possible for the vertices 4 and 6). If the second coin has also been placed, you can start to make assumptions about the third coin; it cannot always be placed in six ways, for if the second coin has been placed on 3, then the third coin cannot be placed on 6, because then neither of the lines 1-6 and 3-6 is free any more. Proceeding in this way, you will find all solutions after much sifting. If what you want is to present the puzzle to others, you can stop as soon as a solution has been found; you then note it down and learn it by heart. However, someone who proceeds thus has not seen through the puzzle, even if he finds a solution. Yet, one might say he has followed the directions about systematic trial and error. Indeed he has, and certainly there are still less efficient ways of solving the puzzle. Many a solver who looks for the solution starts pushing at random, and if, after much pushing, he has been able to place the seven coins, he has the bad luck of not remembering how he did it. This, of course, is puzzle solving of the worst kind.

Solution :

In order to understand the puzzle completely, you should observe that you ought to try to retain as many free lines as possible, and hence you should make as few lines as possible useless. Placing the first coin on 1 makes two lines useless: 1-4 and 1-6. Placing the second coin makes two more lines useless, except when this coin is placed on 4 or on 6. The correct continuation is, therefore: on 4 or on 6; for example, by occupying 4, only the line 4-7 is made useless, because 4-1 had already dropped out; the third coin should now be placed on 6 or 7, etc. The coins should always be put on adjacent vertices (such as 3, 6, 1, 4); by adjacent vertices we mean vertices that are connected by a line, like 3 and 6.

The puzzle can be simplified considerably by observing that the eight lines form an octagon, and that the shape of the octagon reduces the complexity of puzzle to be an obvious easy one.

Post-analysis :

Now the seemingly complex problem is reduced to a very simple puzzle which everyone can solve instantly. It’s a very nice and handy technique of solving puzzles but it needs a lot of visualization and imagining capabilities.