text: (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
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