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Fri 9 Dec, 2005 07:02 am
On the planet Zeta, time is measured in days, weeks, and years, where a year consists of 22 weeks that each has the same number of days (greater than 5). A single grid that contains the whole year is always used for their calender, so every day is numbered consecutively. Suppose a five-by-five square of 25 grid days is chosen at random from this calender. Consider the products of the day numbers that appear in the middle of opposite sides of the square. What is the absolute value of the difference between these products, regardless of where you place the 5 by 5 square?
I'm not sure I completely understand the question, but here is my take.
Let's say there are N days in a week and the calendar has N columns and 22 rows. If the start of your 5x5 box is day X, then the first row is
X, X+1, X+2, X+3, X+4
The next row is
X+N, X+N+1, X+N+2, X+N+3, X+N+4
You can extrapolate from that what the entire box looks like. If I read the questions correctly, you are being asked what the difference is between the products of the top middle and the bottom middle vs the right middle and the left middle. The top middle is X+2. The bottom middle is (X+4N+2). The left middle is (X+2N) and the right middle is (X+2N+4).
If you take the products and substract them, you are left with 4N^2 - 4.
If I had a picture of how the grid is arranged, I could do better. It might be set up so that N is 22 and then you could produce a concrete result. If you have more insight into the question, you should be able to use the approach above to answer the question.
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