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
Every Sunday morning, Alice and Beth bake a huge pie together. After it is ready, Alice chooses a positive integer less than 10 and declares it to Beth, then Beth chooses a positive integer less than 10 and declares it to Alice. They slice the pie by making straight cuts with a knife. The number of straight cuts they make is equal to the sum of the two numbers they have chosen. They slice it carefully in order to get the maximum number of slices possible.
After slicing, Alice and Beth take turns eating a few slices of the pie. The number of slices each eats in her turn is never more than 6.
Beth ensures that she gets to eat the last slices on the plate. How does she ensure this?