As a kid, I watched Sesame Street. One segment was on sharing. Bert has a cookie, and Ernie comes and asks him “will you share your cookie with me, Bert?”
Bert says “what’s ‘sharing?'”, and Ernie explains. While he explains, he takes half of Bert’s cookie and eats it. Bert seems a little unconvinced, however. At the end of the segment, Bert has half a cookie, and Ernie asks him “will you share your half-cookie with me, Bert?”
It stuck in my childlike as a very funny end to the segment – clearly, Bert would end up with no cookie at all, as Ernie ate first 1/2, then 1/4, then 1/8, then 1/16, and so on.
If you ever study “geometric series”, one thing you’ll learn is that 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … equals 1.
There are other series that add up to 1, and we can figure out some of them the same way. If Ernie had only taken 1/3 of Bert’s cookie fragment each time, he would have eventually eaten the whole cookie, but as 1/3 + 2/9 + 4/27 + 8/81 + 16/243 + …., so this infinite sum must also equal 1.
The other day, my wife and I went to an Indian restaurant we both love. At the end, I’d eaten my prata (a flat bread), but my wife hadn’t started hers. So she tore off half and gave it to me. I didn’t want to be greedy, so I tore off half of her half and gave it back. Eventually, she still wasn’t eating it, so I took the quarter prata I’d given back to her, tore it in half, and took half of the quarter. What if this had gone on forever?
Confused yet? Imagine then, that Ernie takes half of Bert’s cookie. Then Bert takes half of Ernie’s half cookie. Then Ernie takes back half of the piece that Bert just took back. Then Bert takes back half of that, and so on forever. You can write this like so :
- Bert starts with a whole cookie.
- Ernie takes half (I take half). Bert now has 1 – 1/2.
- Then Bert takes back a quarter, so Bert now has 1 – 1/2 + 1/4.
- Then Ernie takes back half of that, so Bert now has 1 – 1/2 + 1/4 – 1/8
- And so on, ad infinitum.
In the “end”, Bert has 1 – 1/2 + 1/4 – 1/8 + 1/16 – 1/32 + … of a cookie. My wife has 1 – 1/2 + 1/4 – 1/8 + 1/16 – 1/32 + … of a prata. But how much is this as a simple fraction?
If you know about geometric series, this is a simple (and not very interesting) problem. However, I’m almost certain that it can be solved without using any geometric series theory. So I present this puzzle to all of you who don’t know about geometric series : What does 1 – 1/2 + 1/4 – 1/8 + 1/16 – 1/32 + … add up to? And if you do know about geometric series : Can you find a way to get the answer that a primary school kid could understand?