Problem of the Month for April, 2008

Problem: A square sheet of graph paper is ruled (by horizontal and vertical lines) into a number of small squares ¼ inch on a side. Along the lines of this grid we cut out another square (this square could be anywhere inside the original square). There are 148 small squares in the piece that remains after we remove the square that we cut out. Is it possible to determine the dimensions of the original square? If so, find the dimensions and carefully explain how you determined the answer. If not, explain why it’s not possible to determine.

Possible Extensions/Generalizations to Consider: What if there are n small squares in the piece that remains after we remove the square that we cut out? Can you find conditions on n that will guarantee that we can determine the dimensions of the original square? Can you give an algorithm for finding the dimensions in such cases? Are there values of n that cannot occur in this set up? What about paper of other dimensions (like rectangles where the height is twice the width)? What about cutting cubes out of cubes? You could do a lot of thinking about a lot of possible generalizations here if you have the time and interest.

Rules

1. You must be a Bethel University student during the given month.
2. Your solution should be written in complete sentences (at least if you want to win) and either typed or written very neatly by hand. Equations and diagrams may be included by hand or by computer as necessary.
3. Your solution must be turned in to P.O. 95 by 4 PM on the last day of classes of the given month.
4. The winner will be the person who does the best job answering the problem as judged by a faculty member of the math and computer science department. If more than one person answers the problem correctly, the person who does the best job in communicating their solution and/or considering generalizations of the given problem will win. If no one answers the problem correctly, the best attempt will win.
5. Do not put your name on your solution paper. Instead, put your Bethel ID number in the top right corner of your solution paper.

Suggestions

1. Be thorough, yet concise. Be sure to answer the question completely and in such a way that clearly communicates your solution, while at the same time being as efficient in your communication as possible.
2. If you think other people will also answer the question correctly and are also good writers, you can increase your chances of winning by considering and writing about possible generalizations of the given problem and the solutions to those generalizations. However, a correct answer to the original problem that does not consider generalizations will beat out an incorrect answer to the original problem that does consider generalizations. In short, make sure your answer to the original problem is correct before considering any generalizations.
3. Neatness counts. Grammar and spelling count. When relevant, pictures are helpful.
4. Explicitly state any assumptions you are making. If you are unsure whether a particular assumption is "allowed", say so in your write-up but then answer the question by either making the assumption in question or by stating why you think you can't or shouldn't make the assumption.

Prizes and Benefits

1. Your picture and a short biographical sketch will be posted, as well as your solution, for all to admire. This will be done temporarily in the math and computer science hallway, and, perhaps, for as long as Bethel exists on the internet. You can inspire and show your accomplishment to your friends, children, grandchildren, your future bosses, and more!
2. You will win a $25 gift certificate from House of Wong restaurant.
3. You will earn some extra credit in your math and computer science courses of the given month (amount to be determined by your professor).
4. You might be able to get an extension on an assignment for your math/cs courses if you are working on the problem of the month (discuss this with your professor).