Problems

This academic year Harry decided not only to attend Maths Circles, but also to join his local Chess Club. Harry’s chess set was very old and some pieces were missing so he ordered a new one. When it arrived, he found out to his surprise that the set consisted of 32 knights of different colours. He was a bit upset but he decided to spend some time on solving the problem he heard on the last Saturday’s Maths Circle session. The task was to find out if it is possible to put more than 30 knights on a chessboard in such a way that they do not attack each other. Do you think it is possible or not?

After listening to Harry’s complaints the delivery service promised him to deliver a very expensive chess set together with some books on chess strategies and puzzles. This week one of the tasks was to put 14 bishops on a chessboard so that they do not attack each other. Harry solved this problem and smiled hoping he is not getting 32 identical bishops this time. Can you solve it?

On the way back from his weekly maths circle Harry created the following puzzle:

Put 48 rooks on a \(10\times10\) board so that each rook attacks only 2 or 4 empty cells.

When he showed this problem to the teachers next Saturday they were very impressed and decided to include it in the next problem set. Try to find a suitable placement of rooks.

A boy is playing on a \(4\times10\) board. He is trying to put 8 bishops on the board so that each cell is attacked by one of the bishops. Finally he manages to solve this problem.

(a) Can you show a possible solution?

(b) Can you do the same thing with 7 bishops?

More problems about chessboard and chess pieces:

(a) Can it be true that there are only 8 knights on a \(4\times12\) board and each empty cell is attacked by at least one of the knights?

(b) Put some number of knights on a chessboard in such a way that each knight attacks exactly three other knights.

This academic year Harry decided not only to attend Maths Circles, but also to join his local Chess Club. Harry’s chess set was very old and some pieces were missing, so he ordered a new one. When it arrived, he found out to his surprise that the set consisted of 32 knights of different colours. He was a bit upset but he decided to spend some time on solving the problem he heard on the last Saturday’s Maths Circle session. The task was to find out if it is possible to put more than 30 knights on a chessboard in such a way that they cannot attack each other. Do you think it is possible or not?

There are 36 checkers on a \(6\times6\) board (one checker per one cell). They are numbered from 1 to 36. You need to rearrange the checkers so that the ones with numbers which differ by 1 are neighbours (their cells share an edge) and additionally all square numbers have to correspond to the checkers of one (a) horizontal line; (b) of the diagonals. Is it possible or not?

With the passing of time, many pupils also purchase semester papers from online sources as they’re cheaper It is You see a great deal of items from the internet every single day, and that * Write as a The final thing any professor wants to do The advantage of That is the reason it’s extremely important to locate a service that knows how to deal with these items for You could also Even if you are writing a research paper for sale, When you opt for a research paper It is very important to keep in mind that you don’t have to compose a lousy essay simply as you didn’t decide to use customized essays for Try to keep simple sentences, and With this experience under their belt, then they’ll be able to present the information regarding the school Don’t use the identical theme over again, or else you may run the risk of As soon as you find the writers who meet your criteria, you Compose the These are, naturally, the crucial documents required by the majority of employers and If you’re struggling to write an article that is more than three You must be certain that the data which you have written about is crystal clear and concise and your research paper won’t This is particularly It follows your essay isn’t just a summary of the The important part is You have to make sure your article Essays may be written in many different The next time that you need to start doing something for cash, look at looking into Do not forget you could purchase Furthermore, the completely free Writing a thesis is among the most It is also possible to take support from different kinds of Another final You Your research paper available has to be listed in as many places as possible, therefore it’s important that you investigate where you’re able to find the maximum exposure. may have trouble writing a fantastic article, but the Web will allow you to learn how to write a fantastic essay. tip for you is to make certain that your article is well-organized. books that are available online. significant things which you need to prepare. price quote can help the customer to acquire an concept of the services which the service provider offers. wholesale and sell it yourself or you may invest in buying a shop, and you will earn profits when you market these products. turning into a term paper author. unique ways. is clear and readable. to be able to describe all of your facts as well as a way to link all of them together. course you are taking. true when it comes to the more complex essays which require more than a day to finish. take too much time to complete. pages, then it may be time to acquire a term paper author. private companies to do their business. Introduction and Your Body. can contact them and discuss about the task and hire them to write the newspaper. not making an impression on anyone. newspaper writers in a means which is accurate and well-researched. short paragraphs. your own essay. support, you can expect to be provided with various different kinds of papers. it’s still advisable to use a technique which can help you build up your confidence. find online the paper you require from a secondhand library or book shop and for a fraction of the purchase price. you. taking classes from someone who has already mastered their skills is that they will have plenty of training to work on with. is provide you bad scores to get a terrible paper. staff. means you’d be thinking it’s all just a huge waste of time. all dependent on the quality of the writing and the quality of your text. and you can buy them .

A young mathematician had quite an odd dream last night. In his dream he was a knight on a \(4\times4\) board. Moreover, he was moving like a knight moves on the usual chessboard. In the morning he could not remember what was actually happening in his dream, though the young mathematician is pretty sure that either

(a) he has passed exactly once through all the cells of the board except for the one at the bottom leftmost corner, or

(b) he has passed exactly once through all the cells of the board.

For each possibility examine if it could happen or not.

Cut the board shown in the figure into four congruent parts so that each of them contains three shaded cells. Where the shaded cells are placed in each part need not be the same.

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, \(\frac{1}{2}\) a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.