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?

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?

How many positive integers \(n\) are there such that \(n! + 1\) is composite?

A group of friends wants to know their average salary such that no individual salary can be deduced. There are at least three friends in the group. How can this problem be solved with a pen and paper?

There are 100 prisoners in a prison. The warden will set them free if they win a game involving red and blue hats. All the prisoners will be made to stand in a straight line. The warden will blindfold all the prisoners, then put either a blue hat or a red hat on each prisoner's head, and finally remove all the blindfolds. Each prisoner can then see the hats of all the prisoners in front of him but he cannot see his own hat or the hats of those behind him. If at least 99 prisoners can correctly declare the colour of his hat, the warden will set them free.

Once the game begins, each prisoner is allowed to utter "red" or "blue" only once to declare the colour of his hat. They will not be allowed to communicate in any other manner. The warden will give them one day to decide a strategy to win this game. What should their strategy be?