depends on your choice of floating point standard, you're right for the current IEEE one

This is a classic example of something that should be fixed point, not floating point. Just represent positions internally as an integer between 0 and some large maximum. You could use the number of screen pixels, or a larger value if you want to allow e.g. precise moves by zooming in.

If the gaps are all the same size, isn't it fixed point rather than floating point?

I mean, if your board has a fixed zoom level you have to fix a certain precision