[ninty]

Tom Spears
The Ottawa Citizen


Monday, June 02, 2003
Dimes, quarters and loonies are less efficient than a money system with coins worth 13 cents and $1.60, says a University of Waterloo mathematician.

These odd coins would allow us to make change more efficiently, so that we wouldn't rattle around with heavy piles of change in our pockets.

And the Americans (who are different because they have a $1 bill) would benefit by replacing the dime with an 18-cent coin.

There's no special use for this knowledge. This is just how mathematicians like Jeffrey Shallit have fun, and stretch their brains a little at the same time.

He is exploring coins of all denominations to replace the existing ones of Canada, the U.S. and the European Union with its euro.

"You realize the suggestion is tongue-in-cheek?" he cautions in an interview.

Not every one has realized; since he published a recent article called What this country needs is an 18-cent piece, some people think he wants the mint to crank out funny money.

His paper ran in the Mathematical Intelligencer, which you won't find at the supermarket. Normally he writes less catchy stuff like Factoring with cyclotomic polynomials and An efficient algorithm for computing the ith letter of phi^n (a).

Mr. Shallit teaches computer science and number theory at Waterloo, and those two fields meet in the problem of how to make change efficiently, which math types call an optimal denomination problem.

First he went looking for the four coins that would make all possible combinations of chance from a penny to 99 cents, with as few coins as possible. That's based on adjusting the current American system of a penny, nickel, dime and quarter (ignoring the rarely-used 50-cent piece).

Once he set up the computer, it took less than a second to calculate that coins worth one, five, 18 and 25 cents would require, on average, 3.89 coins to make change from a dollar.

One other combination was tied (a 29-cent coin in place of the familiar quarter). All the rest would take on average more coins to make all the amounts of change -- including the mix Americans use today, which uses an average of 4.7 coins to change a dollar.

But Canada's problem is tougher. Since we have no $1 bill our six coins have to make every amount up to $4.99.

Mr. Shallit's computer has been mulling over this one for a month at the back of its electronic brain while he does other work. So far his best result is one cent, eight cents, 13 cents, 69 cents, $1.10 and $1.60, but Canada's small change is still a work in progress.

Now, what should we call a new $1.60 coin?

"Maybe a cultural icon," he suggests. "Maybe a Gordie, or an Elvis. I'm not sure."

Behind the fun there are some serious math issues.

"Number theorists have been writing papers about how to do this efficiently for years," he says. "The real interest in my article is the mathematics at the end, but I was looking for a catchy way to express the ideas." And that meant inventing funny coins.

Besides, big pocketfuls of change bother him.

Eventually his computer will stop, but Jeff Shallit won't be finished. He's also taking a look at stamps.