I am sharing three examples of things that I was impressed to find existing. As they exist, they are clearly not impossible; a more accurate word might be inconceivable. Until I came across them, I had no conception of this stuff, and learning about them simply makes me glad. It also reminds me not to assume that something’s impossible just because I’d never heard about it.
Drilling a square hole
It turns out that you can drill a square hole, if you use a drill bit that’s based on a Reuleaux triangle and mount it on a special chuck. Such a thing was built by a guy called Harry Watts in 1914 and apparently you can still get them from the Watts Brothers Tool Works. The resulting hole has slightly rounded corners for practical reasons, but it still has four straight edges at 90 degrees to each other.
Assemble “Stonehenge” without a crane
A retired carpenter has shown on his site how he was able to assemble two vertical pieces and a capping piece (a la Stonehenge) by himself and without a crane. He also demonstrates some techniques that might have been used to move heavy stones in ancient times for other projects. Exactly how they did this will be a mystery, since they didn’t document it and aren’t around anymore, but it’s interesting to see simple techniques that would have made it straightforward.
Sharing a cake fairly
Of course, it’s easy to share a piece of cake two ways, while maintaining fairness (or “envy free”, i.e. no one feels someone else has a bigger piece) – one cuts, the other chooses. But, how to do it for more than two people? Well, in 1995, Brams and Taylor published a procedure for sharing between any number of people, involving cutting more pieces than necessary and taking turns trimming them. Assuming the people involved understand the proof, they should be happy that a fair distribution of the cake has been made, even if they each risk ending up with multiple pieces of different sizes.