Problem of the Week: Skipping Around
Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s). Each week’s entries are scored by a faculty judge, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button. The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.
THE RULES:
The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited. Send solutions to bkennedy@gettysburg.edu. This problem was posted on Friday, March 18 and solutions are due on Friday, March 25 by 5:00 p.m.
THE PROBLEM:
Consider the following game. We start at the number 1. Each turn, we go to a new number by making our choice of the following two moves:
- we multiply our current number by the positive whole number a; or
- we reduce our current number by the positive whole number b.
So, for example, if a = 3 and b = 4, the first four numbers we reach could be
1, 3 (multiply 1 by 3), 9 (multiply 3 by 3), 5 (reduce 9 by 4).
QUESTION:
Can you find a choice of a and b (and a sequence of moves) so that the first ten numbers we reach are (in any order) exactly the numbers
1, 2, 3, 4, 5, 6, 7, 8, 9, and 10?
If this is possible, say what a and b are, and give the sequence of moves. If this is not possible, explain why.
BONUS QUESTION:
Exactly the same question as above, but this time start at the number 2 instead of the number 1.
