<abstract xmlns="http://www.w3.org/1999/xhtml"><p>A <italic>dragon king</italic> is a shogi piece that moves any number of squares vertically or horizontally or one square diagonally but does not move through or jump over other pieces. We construct infinite families of solutions to the <italic>n + k dragon kings problem</italic> of placing <italic>k</italic> pawns and <italic>n + k</italic> mutually nonattacking dragon kings on an <italic>n</italic>×<italic>n</italic> board, including solutions symmetric with respect to quarter-turn or half-turn rotations, solutions symmetric with respect to one or two diagonal reections, and solutions not symmetric with respect to any nontrivial rotation or reection. We show that an <italic>n + k</italic> dragon kings solution exists whenever <italic>n</italic> > <italic>k + 5</italic> and that, given some extra conditions, symmetric solutions exist for <italic>n</italic> > <italic>2k + 5</italic>.</p></abstract>

A dragon king is a shogi piece that moves any number of squares vertically or horizontally or one square diagonally but does not move through or jump over other pieces. We construct infinite families of solutions to the n + k dragon kings problem of placing k pawns and n + k mutually nonattacking dragon kings on an n×n board, including solutions symmetric with respect to quarter-turn or half-turn rotations, solutions symmetric with respect to one or two diagonal reections, and solutions not symmetric with respect to any nontrivial rotation or reection. We show that an n + k dragon kings solution exists whenever n > k + 5 and that, given some extra conditions, symmetric solutions exist for n > 2k + 5.

Reflections on the <italic>n</italic> +<italic>k</italic> dragon kings problem

Recreational Mathematics Magazine

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Reflections on the  + dragon kings problem

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A dragon king is a shogi piece that moves any number of squares vertically or horizontally or one square diagonally but does not move through or jump over...

Reflections on the n +k dragon kings problem

Published Online: Dec 31, 2018

Page range: 39 - 55

DOI: https://doi.org/10.2478/rmm-2018-0007

Keywords
shogi, combinatorics, symmetry, -queens problem

© 2018 Doug Chatham, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Reflections on the n +k dragon kings problem

Published Online: Dec 31, 2018

Page range: 39 - 55

DOI: https://doi.org/10.2478/rmm-2018-0007

Keywordsshogi, combinatorics, symmetry, -queens problem

© 2018 Doug Chatham, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
shogi, combinatorics, symmetry, -queens problem