It all started with a bag of peanut M&Ms. There were three of us and we would take the M&Ms out of the bag three at a time; Zoe would pick her favorite color out of the three, then Katie would do the same, and I would get the one that was left over. Finally, we took out the last three and there were none left in the bag. It worked out perfectly. And all of a sudden, Zoe said that she wasn’t feeling too well and she did not want her last M&M.
“So what do we do now?” I asked Katie.
“I know!” she replied immediately. She took one of the M&Ms, bit off half, and then gave me the remaining half.
“Now there is one left for each of us and it’s all fair,” she continued.
“So how many did each of us get in total?” I asked.
K: One and a half.
Me: And if there were 5 of them, how many would each of us get?
K: Two and a half. And if there were 7 of them, then we’d each get three and a half.
At this point Katie paused and thought for a moment.
K: You know, I am 7 right now and Zoe is three and a half. That means that Zoe is half my age!
Me: That’s a nice observation, but it’s not fully true because you are actually seven and a half.
K: I guess that’s true… Wait, but next year she will be 4 and I will be 8 so then she’ll be half my age.
Me: Nice. Will she ever be half your age again after that?
K: (confidently) No.
Me: Why is that?
K: (less confidently) Well…when she is 5 I’ll be 9 and twice 5 is 10, then when she’s 6 I’ll be 10, and twice 6 is 12, when she’s 7 I’ll be 11, and twice 7 is 14. You see, twice her age is getting further away from my age.
Me: So what is special about 4? Why will she be half your age only when she is 4?
K: Because then I’ll be 8.
Me: Right, but why does it only happen with 4 and 8. Why will you not be twice her age when she is 5, 6, or 7?
Katie thinks about this for a moment, then replies, “Oh, it’s because I am 4 years older than she is!”
And it all started with just a bag of M&Ms…