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We could simulate this by imagining the game to be played on a "soft gameboard." When a piece is on a square, it would sink into the material and deform the surface. The second game's pieces would thus be "informed of its presence," without being able to interact with it, take it or be taken by it.
A double game of checkers on a soft gameboard. The grey piece, belonging to one of the games, "informed" of the presence of a checker participating in the second game through the deformation of the material.
Games are often a good basis for reflection and a source of didactic models.
We note in passing that we can envisage a game of checkers in three dimensions. It suffices to remove one cube out of two onto a layer of cubes and stack them, making a translation each time. The "full" cubes become the "black squares" and the empty cubes the "white squares."
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The "game of checkers in three dimensions."