(a) We need to protect from the rain a cake that is in the shape of an
equilateral triangle of side 2.1. All we have are identical tiles in
the shape of an equilateral triangle of side 1. Find the smallest
number of tiles needed.
6 tiles are needed. Since each edge is more than 2 units long, there
must be three tiles on each edge. Allowing one triangle at each corner
to be on two edges, there must still be 6 tiles touching the edges.
And 6 tiles are sufficient; put one tile exactly in each corner, and
the remaining part is a truncated edge-1.2 triangle.
3 tiles are sufficient for covering any triangle up to edge 1.5, by
putting one in each corner. 4 tiles cover up to edge 2, and in
general, n^2 tiles cover up to edge n. By putting these patterns in
the middle and one triangle at each corner, 6 tiles can cover up to a
triangle of edge 2.25 and 7 can cover up to a triangle of edge 2.5.
(b) Suppose the cake is in the shape of an equilateral triangle of
side 3.1. Will 11 tiles be enough to protect it from the rain?
Yes, cover a size 2 triangle in one corner with 4 tiles and a single
tile in each other corner. In the two edge gaps of length 0.1, place
triangles so that they just touch the other triangles, with a triangle
of size 0.1 hanging off. The part left over is a hexagon of sides 1.1,
0.1, 1, 0.2, 1, 0.1. This is a truncated triangle of edge 1.3, which
can be covered with 3 tiles. Indeed, this works if the last triangle
is up to size 1.5, which corresponds with an overall triangle size of
3 and 1/6. With 12 tiles, the last triangle can be up to edge 2, and
so the overall triangle can be up to edge 3 and 1/3.
What about 8 tiles? If you put 4 in one corner, and one in each of the
other two corners, the space in the middle can be covered with two
outward-pointing triangles up to a certain limit. At the limit, two
projecting parts of these triangles (of size X) do not themselves
overlap, and at the same time the corner triangles overlap the
4-triangle group by X. So an edge = 2 + 2X = 3 - X, so X = 1/3 and the
edge is 2 and 2/3.
tiles largest triangle coverable
1 1
3 3/2
4 2
6 9/4
7 5/2
8 8/3
9 3
11 19/6
12 10/3
I don't see any real pattern here except at the whole number triangle
size values.
Joe Devincentis