December 7, 2020
What is the greatest amount of area that can be created out of an isoceles right triangle from a cookie cutter cutting the shape three times? Each shape must be congruent to one another, stay entirely within the big triangle, and not overlap each other.
Answer: Approximately 94.1% of the area of the big triangle.
Explanation:
The achievability is provided courtesy of AZ, which I thought about besting it at first, but after a few trials realized some patterns that is strongly pointing to it being the optimal solution. My explanation is as follows: first, the best cookie cutter has to cut a shape that share some edge(s) of the big triangle. If the shape is entirely within the triangle, then it can be extended. This also follows why the cookie cutter has to be a polygon. Secondly, the cookie cutter's shape must be convex, otherwise we can also extend the area by taking the convex hull of the cookie cutter. So we are looking for a convex polygon that will cut along the edges of the big triangle to get our three cookie doughs.
I am approaching the problem from the other way around: what is the minimum area of dough we must throw away? From the proposition earlier, since the cookie cutter is convex and cuts along the edges of the triangle, the area of the throw-away dough must also be polygons. We already have AZ giving a lower bound where when we throw away *two* corners of the original big triangle, we get the construct provided (and this is unique, by the way). The upper bound is zero, since we were already given that some dough must be wasted. So can we do better with just one corner of the triangle removed? If not, then what AZ provided will be the best cookie cutter shape.
If we throw away just one corner of the big triangle, we get a quadrilateral containing at least one 45-degree angle. Assuming no more pieces are to be thrown away, we would need the cookie cutter shape to contain an angle less than or equal to 45 degrees. At best, one of the cookie cutter needs to cut a shape containing the original 45-degree in its entirety, which means we would only have two more of such shapes each containing an angle at most 45 degrees, combining to be at most 90 degrees. However, the quadrilateral will contain an angle greater than 90 degrees. You can see this in two cases. First, when the right angle is sliced away, you end up with two new angles, one of which will be greater or equal to 135 degrees. Second, if you slice away one of the 45-degree angles, then the resulting quadrilateral will contain the original right angle, the original 45 degrees angle, and two angles adding up to 225 degrees, therefore one of them must be greater than 90. In any case, just by slicing away one corner of the big triangle, we cannot cover the remaining quadrilateral with three congruent shapes without overlapping or going outside the dough.
Which is why, we need to slice at least two corners of the triangle. When we do that, we get a unique solution. Since slicing just two corners minimize the amount of dough wasted, the achievability solution provided by AZ yields an answer of 94.1%.