Puzzle of the week

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?


Previous puzzle

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

For a given even positive integer, we create two lists of integers such that every positive integer less than or equal to the given integer belongs to exactly one of the two lists and both lists contain an equal number of integers. There is a third list which is initially empty. Then, at every step, we remove the smallest number from the first list, the largest number from the second list and insert the positive difference between the two numbers into the third list.

What do we get if we divide the square of the given even positive integer by the sum of all the numbers in the third list when the first list becomes empty?


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