A boardgame board consists of 10 squares in a row that are numbered 1 to 10. On some square there is a button. In one move it is allowed to move the button to a square whose number is either smaller by 2 or 2 times bigger. Does there exist an initial location for the button that allows the player to visit all squares of the board? It is allowed to visit one square several times.
Solution
No move allows the button to be placed to the square number 9. Therefore the button should start from there to have any hope. If on some later move the button is placed on an even-numbered square, then it will also stay on an even-numbered square on every move that follows. Therefore all the odd-numbered squares must be visited right in the beginning, i.e., the button must be moved to 7, 5, 3, 1. On the next move there is no other option but to step to square number 2. But now it is impossible to reach square number 5, since it is odd-numbered, and therefore it is also impossible to reach 10. Therefore it is not possible to visit all the squares.
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