The Linear, One-dimensional Game

Origin of the Game
There seems to be no specific paternity attributed to the Hop-Over Game, only an approximate birth date around the beginning of the XXth Century. This is really a great game for young children (5- or 6-year olds) and on a par with The Tower of Brahma to get them started.

Description
The Game is quite easy to make.
The board consists of an odd-numbered row (here 9) of 1 inch squares,
with alternate colors (as above) being a bonus to place the Hops,
and the tokens can easily be checkers (or coins) of 2 different colors.
The middle square is left unoccupied, and tokens are placed in the other squares,
one color on the left and the other color on the right.

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The Rules
The positions of the tokens must be inverted, those on the left sent to the right, and vice-versa,
(a) by either Sliding a token into the adjacent empty square, or
(b) by Hopping a token over one of the opposite color into the empty square.

Mathematics
To see the mathematics involved in the game.

What permits the all-important Hops in the game
is keeping the colors alternated and never having 2 tokens of the same color together.
The Slides are only used to get a token started or stopped.
Once a token has Slid once, out of its old "home",
it will not Slide again, unless to get into its new "home" (on the other side).

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START WITH ONLY 2 TOKENS ON EACH SIDE OF THE CENTRAL EMPTY SQUARE.

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The FIRST MOVE move is evidently limited.
A token of either color (here red) can only Slide (out of its "home") into the empty square.

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The SECOND MOVE is not too dangerous.
One will usually Hop over (with blue) the token which has just Slid (red),
placing it between 2 reds where it really belongs.
Sliding another token (out of its "home") would place 2 of the same color together.

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The THIRD MOVE is the tricky one.
It is important NOT to Slide the first token (red) again out of its central position
(it would not be Sliding into its new "home")
because Hopping over it will also place 2 tokens of the same color together.
The correct move is to Slide from the exterior (blue out of its "home") into the empty square,
surrounding the token in the middle (red) with tokens of the other color (blue).
The 4 tokens are now perfectly disposed for multiple Hops.

THERE WILL NOW BE :

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2 Hops in a row (both with red),

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a Slide (with blue) into the new "home",

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a last Hop (with blue),

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and a last Slide (with red) into the new "home". (8 moves)

Gradually increase the number of tokens on each side of the central empty square.
Children will quite quickly get the knack of moving a larger number of tokens.

Generating Major Triads

NOW WITH 3 TOKENS ON EACH SIDE OF THE CENTRAL EMPTY SQUARE.

 F C G D A E B

Place the 7 natural notes in order of fifths, with the "D" in the empty square,

 F C G A D E B

Slide the A,

 F C D A G E B

Hop the G Over the A,

 F D C A G E B

Slide the C,

 F A C D G E B

Hop the A Over the C,

 F A C E G D B

Hop the E Over the G,

 F A C E G B D

Slide the B.
We now have 3 major triads, F (F-A-C), C (C-E-G), and G (G-B-D).

Generating Minor Triads

 F C G D A E B

Place the 7 natural notes in order of fifths, with the "D" in the empty square,

 F C D G A E B

Slide the G,

 F C A G D E B

Hop the A Over the G,

 F C A G E D B

Slide the E,

 F C A D E G B

Hop the G Over the E,

 F D A C E G B

Hop the C Over the A,

 D F A C E G B

Slide the F.
We now have 3 minor triads, Dm (D-F-A), Am (A-C-E), and Em (E-G-B).

Mathematics
To see the mathematics and the music involved in this generation.

The Square, Two-dimensional Game

This Two-dimentional Game is not unduly difficult for 10- to 12-year-olds.

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Description
We now have square composed of an odd-number of lines and columns (here 5).
1. The central position is empty with red tokens on the left and blue tokens on the right.
2. The 2 top lines are filled with red tokens, and
the 2 bottom lines are filled with blue tokens.

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The Rules
The rules are very much the same as they were for the Linear Game -
the positions of the tokens must be inverted, those on the left with those on right, and
those above with those below,
(a) by either Sliding a token into the adjacent empty square, or
(b) by Hopping a token over one of the opposite color into the empty square.
All moves are made either (a) horizontally or (b) vertically.

Mathematics
See mathematics for the number of moves.

The 5 vertical games are played whenever the horizontal game permits.

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The central vertical game can be played right at the start (in column 3).

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When the horizontal game has made the first Slide (with red),

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a second vertical game can be played (in column 2).

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After a Hop in the horizonal game (with blue),
a third vertical game can be played (in column 4).

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After a Slide in the horizonal game (with blue),
a fourth vertical game can be played (in column 5).

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And after 2 Hops in the horizonal game (both with red),
the fifth vertical game can be played (in column 1).

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The horizontal game can now be terminated
with a Slide (blue), a Hop (blue), and a last Slide (red). (48 moves)

The Diagonal Game

This Diagonal Game is a Sam Loyd creation and is fiendishly difficult !

The Rules
The rules are very much the same as they were for the Square Game -
the positions of the tokens must be inverted,
those on the top-left with those on the bottom-right,
(a) by either sliding a token into the adjacent empty square, or
(b) by jumping a token over one of the opposite color into the empty square.
All moves are made either (a) horizontally or (b) vertically, no diagonal moves.

Mathematics
See mathematics for the number of moves.

GOOD LUCK WITH THIS ONE !