A math teacher who cares

Democratic election revisited

Almost a year has passed since we posted the problem "Democratic election" and no one has solved it. Perhaps we should suggest a hint at this point. The answer is "yes" it is possible to organize the election process in such a way that the President Tree gets elected even though only a minority supports him. Indeed, look please at the picture:



It symbolizes a two-stages election. Even though the "blue" candidate is supported by a majority (5 out of 9 voters) - he/she doesn't get elected if the election is organized as indicated. That is at the first stage there are three cells each of which vote for one person to vote at the next stage. Two cells will vote for the "black" candidate and the third for the "blue" one. Ultimately, at the second stage the voters vote for the "black" candidates, even though only a minority (4 out of 9) support him/her.

The President Tree can in principal organize something similar, if the election process is organized with enough initial cells and enough stages. Please complete the solution.

Democratic elections

The following problem has been inspired by the upcoming presidential elections in the USA.

In the country called Anmeria, ruled by President Tree the time for the next presidential elections has arrived. The country has precisely 20 millions of voters, of which only one percent (the regular army of Anmeria) support President Tree. He naturally wants to be reelected. However, on the other hand he wants the elections to appear "democratic". By "democratic elections" Tree means the following: All voters are divided into several equal groups. Next each of the groups is in turn divided into a certain number of (smaller) equal groups, and so on; each of the smallest groups so obtained elects a group representative - the elector. Then the electors elect electors for bigger groups and so on; finally, the representatives of the biggest groups elect the president. President Tree can divide the voters into the groups the way he wants. He can also instruct his supporters how to vote.

Question: Can President Tree organize the "democratic elections" in such a way that he gets reelected?

A week without food and sleep: Solution

In this post we publish a solution to the logical problem published on October 6, 2008.

A person cannot eat and sleep at the same time. Therefore the moments "seven days without sleep" and "seven days without food" correspond to different points in time. So the answer is this: The person must do what he or she did first seven days before. If he or she ate first and then slept, then the person must eat first and the other way around.

A week without food and sleep

Let us assume that if a person does not eat or sleep for seven days, then he or she will die. Then what if a person haven't eaten for seven days and haven't slept for seven days, what he or she must do first by the end of the seventh day in order to survive: eat or sleep? [In spite of a curious nature of the problem it has a well-defined and unique solution.]

Basic Trigonometry

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