The bathroom floor consists of square tiles with a side length of 10 cm. On it is a translucent carpet that consists of a grid rectangles. If I position the carpet just right, some intersections on the carpet grid match intersections on the floor tiling. If three intersections match, is that enough information for me to figure out the the dimensions of the rectangles of the carpet?

The bathroom wall has square 15 cm tiles whereas the floor has 10 cm tiles. Between each tile is a width of silicon paste, as wide on the floor as on the wall. I'd like the 308 cm width of the room to start and end with paste, the width of paste between each tile be a constant, for both the floor and the wall to consist of unbroken tiles, and for the paste width between each tile to be roughly 4mm. How close to that can I get if I have the same with of paste on the floor and the wall? How about if I don't require the same exact width of paste for the wall tiling that I use on the floor tiling?

Originally posted by talzamir The bathroom floor consists of square tiles with a side length of 10 cm. On it is a translucent carpet that consists of a grid rectangles. If I position the carpet just right, some intersections on the carpet grid match intersections on the floor tiling. If three intersections match, is that enough information for me to figure out the the dimensions of the ...[text shortened]... don't require the same exact width of paste for the wall tiling that I use on the floor tiling?

Originally posted by talzamir The bathroom floor consists of square tiles with a side length of 10 cm. On it is a translucent carpet that consists of a grid rectangles. If I position the carpet just right, some intersections on the carpet grid match intersections on the floor tiling. If three intersections match, is that enough information for me to figure out the the dimensions of the rectangles of the carpet?

Is the floor covered exactly with 10cm tiles, with no cutting and assuming no grouting in between?
Is the floor a quadrilateral?
(you can do it very easily if the "floor" is made of three separate pieces)

Step 1:
Assuming the floor has an even number of tiles on two opposite sides, then there are various ways to take an intersection on one side, and intersection in the middle, and an intersection on the opposite side, such that these intersections are in a straight line, the middle intersection is exactly mid-way between the other two and there are no other intersections on the line.

Step 2
Make half the line length the side of a rectangle in the translucent carpet, and now choose the other rectangle side to be an irrational number like pi so that there are no other intersections.

You now have three shared intersections and, because there are several choices for the line gradient in the first step and an infinite number of choices for the irrational number in the second step, the exact carpet dimensions cannot be determined.

If the floor has an odd (and large enough) number of tiles per side, the same argument can be used by simply ignoring the row of tiles along one edge and performing the same construction.

If the floor tiles have an irrational side it still works as long as a different irrational number (or a rational number) is chosen in step 2.

Or with arbitrarily sized floor.
Take an even number N, larger than 2
Assume floor has either N by N, or N+1 by N+1 tiles

Drawn the line from 0,0 to N,N (let us say these are two intersections with our rectangular tiles)
mark the point (N/2,N,2), let us say this is also rectangular carpet intersection, now we have three intersections, and one side of out rectangle is N/sqrt(2)

choose the other side of the rectangle to be an transcendental number T, this number is not the root of a polynomial equation with rational coefficients. As the tile intersections are M.sqrt(2)N away perendicular to the original line and the polygon intersections are K.T away, where M, K, N are integers the there can be no other intersections otherwise T = M.sqrt(2).N/K which would mean T was not transcendental.