Alex Bellos had three geometry puzzles today, which I'd normally ignore, because they can be done with boring math, without the need for logic, but today's column had a fun twist at the end, because the writer actually screwed up. First, let's start with the easiest:
The large triangle is equilateral (all sides and angles are equal). The red triangle is made by connecting the midpoints of two sides and one of the vertices of the third. What is the area of the red triangle? (You don't actually even need math for this one.)
Two:
The four semicircles are all of radius 2. What is the area of the red square? (Not going to lie; I screwed this one up, and kicked myself when I read the answer.)
But now the fun one:
The red shape is a quarter circle of radius six. There are two semicircles and a smaller full circle inside, and all shapes are tangential to each other (i.e., just as it looks, the edges touch, but don't overlap). What are the radii of the smaller semicircle and the even smaller full circle?
This last one turned out to be a riot, because you can actually solve it with trial and error, but when Bellos published the answers today, he purported to have a simple solution that actually turned out to be faulty, because it rested on an assumption that wasn't justified by the information given in the puzzle. It took one of the Guardian's astute readers to come up with a proof that his assumption was valid. See if you can figure it out. (For my part, I came up with the answers by trial and error pretty easily, but never came close to figuring out the mathematical proof necessary to confirm the answer.)
The large triangle is equilateral (all sides and angles are equal). The red triangle is made by connecting the midpoints of two sides and one of the vertices of the third. What is the area of the red triangle? (You don't actually even need math for this one.)
Two:
The four semicircles are all of radius 2. What is the area of the red square? (Not going to lie; I screwed this one up, and kicked myself when I read the answer.)
But now the fun one:
The red shape is a quarter circle of radius six. There are two semicircles and a smaller full circle inside, and all shapes are tangential to each other (i.e., just as it looks, the edges touch, but don't overlap). What are the radii of the smaller semicircle and the even smaller full circle?
This last one turned out to be a riot, because you can actually solve it with trial and error, but when Bellos published the answers today, he purported to have a simple solution that actually turned out to be faulty, because it rested on an assumption that wasn't justified by the information given in the puzzle. It took one of the Guardian's astute readers to come up with a proof that his assumption was valid. See if you can figure it out. (For my part, I came up with the answers by trial and error pretty easily, but never came close to figuring out the mathematical proof necessary to confirm the answer.)
