MathNovatory/Applied Maths/Game Maths/Hop-Over


Hop-Over

Numbers Of The Game
Where "n" represents the number of tokens of one color -
     There will be 2n + 1 squares on the board.
     Each token must move n + 1 squares, for a total of 2n(n + 1) square-moves,
          partly Slides (1 square) and partly Hops (2 squares).
     If n + 1 is odd, each token will make only 1 Slide, alternately at the beginning or at the end.
     If n + 1 is even, alternate tokens will make 1 Slide at the beginning and 1 at the end, or no Slide at all.
     The total number of moves (Slides and Hops) is represented by 2n + n2,
          where 2n represents the number of (1 square) Slides
               and n2 represents the number of (2 square) Hops.
     When n = 5, there will be 10 Slides and 25 Hops, 35 moves in all.
     When n = 4, there will be 8 Slides and 16 Hops, 24 moves in all.
     When n = 3, there will be 6 Slides and 9 Hops, 15 moves in all.
     When n = 2, there will be 4 Slides and 4 Hops, 8 moves in all.
     When n = 1, there will be 2 Slides and 1 Hop, 3 moves in all.

Temporal Symmetry

This game unfolds in time in a perfectly symmetrical manner.

If we examine each move of the "n = 2" game,
          with its 8 moves, we see that
     the moves 1 and 8 are Slide red,
     the moves 2 and 7 are Hop blue,
     the moves 3 and 6 are Slide blue, and
     the moves 4 and 5 are Hop red.

The 9 positions are also symmetrical,
     positions 1 and 9 with everybody home
          and the space in the center,
     positions 2 and 8 with the space between the reds,
     positions 3 and 7 in alternation with the space inside,
     positions 4 and 6 in alternation with the space outside, and
     position 5 in perfect symmetry with the space in the center.

Positions 1, 4, 6, and 9 are particularly pertinent,
     from one "home",
     through the 2 forms of alternation,
     to the other "home".

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The Two-dimentional Game
There is 1 central horizontal game plus 5 vertical games.
     These 6 games are of size "n = 2", with 8 moves each, a total of 48 moves.

The Diagonal Game
This version is so complicated that no one seemed sure of the minimum number of moves -
     Most of the game books spoke of 52,
          until Sam Loyd himself divulged that he could do it in 47 moves,
               only to be outdone by Henry Dudeney who managed the feat in 46 moves !