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

[triominoes diagram] There is a square board made of at least 4 equal-sized squares. The number of squares in the board is a power of 2. An unlimited number of L-shaped triominoes are given. Each triomino is made of 3 equal-sized squares. Each equal-sized square that makes a triomino has the same dimensions as that of each equal-sized square that makes the square board.

The square board needs to be tiled with the L-shaped triominoes such that each square is covered at most once without parts of each triomino extending beyond the 3 squares it covers.

What is the minimum number of squares that can't be covered with the triominoes?


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