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
A mathematics teacher narrated the most important story of his life to his students one day, "Seven years ago, I was sentenced to an unusual punishment after having been convicted of theft. I was taken to a closed space with many doors in front of me. I was told that all but one of them had hungry lions behind them. If I could open the one without a lion behind it, I would be free. If I opened the wrong door, I would be killed."
He continued, "Only the king knew which one of those doors would lead me to freedom. He allowed me to ask two yes-no questions under the condition that if I managed to get out that place alive, I would have to serve his kingdom as a mathematics teacher for the rest of my life. The king promised me that he would answer both the questions truthfully if he knew the answers. In case he didn't know the answer to a question, he would clearly say so. I was vey lucky. The number of doors was just right. Had there been one more door, I might not have got out of that place alive."
What two questions might he have asked?