Cut up a square into a finite number of identical tiles.
Here is one possibility:
How do I prove that the tiles could never be rearranged to form an equilateral triangle (with filled interior and no overlapping tiles)? This is easy to see for the cut drawn above (though I intentionally chose 60 degree angles), but how do I prove it in general?
As for my attempts at proving this, I think this becomes a question of listing the ways that a square can be cut into equal pieces. Instead of the cut drawn above, one could cut the square into "rational rectangles" and halve the rectangles into triangles, but all angles then must have rational tangents and therefore can never sum to a 60 degree angle. Also, by equating the area of the square and the area of the equilateral triangle, we know that some tile edge lengths must irrationally divide the square edge length. Beyond these two conclusions, I am stuck, but have a feeling that there is a very easy proof here which will leave me feeling foolish.
To be clear, all tiles must be used in the rearrangement and no tiles can be flipped over, but I think I could relax these two conditions also and still never form an equilateral triangle.
Comment:
In a simple case, suppose the areas of square is equal to area of triangle. As can be seen in figures a and b we can tile isosceles triangles as an square with equal area , provided the side angles are a multiple of $15^o$.
In figure c the area of equilateral triangle Q'UV is equal to area of rectangle $RR'_1S'_1D_1$. Now we have to construct an square equal in area with this rectangle. But this is impossible because the product of two unequal integers can not be equal to square of an integer.