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
Two mathematicians perform a trick with a shuffled deck of distinct cards. One mathematician asks a member of the audience to select five cards at random from the deck while the other mathematician is blindfolded. The audience member hands the five cards to the first mathematician who examines the cards, hands one of them back to the audience member, arranges the remaining four cards and places them face down into a neatly stacked pile on a table. The audience member is then allowed to move the pile on the table or change its orientation without disturbing the order of the cards in the pile. The second mathematician now removes his blindfold, examines the four cards on the table and determines the card held by the audience member from these four cards and their order in the pile.
If there were one more distinct card in the deck, the mathematicians cannot perform this trick. How is this trick done?