Where: side of the house

When: this afternoon

# Sums of Cubes

Cathy O'Neil (THE Mathbabe) flagged a proof-by-picture of Nichomachus's theorem yesterday and suggested it would be a fun discussion with kids. J2 and I started exploring it today. My current favourite introduction is to say: "a friend thought you might be interested to explore ..." In this case, to explore patterns from adding up perfect cubes.From past conversations, his natural inclination was to start with 1

^{3}, then 1

^{3}+2

^{3}, etc and look for patterns. Initially, he confused 2

^{3}and 2

^{4}, so we clarified that and he embarked on a bunch of calculating. I kept notes for him. To give an easy extra term for his pattern seeking, I started with zero cubed.

**First conjecture**

We built our table to this level:

0^{3} | = | 0 |

0^{3}+1^{3} | = | 1 |

0^{3}+1^{3}+2^{3} | = | 9 |

0^{3}+1^{3}+2^{3}+3^{3} | = | 36 |

At that point, J2 noticed we had squares and guessed that we were going to get every other square. Two nice conjectures, one of which already wasn't quite true, but that was more obviously clear with the next term:

0^{3} | = | 0 |

0^{3}+1^{3} | = | 1 |

0^{3}+1^{3}+2^{3} | = | 9 |

0^{3}+1^{3}+2^{3}+3^{3} | = | 36 |

0^{3}+1^{3}+2^{3}+3^{3}+4^{3} | = | 100 |

0^{3}+1^{3}+2^{3}+3^{3}+4^{3}+5^{3} | = | 225 |

J0: so, are we still getting squares?

J2: ... yes. That's 15 squared

J0: hmm, shall we write that down?

J2: yes, daddy. write down 0 = 0

^{2}, 1 = 1

^{2}, 9 =

^{3}(etc)

J0: ok, so:

0^{3} | = | 0 | = | 0^{2} |

0^{3}+1^{3} | = | 1 | = | 1^{2} |

0^{3}+1^{3}+2^{3} | = | 9 | = | 3^{2} |

0^{3}+1^{3}+2^{3}+3^{3} | = | 36 | = | 6^{2} |

0^{3}+1^{3}+2^{3}+3^{3}+4^{3} | = | 100 | = | 10^{2} |

0^{3}+1^{3}+2^{3}+3^{3}+4^{3}+5^{3} | = | 225 | = | 15^{2} |

**More testing**

For the rest of the conversation, he talked about what he expected the next terms would be, then he did the calculations to check. He didn't remember 6 cubed or 7 cubed, so we had diversions to talk about strategies to calculate them. At the end, he was very excited to see that the conjecture was still working, our cubes were adding up to squares of triangular numbers.

**To be continued**

Frankly, I think it will be a while before he can attack the wallet proof on his own (or with my minimal guidance). In the next couple of weeks, if we have access to some blocks construction sets, though, I'm hoping we can work together to actually do these transformations, rearranging the cubes into squared triangular numbers and vice versa. I expect even this will be a bit difficult, but it should be fun!

# The avocado

Our original avocado project was supposed to run for a whole year, with the kids making observations periodically and tracking the progress. The plant (and kids) have grown well during the last several months. Unfortunately, I am pessimistic about the future prospects of our plant as we enter the hot season. We'll see at the next update.How tall is it? Shoulder height |

Key observation: these new leaves are very shiny |