Skip to main content

Printing Money: A Better Way

20 Dollars art4Image via WikipediaSince the United States really likes to print money, thought I would do our civilization a favor and calculate the optimal set of bank notes to produce the fewest numbers of notes needed for any amount of cash under $1000. (You're on the edge of your seat, aren't you?)
Current State of AffairsIn the United States, there are six bills currently in fair-to-wide circulation: $1, $5, $10, $20, $50, $100. If you were to always use the biggest bills possible, then the average number of bills you would use on a given transaction where the total was less than $1000 (this is also assuming that all dollar amounts are equally likely) is 8.71. The most number of bills you would need would be for $999 => 9 $100 bills, 1 $50 bill, 2 $20 bills, 1 $5 bill, and 4 $1 bills. That's 17 bills. We can take that number down to 13 by introducing the $500, which I did for comparison's sake.

The Currency Mapper

So like any devoted computer programmer, I created a webpage to do this for me. You can play with it yourself at (it's written in Javascript, so you can see the code yourself by right-clicking and selecting 'View Source' when you get to the page). What you do is enter the denominations that you want to test in comma-delimited-no-spaces fashion. Like this: '1,5,10,20,50,100,500' or '1,2,3,4,5,6,7'. Then, click 'Go' and it will display your efficiency rating along with a breakdown of which bills would be necessary for which dollar amounts. Here is an example of the output for '1,5,10,20,50,100,500', or the current US currency:
As you can see, the average number of notes for that set-up is 6.702. The 'Efficiency' number is just averagebills * totaldenominations. In this case: 6.702 * 7 = 46.914. And, as said earlier, Max Notes is 13 since 5 $100 bills can be replaced with 1 $500 bill.

We can do better!

Of course, there is room for improvement with anything, even in a monetary system crafted and propagated without competition or accountability by a pseudo-governmental entity over the past 100 years.

Here are the top five denomination schema that I discovered. Consider them carefully:

#5 - $1,$5,$10,$20,$50,$100,$500

Our standard system doesn't do so bad, especially for small values. While it is still the least efficient of the five, it averaged only 4.21 bills for any dollar amount $100 or less. Not bad.

#4 - Powers of 3: $1,$3,$9,$27,$81,$243,$729

Aside from this lowering the average number of bills across the board, it also makes the dollar bills more uniform, making it easier to divide things out consistently. For example, if there were three of you, and you needed to divide $27 between the three of you. Sorry, you have four? My theory totally falls apart.

#3 - Squares: $1,$4,$9,$25,$64,$144,$289,$529

What about dollar amounts being perfect squares? The squares that I picked worked a little better for values under $100, but still outperformed the standard at both levels.

#2 - Fibanocci A: $1,$3,$8,$21,$55,$144,$377

Then, the Fibanocci numbers occurred to me as being a viable option since they were an arithmetic sequence as opposed to a geometric sequence. Using all of them up to the necessary 300 or so was too many, so I tried every other.

#1 - Fibbanocci B: $1,$2,$5,$13,$34,$89,$233,$610

And the winner is: The other every other Fibbanocci number! This had an amazingly low max, standard deviation, and overall average. On average, you would only need 5 bills from here to $1000. Amazing. Under $100, it also performed the best averaging only 3.34 bills per dollar amount.

Since I know that you are wondering, here is the statistical breakdown of each schema. The right-hand column is all the statistical values multiplied together for comparison's sake:


Standard DeviationMeanMaxMedianMode


1,5,10,20,50,100,500To $10002.300540912152726.70213779821.40944809868


To $1001.736680916593664.21844935.862612333992


1,3,9,27,81,243,729To $10001.946050130677815.97412665022.30390364911


To $1001.536229149573723.94844774.751084713018


1,4,9,25,64,144,289,529To $10001.576375894563535.2829551873.44393189403


To $1001.315755342460033.69644466.093172513041


1,3,8,21,55,144,377To $10001.524714975251365.4139662674.06342017554


To $1001.215181742237213.59644418.800235644632


1,2,5,13,34,89,233,610To $10001.366798865648915.0278551374.17957952341


To $1001.12114659923213.34534224.677778486113

I'm sure that no one will mind needing to pull out an $89, 2 $5s, and a $1 to give a friend $100. Hey, it's more efficient!
Enhanced by Zemanta


Popular posts from this blog

How Many Teeth Does The Tooth Fairy Pick Up Each Night in Utah?

Somebody asked me a question about my Tooth Fairy post the other day that got me thinking. How many baby teeth are lost every day in Utah?

I began with Googling. Surely someone else has thought of this and run some numbers, right? Lo, there is a tooth fairy site that claims that the Tooth Fairy collects 300,000 teeth per night.

That's a lot; however, when I ran the numbers, it started to seem awfully low.

Let's assume that the Tooth Fairy collects all baby teeth regardless of quality and we assume that all children lose all their baby teeth. The world population of children sits at 2.2 billion, with 74.2 million of them in the United States. Of those, approximately 896,961 of them are in Utah. This means that somewhere around .04077% of the world's children are in Utah.

If we assume that kids in Utah lose teeth at the same rate as all other children in the world and that each day in the year is just as likely as the rest to lose a tooth, then we have that of the alleged …

I don't know you from Adam OR How to Tie Yourself Back to Adam in 150 Easy Steps

Last Sunday, I was working on my genealogy on, a free site provided by The Church of Jesus Christ of Latter-Day Saints for doing pretty extensive family history. While looking for information about a Thomas Neal, I found an individual who had done a bunch of work on his family including is tie into the Garland family, which tied in through Thomas's wife.

So, while I was pondering what to do about Thomas Neal (who's parents I still haven't found), I clicked up the Garland line. It was pretty cool because it went really far back; it's always fun to see that there were real people who you are really related to back in the 14th century or what not.
As I worked my way back through the tree, I noticed it dead-ended at Sir Thomas Morieux, who, according to the chart, was the maternal grandfather-in-law of Humphy Garland (b. 1376).  The name sounded pretty official, so I thought I'd Google him. I learned from Wikipedia that Sir Thomas Morieux married Blanc…

Twas the Night Before Pi Day

Twas the Night Before Pi Day
by Joshua Cummings

Twas the night before Pi Day, when Archimedes, the muse,
Went to pay me a visit whilst I took a snooze.

I'd visions of carrot cake, candy, and cheese
When dashed open my window and entered a breeze
That stirred me to consciousness, albeit in time
To see my face plastered in pie of key lime.

And once I'd removed the fruit from my eyes
And put on my spectacles did I realize
That before me presented a most divine spectre
Who clearly possessed the Key Lime Projector.

"It's a fulcrum, you see!" he began to explain,
"All I use is this crank to cause the right strain,
"Then releasing its fetter it launches sky high
"The juiciest pie of key lime in your eye!"

I sat there immobile for what seemed a year,
As the spectre protested I his genius revere,
When clearly it came, the fine revelation,
Of his piety, honor, achievements, and station.
With his little old catapult, so lively and quick,
I knew in a moment that this must be …