To place a tile, move the sample to the position you want, then use the slider for the proper rotation, and finally click the "store new tile" button.
The first three examples in this Demonstration deal with the problem of packing rectangles into a rectangular box. In 1980 the author showed (in the Journal of Recreational Mathematics) that the ( box is the smallest box that allows a fault-free packing of tiles.
The author also showed how in general a ( box can be packed with tiles, and he conjectured that this is indeed the smallest box of this type for any numbers and with no common divisor. This interesting result answered a question posed by Martin Gardner.
Example 4 is taken from the author's book A Puzzling Journey to the Reptiles and Related Animals, 1987, published privately.
Reference: M. Gardner, New Mathematical Diversions from Scientific American, New York: Simon and Schuster, 1966.