Penny Doubled Daily - Exponential Growth of One Cent

A comments discussion of how a church grows akin to cell division over at The Original Mud Puppy turned to that old mathematics trick of penny doubling based on exponential growth.

Penny doubling is when you start with a single penny ($0.01) and you double the value each day. So, on the first day you get $0.01, on the second day you get $0.02, the third day gets you $0.04, with $0.08 on the fourth day, and so on.

Back in the mid 80's, in high school, I had a computer course (using old TRS 80s) where we had to construct a BASIC program that found the exponential total of the penny doubled daily. I think we had to go to 30 days. Huge number. $10, 737, 418.23

What is odd is that the aggregate cumulative total is always 0.01 (or 1 if using whole numbers) less than the next doubled amount. In the above, if we went to 31 days, the next number would be 10737418.24 to be added for a total of $21,774,836.47

There is a classic riddle involving this process where a child is given two choices: a linear progression of $1 a week allowance, or an exponential progression starting at $0.01 and doubling each week. We did something similar to the kids once, as an object lesson about thinking things through before making snap choices.

They get $1 a day allowance, paid every two weeks, for a total of $28 a month. (Out of this they are responsible to manage their money and purchase anything they want, as well as gifts for celebrations, snacks at school, movies, skating, etc, and replacement materials for things they break or misuse. We hope it teaches them some fiscal responsibility and the value of money).

We told them that for the next two weeks, instead of paying them bi-weekly, we would give them their allowance for the day, each day. They could either get $1.00 a day, or they could start out at a penny ($0.01) and we would double it each day. We explained how that works, much as I did in my second paragraph above.

The youngest (just turned 8 at the time) immediately sat down and tried to figure it out. He bent over a piece of paper, pencil in hand, and furiously scribbled figures down for a minute or so. For whatever reason, he stopped after only 7 days and declared it was a rip off and wanted the $1.00 a day. We shrugged and agreed, handing him his first dollar which he happily snatched and stuffed in his piggy bank.

His older brother, (12 at the time) tried to be a show-off and was working the figures in his head. I could see he was having trouble, but had no clue how far he had figured it out. When his little brother disgustedly dropped his pencil and called the penny doubling a rip-off, you could see his eyes sort of flicker in relief and he also stated he would rather have the dollar per day.

He got his dollar, stuffed quickly into his jeans' pocket, and they both went their way. I started a chart on the whiteboard we use for chore lists and calendar. I made two rows of 14 boxes in a line and wrote $1.00 in the first top box, $0.01 in the first bottom. Each day I'd write the next amount to be given, as well as the total to that point.

It was on day 9 when the oldest boy noticed the penny doubled row was now at $2.56 for the day, and a cumulative total of $5.11. His brow furrowed in deep thought; his fingers flicked, counting phantom numbers his lips silently mouthed. Suddenly he smacked his hand on his forehead (I hate when he does that;, little drama queen sometimes, hehe) and yelled to his brother they were getting ripped off.

Yes, he had figured out that even though as of that day, the penny double was at a total of $5.11, the next day would add $5.12 to that be a total of $10.23. A whole $0.23 more than they each would have earned by that day. He grabbed some paper and a pencil and figured out for a couple more days and threw a fit.

The both cried unfair, we tricked them. We didn't tell them they would end up with so much money at the end of 14 days. We shrugged and told them they should have worked it out. I would have helped them with the math, had they asked. And, yes, we would have made good on the payment if they chose the right one.

Anyhow, this all got me to thinking today ... what if someone had made that deal with you for an entire year?

I know, pure nonsense. No one has that kind of money, but it is an interesting fantasy. And, just how much money would that be?

To find out, on each daily blog entry, at the bottom, I will write the new total earned if it were a real deal. Since I'm starting on February 1st, I'll be using a figure of 32 days. I am also creating a new post which will show the entire process from $0.01 to as high as I am able before my brain explodes and my computers melts to plastic slag. Each daily blog total will link to that consolidated information page. (I'll have that page up tomorrow sometime ... hehe)

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Today's Penny Doubled Daily Cumulative Amount for 32 days is:

$42,949,672.95