Update 12/21/2006: There is an error in this entry. I've done the math incorrectly. The entry still contains my original idea, but for now, it's incorrect. I'm going to work on it and update when I fix it. Information in this comment.

Today at work, I had another realization: "We need a two cent piece."

I had been making change for cash transactions all day and I noticed that with a two-cent piece, I could make change faster on average, and would give out less coins. This would make ME faster as a cashier, and it would have the added benefit of weighing less in a purse or pocket. I wouldn't be satisfied with an ambiguous result though - I came home and wrote a PHP script to figure it out for me. The results were rather astonishing.

The Script

First the script iterates through making change for a 100 cents using coins, for each 'new' coin (such as a 17¢ piece). Then it creates an average based on those numbers. From there I spit out the data into a comma delimited file and used Open Office's Calc program to import the averages and create a graph. The graph is shown below (click to get full-size version).

The Results

The script tells me that using our current coinage, the average number of coins when making change is 4.74. (Important! This assumes that each change amount is equal! There may be different results in real-world data.) Now for the amazing part: If we introduced a two or three-cent piece, the average number of coins when making change would drop to 3.94 - almost an entire coin!

It's very interesting to see trends in the graph - as you would expect, there are 'peaks' after each current denomination. This is explained by the need for pennies to fill the gap between the denomination and another higher denomination (such as nickels). There are some 'anomalies' in the smaller denomination of coins and not in the higher denominations. I attribute this to the fact that new coins that are near our current denominations would thwart the efficiency of current coins. For example, the only time a six-cent piece would be advantageous is when change needed ends in 6 or 12. If you are trying to make 14 cents, using a six-cent piece would only allow you a nickel, the six-cent piece, and three pennies - hardly an advantage to a dime and four pennies. In the larger denominations (over 50 cents), they are only able to be used once, so they 'normalize' back to the efficiency of our current denominations.

But why would a two-cent piece not bring the average down a whole coin? If you look at the averages, they differ by .2 coins, or twenty percent of a coin. Where does this come from? It comes from the exactly twenty situations when a penny is better than a two-cent coin - on all multiples of 5, plus one (6, 31, etc.).

I don't know the economic implications of introducing another coin. I don't know what resources (monetary and other) it would take, nor do I know the cost of creating another coin. However, I'm compelled by the data to believe that another coin in the two or three-cent denomination would be beneficial, to some degree.

Files

I've uploaded several files for your investigation.

• coinage.html - The raw data.
• coinage.php - the script that renders the html above.
• coinage.txt - the text version of the script above. Check my work! I may have made a mistake!
• coinage_data.ods - the OpenDocument format spreadsheet (includes graph object).
• coinage_data.xls - MS-friendly spreadsheet file.