Question

What is the minimum cost? Can someone on here finally help me with solve this massive problem? See picture, Thanks!

Answer 1

30 large tiles, 0 small tiles, for a minimum cost of `$135`.

Explanation:

This problem would be trickier if the tiles had different dimensions or if the allotted space had different dimensions. Fortunately, the tile cost, tile size and mosaic dimensions are all perfect, so as to give a trivial solution.

First, note that a large tile is significantly more cost efficient than a small tile. The small tile costs `$3.50` and covers 16 square inches, for a cost of about 22 cents / square inch. The large tile costs `$4.50` and covers 72 square inches, for a total of about 6 cents / square inch. Clearly the artist should attempt to use large tiles whenever possible, resorting to small tiles only if necessary.

We further note that the mosaic must minimally be 3 feet tall and 5 feet wide. If the artist can tile exactly this area with large tiles, he has clearly minimized the cost as a) large tiles are less expensive, per square inch, and b) the smallest possible sized mosaic has been completed. (Meaning there cannot be a different arrangement of tiles which takes up less space.)

And we see that the artist can tile a 3ft tall by 5 ft wide space with only large tiles. It will require an array of large tiles 5 tiles wide and 6 tiles tall, for a total of 30 tiles, costing `$135`.

If the dimensions were different, this would become a "packing problem" and such problems are very hard. Many of them are still unsolved, in fact.