KA-0086Tiling and tessellation
3 points, difficulty 1 of 3Level 3–4about 45sA floor measuring 6 by 4 units is covered exactly with tiles measuring 2 by 1, with no gaps and no overlaps. How many tiles are needed?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
How much floor does one tile cover?
2The strategy
Every tile covers the same area and none overlap, so the number of tiles is the floor area divided by the tile area. Check that the division comes out whole, otherwise the tiling is impossible.
3The full solution
The floor is 6 x 4 = 24 units and each tile is 2 x 1 = 2, so 24 divided by 2 = 12 tiles.
Solution— no shortcut on this one
The floor covers 6 x 4 = 24 unit squares and each tile covers 2, so 24 divided by 2 = 12 tiles. Because 24 divides exactly by 2 the count is at least possible; laying six tiles across each of two double rows shows it really works. The transferable idea: dividing the areas gives a necessary check, never a sufficient one — a shape can pass it and still be untileable.
Why this is on the test: It sets up the area check that harder tiling questions rely on, including the ones where the area divides but the tiling is still impossible.