Powers of two in allowances

Does this guy look like he's worried about paying his kid's allowance?

The following is one of my favorite childhood memories:

I was about 8 years old, sitting with my mother at the dining room table after dinner. I was lobbying for an allowance--and also possibly procrastinating about clearing the dishes away.

An allowance was clearly not in the cards for me at the time, though I did eventually get one a couple years later. (This was the early/mid-sixties--my first allowance wound up at 35 cents a week. Neighbors also hired me as a babysitter at the going rate of 25 cents an hour!)

Perhaps to distract me, my mom decided to tell me about a classic brain-teaser that her own father had shared with her when she was a little girl, back in the Great Depression.

Suppose you could have a choice between two allowance plans.

Plan A: you get a dollar a week allowance.

Plan B: you get one cent allowance the first week, two cents allowance the second week, four cents allowance the third week, and so on, doubling every week after that.

Inevitably, I fell for the trap and said, "I'd take Plan A, of course!"

But my mom pressed me to think more carefully about the choice, and I was amazed to discover that Plan B would give me over a thousand dollars in less than four months!

In fact, the cumulative total would reach a million dollars around the 6-month mark, a billion dollars around the 9-month mark, and it would be over 45 trillion dollars by the end of the year.

Don't believe me? Here's a spreadsheet showing my calculations.

I've posed this mind-boggling problem to many children over the years since my mother passed it on to me, and it invariably inspires gasps when they work out the numbers for themselves.

As a child, I worked out the calculations by paper and pencil, and I still encourage kids to do a certain amount of this themselves. I like handing out long strips of adding machine paper tape for the purpose, so they can just keep doubling and doubling and doubling until they get tired of writing.

One 7 year old girl was inspired to take her tape home and then continue on larger pieces of paper until she'd figured it out for a whole year. I did suggest she might want to switch to a calculator or spreadsheet program at some point, but she wanted to see it for herself with pencil and paper calculations. The following week, she came back in proudly brandishing her total. She had figured out that even Bill Gates, the richest man in the world, could not have paid her Plan B allowance by the end of the first year!

Post script: That was twelve years ago. That 7-year-old girl is now 19 and majoring computer science at MIT. She worked as a Microsoft intern this past summer. MicroSoft pays its interns very well (and even buys a bicycle of their choice for commuting, which they get to keep afterwards.) But, even with twelve years of inflation and growing Microsoft stock prices, Bill still can't pay her allowance for a full year under Plan B! Even in his peak net worth year, a few years back, when his estimated wealth approached $100 billion, it would have only held out through week 43.