Copper Thoughts

It’s been cold.

We’re having a warm spell right now, actually. It’s a warming -15º before, of course, you factor in the wind chill.

But I try to look on the bright side: it keeps my brain sharp, my step brisk and my expectations low.

I’m writing to share a little mathematical proof I was reminded of today. Stay with me here, even if math’s not your thing. You might find it interesting… it goes like this:

Starting with the number 1, add consecutive odd integers. Simple, huh?

A pattern begins to emerge from this… see if you can find it.

a = 12, b = 22, c = 32, d = 42 ...

If this is true for as far as you go with the pattern (that is, for 1 + 3 + 5 + … + the next odd integer = number of numbers on the left squared), that’d be sort of cool. But we haven’t proven that. Of course, we could prove it up to any particular odd number we wanted by actually doing the computation, but if I were to ask you to do that for the number 10000000009, you might not be inclined to write that out. So let’s prove it for all numbers. And let’s do it without any more words (aside from my brief explanation of the cryptic symbols I’ll use to “draw” the proof)…

[X and O are  representations of a unit square. For all intents and purposes, X = O, but I use two different characters to make the diagram easier to read.]

X = 1

XO
OO = 4

XOX
OOX = 9
XXX

etc...

1357...
XOXO ...
OOXO ...
XXXO ...
OOOO ...
....
....
....

You can hopefully see from these crude little drawings that if you add an odd number of X (or O) to the square, you get another square. And that other square happens to be the next possible square you could make. You can also see that it only requires two more X (or O), plus the number added before it, to make another square. Since odd numbers are spaced two apart (1 + 2 = 3, 3 + 2 = 5, etc), we cover all odd numbers in this fashion, and all squares.

Time to burrow into 6 layers of blankets…