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Six Frogs

The six educated frogs in the illustration are trained to reverse their order, so that their numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its present position. 
They can jump to the next square (if vacant) or leap over one frog to the next square beyond (if vacant), just as we move in the game of draughts, and can go backwards or forwards at pleasure. 

Can you show how they perform their feat in the fewest possible moves? 

It is quite easy, so when you have done it add a seventh frog to the right and try again. 
Then add more frogs until you are able to give the shortest solution for any number. 
For it can always be done, with that single vacant square, no matter how many frogs there are.

See answer





Math Genius