Tile flooring has several potential advantages over wood or carpet alternatives. Unlike carpet, it doesn't contain any fibres that can hold stains or, worse, pathogens. Wood (faux or real) doesn't either, but it also requires frequent care to maintain its gloss and condition. In addition to these practical aspects, tile can also be a good flooring choice because of its versatility -- you can arrange tile in any pattern you choose, including a "chess board" pattern that uses three colours instead of two.
Lay out the main "chess board" pattern using a single colour, whichever one you prefer to be dominant. Set out a single tile, then place one auxiliary tile at each of the tile's four corners, ensuring that each tile's vertical edges are perpendicular -- in other words, one goes straight up and down, the other straight left to right -- to the main tile's horizontal ones and that the horizontal ones are perpendicular to the vertical ones. If you've done this correctly, you should have a large three-by-three square on your floor, the first row of which will have (in order) a tile, an empty space and a tile; the second row an empty space, a tile and another empty; the third identical to the first.
Add the second colour of tile, either in the empty spaces just above and below the original tile or to the left and the right of it. If you want to maintain a checkerboard look, it's important that like colours be opposite one another (rather than adjacent or nearby), lest the pattern look messy rather than put-together.
Insert the third colour of tile into the two spaces that remain. For example, if your main colour is black and you put white tiles above and below the original tile square, you should put your red tiles to its left and right.
Expand upon your main tricolour chess board by placing new tiles of the main colour at the corners of the original tiles and repeating the pattern you made with the auxiliary colours. Continue expanding until you reach the furthest edges of the room.
Adhere your tile to your prepared and primed floor as you usually would once you have set the pattern you want. Double-check this pattern before you prepare mortar, as it will be troublesome to remove any tile you lay improperly.