In finding coin denominations that minimize the average cost of making change, Shallit assumed that every amount of change between 0 and 99 cents is equally likely. For the current four-denomination system, he found that, on average, a change-maker must return 4.70 coins with every transaction.

He discovered two sets of four denominations that minimize the transaction cost. The combination of 1 cent, 5 cents, 18 cents, and 25 cents requires only 3.89 coins in change per transaction, as does the combination of 1 cent, 5 cents, 18 cents, and 29 cents.

U.S. quarter.

"We would therefore gain about 17 percent efficiency in change-making by switching to either of these four-coin systems," Shallit says. "The first system possesses the notable advantage that we only need make one small alteration in the current system. We could speed up customer service just by replacing the dime with an 18-cent piece."