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
Some perfectly circular coins are placed in a nearly circular chain. Each coin touches two other coins in the chain. There is room for least one more coin inside the chain.
A new coin is placed inside the chain and rolled around the chain once completely. The rolling coin touches at least one coin in the chain throughout the roll. It rolls without slipping. Then this coin is placed outside the chain and rolled around it in a similar manner in the opposite direction. All coins have the same radius. There is enough room for the rolling coin to touch each coin at least once in each roll.
How many times should we perform these two rolls so that the number of rotations made by the coin while rolling is twice the number of coins in the chain?