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
There are two sequences of natural numbers. The first sequence consists of all natural numbers that do not contain a particular digit, in ascending order. The second sequence consists of all the remaining natural numbers in ascending order. Every natural number occurs exactly once in exactly one of the sequences.
How many terms are there in the first sequence such that each of those terms is greater than the corresponding term at the same position in the second sequence?