Cute NumbersDate: 06/05/2001 at 05:05:16 From: Emily Subject: Cute numbers If a square can be cut into n squares of at the most two different sizes, then n is called a cute number. For example, 4 and 10 are cute numbers. _______ | | | |___|___| | | | |___|___| _______ |_|_| | |_|_|___| |_|_| | |_|_|___| a. Show that 6 is a cute number. I have already succeeded in doing so by having 1 square of 2 units squared and 5 squares of 1 unit squared. b. Show that 2001 is a cute number. c. Show that every integer greater than 5 is a cute number. Date: 06/05/2001 at 12:44:17 From: Doctor Peterson Subject: Re: Cute numbers Hi, Emily. This is a "cute" problem to play with on your own, so I don't want to spoil it by giving any direct help. The next step after solving part a is to try extending the solution you discovered there to larger numbers. See what kind of numbers you can show to be "cute" by using the same way of dividing a square. Then you will have to keep playing, trying other numbers and other patterns, and looking for a way to make the numbers you couldn't make yet. (It's very helpful to have the last question, which lets you know that all larger numbers CAN be done, so this is worth the effort.) Once you have found a pattern that works for 2001, you will have the third part; and by then I think you will also know how to handle any number. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 06/05/2001 at 12:52:15 From: Doctor Rob Subject: Re: Cute numbers Thanks for writing to Ask Dr. Math, Emily! Every even number from 4 up is a cute number, because of the following kind of dissection: 2*n o---o---o---o---o---o---o | 2*n-2 | |2 o o---o | | |2 o o---o |2*n-2 | |2 2*n o o---o | | |2 o o---o | | |2 o---o---o---o---o---o---o | | | | | | |2 o---o---o---o---o---o---o 2 2 2 2 2 2 This cuts a square up into 2*n smaller squares, for any whole number n >= 2. (Why are there 2*n smaller squares in the above picture?) Now if 2*n is a cute number, so is 2*n + 3. You get this by taking the above dissection of a square into 2*n smaller squares, and cutting the large square into quarters. (Why does this work?) This is enough to show that all whole numbers greater than 5 are cute. (How can you show this?) I leave the rest to you. By the way, another way to show that 2001 is a cute number is to take a square of side 45, and cut it up into 1 square of side 5 and 2000 squares of side 1. There are LOTS of ways to do a dissection for large numbers. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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