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I would like to know why the coordinates are the way they are. I know it is a made up thing but I'm curious to know if it were setup any other way, would it still work?

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Yes, chess notation is a well-established convention but many other systems of coordinates could work.

For instance, in Shogi, a game similar to chess (actually a distant cousin), we use either :

  • numbers for files and letters for rows (the opposite of chess)

or

  • arabic numers for files and japanese numbers for rows (as in the Takahashi-Fortin game displayed as example on wikipedia),

but the algebraic notation works otherwise the same way as in chess, and this works just fine.

Even in chess, for centuries games were replayed and books were published using the descriptive notation, which doesn't need cartesian coordinates at all.

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Just as a little addition to Evargalos answer, we have:

  • Forsyth notation to describe a whole position. A typical part (a single row) of a Forsyth string might be .../4k2Q/..., i.e. four empty, a black king, two empty, white queen. Add all eight rows from 8 to 1 and separate by /. Surplus support for ep, castling, move numbers and even Chess 960. Also used in...
  • PGN. Does not describe the position but the moves.
  • ICCF numeric notation. Simply numbers instead of letters. For correspondence chess.
  • Telegraphic support: Gringmuth, Uedemann and Rutherford. Completely obsolete nowadays. Can be used for teh lulz since they convert squares into a speakable nonsense language - e.g. "e2-e4" becomes "gagisasi", if I translated correctly.

For details especially of the latter (I never heard of Rutherford before) see the Wiki page.

A little history course.

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