Departmental Undergraduate Scholarships 2009-10

• £1,000 in the first year of study in the Computer Science Department at RHUL.
• renewable at the rate of £500 in years two and three.

Please read the Rules of the Challenge.

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These are the four questions that you must answer as part of your scholarship entry.

The Four Questions.

These questions can all be solved without the use of a computer. Your answers should be submitted in plain text (txt), Word (doc), pdf, rich text format (rtf) or the Open document format (ODF). You should justify your submission by explaining carefully how you arrived at each of your answers.

1. Find four different whole numbers a,b,c,d so that when they are placed at the corners of a square the sum of the numbers on each side is a perfect square. That is, a+b, b+c, c+d and d+a are all perfect squares.

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2. How many rooks (castles) can be placed on a chess board so that every rook is attacked by precisely one other rook.

   This question is similar to the question asked in the programming challenge. It is easier and can be done by hand.

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3. On some trains and many airlines there is a seat reservation system.

   Suppose that there are 23 seats in total, numbered 1 to 23. The first person to arrive sits in a random seat (chosen uniformly from 1 to 23, without looking at their ticket).

   Each subsequent person, when they arrive, sits in their reserved seat if it is available, otherwise they sit in a random empty seat.

   You are the last to arrive, so there is precisely one empty seat left when you arrive.

   What is the chance that you sit in your reserved seat?

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4. You have two sets of four numbered wooden blocks. Each set contains blocks with the numbers 1,2,3 and 4.

   You have to place all eight blocks in a tower so that there is precisely:

   1. one block between the two blocks numbered 1;
   2. two blocks between the two blocks numbered 2;
   3. three blocks between the two blocks numbered 3;
   4. four blocks between the two blocks numbered 4.
   How many ways can you do this?