Problem of the Week: Titans of Finance
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 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 email@example.com. This problem was posted on Friday, November 12 and solutions are due on Friday, November 19 by 5:00 p.m.
Professors Campbell Hetrick and Glass were at the newsstand browsing math journals when Professor Campbell Hetrick decided to buy a pack of gum for 19 cents. Both professors were annoyed by the fact that, to make 19 cents in exact change, you need at least six individual coins (one dime, one nickel, and four pennies).
Accordingly, Professors Glass and Campbell Hetrick are launching a new currency called the Glatfelter. Glatfelters will be issued in coin form only, and each coin will be worth a positive whole number of Glatfelters.
Professors Glass and Campbell Hetrick want people to be able to make any whole number of Glatfelters from 1 to 20 using no more than three individual coins (not necessarily of different values).
We could achieve this objective, for example, with seven different values of coins.
If we create coins with values (say) 1, 2, 4, 5, 7, 9, and 10, we can make any whole number from 1 to 20 using no more than three individual coins: 1 = 1, 2 = 2, 3 = 1 + 2, · · · , 18 = 10 + 7 + 1, 19 = 10 + 9, 20 = 10 + 10.
What is the smallest number of different values of coins that Professors Glass and Campbell Hetrick could create to achieve their objective? In your solution, make it clear both that your answer does work and that any smaller number does not work.