Imagine Magazine - Johns Hopkins - March/April 2011 - (Page 47)

knossos games by Tim Boester Unusually Ordered Sets The number systems we normally deal with are well ordered; that is, we may compare any two numbers in the set to one another. However, sets can be ordered in other ways. In a partially ordered set, we can compare a given number only to some of the other numbers in the set, according to the rules of that set. For example, in a partially ordered set in which only numbers with the same number of digits may be compared, we could say that 2 is less than 3, and that 12 is less than 22, but not that 7 is less than 12. Those last two numbers simply cannot be compared because they do not have the same number of digits. This puzzle is based on a more complicated ordering of a set, in which numbers may be compared with numbers that have the same number of digits, one more digit, or one fewer digit. For this puzzle, move from hexagon to hexagon, following the black lines, starting with the number marked by an S and finishing with the number marked by an F. The first time you land on a hexagon, use the top number on that hexagon. If you revisit the hexagon, use the bottom number. Each number you use to move from the starting number to the finishing number must be “lower” than the one before, by the following rules: 1. You may move between numbers with the same number of digits if one and only one digit is lower in the second number. Example: You may move from (3, 7) to (3, 5) or (2, 7), but not to (2, 5) or (3, 8). You may move from (3, 7, 2) to (3, 4, 2) but not to (3, 4, 1). 2. You may move from a number to a number with one more digit, if the first digits are exactly the same. Example: You may move from (3, 2) to (3, 2, 7) but not to (3, 1, 7). 3. You may move from a number to a number with one fewer digit, if one and only one of the remaining digits is lower in the second number. Example: You may move from (4, 6, 2) to (4, 3) or (3, 6) but not to (4, 6), (5, 6), or (4, 7). Solution to Knossos Games 18.3: Here’s the solution path to the small example to help you get the feel of moving according to these rules: S:(3, 6) – (3, 4) – (3, 4, 7) – (3, 4, 3) – (3, 1, 3) – (2, 1, 3) – (2, 0):F example Tim Boester is an Assistant Professor in Mathematics Education at Wright State University. You can find more puzzles on his website at homepage. mac.com/ boester www.cty.jhu.edu/imagine imagine 47 VEEr Puzzle http://homepage.mac.com/boester http://homepage.mac.com/boester http://www.cty.jhu.edu/imagine

Table of Contents for the Digital Edition of Imagine Magazine - Johns Hopkins - March/April 2011

Imagine Magazine - Johns Hopkins - March/April 2011
Contents
Big Picture
In My Own Words
Problem Solving
Orange County Math Circle
Number Theory
Count Me In
National Mathematics Competitions
Math at the Science Fair
Just My Speed
MathPath
When Origami Meets Rocket Science
Selected Opportunities & Resources
Making a Difference
High School Options for Gifted Students
Off the Shelf
Word Wise
Exploring Career Options
One Step Ahead
Planning Ahead for College
Students Review
Mark Your Calendar
Knossos Games

Imagine Magazine - Johns Hopkins - March/April 2011