Looking at that graphic... it almost seems obvious. The outer corner radius would be relative to the inner curve radius in some fixed relationship, wouldn't it? The shallower the inner curve, the larger the outer curve has to be. A completely convex outside could have a flat inside. An inside that was concave from end to end could have a flat outside. Kudos to the guy for writing a proof! I wish this article explained better why it took 60 years to solve this...
The "obvious" solution that I think you're describing is the Hammersley sofa (1968), which has A=2.2074.
For a long time, it was thought this might be the optimal shape, but it was never proven. And it couldn't have been because it turns out that you can do better: the Gerver sofa (1992) is a more complicated shape, composed of 18 curve segments and has A=2.2195.
Nobody knew whether there might be an even better shape until now (assuming the proof holds up).
Here's a silly one: since 1, 3, 5 and 7 are primes, it almost seems obvious that all odd numbers are prime. Naturally, they are not, and there are countless proofs about various prime number generators to show that they can generate prime numbers, which are really prime.
I agree, modern definitions exclude 1 since "we lose" unique factorization. It's interesting to note [1] that this viewpoint solidified only in the last century.
No, 1 is excluded for reasons closely related to, but not conceptually identical with, the one you mention.
The "intuitive" argument that 1 is prime is that, as with prime numbers, you can't produce it by multiplying some other numbers. That's true!
But where the primes are numbers that are the product of just one factor, 1 is the product of zero factors, a very different status. The argument over whether 1 should be called a "prime number" is almost exactly analogous to the argument over whether 0 should be called a positive integer.†
It's more broadly analogous to the argument over whether 0 should be called a "number", but that argument was resolved differently. "Number" was redefined to include negatives, making 0 a more natural inclusion. If you similarly redefine "prime number" to include non-integral fractions (how?), it might make more sense to consider 1 to be one.
† Note that there is no Fundamental Theorem of Addition stating that the division of a sum into addends is unique. It isn't, but 0 is the empty sum anyway.
3 is also the product of the sets {3, 1}, {3, 1, 1}, etc.
We’re excluding the unit when defining these factor sets (ie, multiplicative identity) because it removes unique factorization.
That 1 is the unit is also why it’s the value for the product of the empty set because we want the product of a union of sets to match the product of a product of sets. But we don’t exclude it from the primes for that reason.
It's usually hard to explain why proving something is hard, because it's often just: existing/known/obvious approaches didn't succeed. Not terribly satisfying. Often you just have to try doing it and see, and even that won't be satisfying.
