The puzzle 14 15

The veterans inhabitants of the country of the riddles will remember that in the 1870s everyone went crazy with a box of mobile blocks that became known under the name of "Puzzle 14-15". The fifteen blocks were willing inside the square box in order, but with the 14 and 15 inverted as seen in the Enlightenment. The problem was to displace the blocks, one at the same time, until the initial position is achieved again but correcting the error of 14 and 15.

The $ 1 prize.000, offered to whoever presented the first correct solution to the problem, has never been granted, although thousands of people claim to have carried out the feat.

People upset with the puzzle, and ridiculous stories about merchants who stopped opening their shops are counted; About a distinguished cleric who remained a whole winter night on the street, under a lantern, trying to remember how he had solved the problem. The mysterious feature of the problem is that no one seemed to be able to remember the sequence of movements through which he had managed to solve it.

It is said that there were pilots who ran off their ships, and machinists who did not stop their trains at the stations. It is known that the farmers abandoned their crops, and one of those cases is the one I chose for the Enlightenment.

It is worth presenting several new problems that developed from the original riddle:

Second problem. Start once again with the blocks in the position shown by the Enlightenment and move them in such a way to arrange the numbers in order, but leaving an empty square in the upper left corner instead of in the lower right corner. See fig. 1.

Third problem. Start with the blocks as before, give the box a quarter of turn and move the blocks until they are in fig. 2.

Fourth problem. Start as before, then move the pieces until they form a "magical square", with the numbers giving a sum of thirty in all vertical and horizontal rows, and in the diagonals.

Solution

The original riddle is impossible to solve, except through a trick that consists in investing blocks 6 and 9.

One of the peculiarities is that any exchange of that class that involves two blocks, immediately converts the riddle into a solution. Actually, any odd number of exchanges exerts the same effect, while a torque number causes the riddle to remain impossible to solve.

The other three problems are solved as follows:
Fig. 1 It can be achieved in 44 movements: 14, 11, 12, 8, 7, 6, 10, 12, 8, 7, 4, 6, 4, 7, 14, 11, 15, 13, 9, 12, 8, 8 , 4, 10, 8, 4, 14, 11, 15, 13, 9, 12, 4, 8, 5, 4, 9, 13, 14, 10, 6, 2, 1.

Fig. 2 It can be achieved in 39 movements: 14, 15, 10, 6, 7, 11, 15, 10, 13, 9, 5, 2, 3, 4, 8, 12, 15, 14, 10, 13, 9 , 5, 1, 2, 3, 4, 8, 12.

The magic square can be achieved in 50 movements: 12, 8, 4, 3, 2, 6, 10, 9, 13, 14, 12, 8, 4, 7, 10, 9, 14, 12, 8, 4 , 7, 10, 9, 6, 2, 3, 10, 9, 6, 5, 1, 2, 3, 6, 5, 3, 2, 13, 14, 3, 2, 1, 13, 14 , 3, 12, 15, 3.