Games played on an Alquerque-style grid
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Alquerque is a traditional game played on the vertices of a square grid on which alternate diagonals have been drawn in. This means that alternate points have 8 neighbors (when diagonals in both directions cross at that point) and 4 neighbors (when diagonals in both directions miss that point).

This pattern has a name: it's the tetrakis square tiling and if you type that into Wikipedia you'll learn more about it than you ever wanted to know. One way to think about it from the point of view of game board design is to realize that if in a square grid you draw in all of the diagonals along just one diagonal direction you get the same adjacencies as a hexagonal grid (the square grid becomes rhombus-shaped, like a Hex board). Here we're alternating the direction of the diagonals, so we get the same density of connections in a slightly different arrangement.

Another way is to rotate yourself 45 degrees so that the diagonals become the orthogonals: then the vertices with 8 neighbors form the regular square grid and the others are extra vertices in the center of each square. Or you can consider the dual tiling: that is, how to draw a board so that playing in the spaces of the new board is equivalent to playing on the vertices of the original. This turns out to be the tessellation of octagons and squares (truncated square tiling): the vertices with 8 neighbors become regular octagons.

In any case, here I'm collecting games with that board topology ... any size. What's odd is that there are numerous traditional games using it, but almost no modern ones. There are games with octagonal tiles such as Keythedral; but to me they don't really count, as the squares and octagons aren't treated symmetrically: you can't put an octagonal tile in a square hole, whereas in Alquerque you can move a piece freely from an "octagon" to an adjacent "square".
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3. Board Game: Octagons [Average Rating:7.00 Unranked]
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Archimedical grid (4-8-8)
 
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4. Board Game: Nam Dinh [Average Rating:7.00 Unranked]
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5. Board Game: Quax [Average Rating:6.97 Unranked]
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A connection game on the truncated square tiling.
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6. Board Game: Triangle [Average Rating:6.50 Unranked]
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Like Phalanx, this is played on the triangles.
 
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Thanks, Mr. Brettie!
 
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15. Board Game: Bagh Chal [Average Rating:5.94 Overall Rank:8368]
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16. Board Game: Phalanx [Average Rating:5.92 Overall Rank:10937]
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I'm clearly cheating here since this is played on the triangular spaces, not the vertices.
 
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18. Board Game: Alquerque [Average Rating:5.63 Overall Rank:11936]
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19. Board Game: Officers and Sepoys [Average Rating:5.50 Unranked]
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20. Board Game: Asalto [Average Rating:5.39 Overall Rank:12998]
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21. Board Game: Bagh Guti [Average Rating:5.38 Unranked]
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23. Board Game: Catch the Hare [Average Rating:5.19 Unranked]
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The game description says it's played on an Alquerque grid, but some of the adjacencies appear to be missing.
 
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24. Board Game: Sixteen Soldiers [Average Rating:5.10 Unranked]
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25. Board Game: Three Tigers Chess [Average Rating:5.00 Unranked]
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