Problem of the Month for January, 2008

Problem: Farmer Ben has forty-three pieces of fencing, each of whose lengths are different. In fact, the lengths are 1, 2, 3, 4, 5, …, 39, 40, 41, 42, and 43 meters. Determine whether or not it is possible to fence a square field using all of these pieces. Explain why or why not. Next, imagine “gluing” pairs of these pieces of fencing together into longer straight pieces (except for one piece which will not be paired with any other…and therefore it will not be glued with any other) so that all of these new glued pieces have the same length (and have the same length as the one original piece that was not glued to any other). Describe all possible rectangular fields that can be fenced with these new pieces. Explain how you know your list is complete.

Possible Extensions/Generalizations to Consider: Imagine going back to the original group of forty-three pieces of fencing with the lengths given above. Explain why any field with dimensions a x b can be fenced, where a + b = half the total length of all the fencing and a ≥ 3.

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).