How can you construct a plane where every point is coloured either black or white such that two points of the same colour are never a unit distance apart?
Puzzle of the week
Alex and Bob work as financial advisors for the same company. They draw equal salaries from the company. They behave well at the office. Both work on similar assignments. Each assignment requires a yes-no decision. The company uses the decisions made by them to make profits.
After the recession hit the company very badly, one of them has to be fired. Both Alex and Bob have worked on almost the same number of assignments in the last ten years. Alex has been consistently taking about 80% decisions correctly every year. Bob, on the other hand, has been taking only about 5% correct decisions every year.
Assuming that the performances of Alex and Bob would remain the same in future, who should the company fire to maximize its profits in the years to come? Why?
Random puzzle from the past
All horses are the same colour; we can prove this by induction on the number of horses in a given set. Here's how: "If there's just one horse then it's the same colour as itself, so the basis is trivial. For the induction hypothesis, horses \(1\) through \(n - 1\) are the same colour, and similarly horses \(2\) through \(n\) are the same colour. But the middle horses, \(2\) through \(n - 1\), can't change colour when they're in different groups; these are horses, not chameleons. So horses \(1\) and \(n\) must be the same colour as well, by transitivity. Thus all \(n\) horses are the same colour; QED." What, if anything, is wrong with this reasoning?