Thursday April 24 (easy)

60 minutes and a few strides.

Let me preface what follows by apologizing to Ian for screwing around with this stuff, but I would like to prove that all numbers are equal to each other.

Consider the series:

S = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - ...

then by fiddling with the order a bit

S = 1 - 1/2 +1/3 + 1/5 - 1/4 + 1/7 + 1/9 - 1/6 + 1/11 + 1/13 - ... = 1

To choose what order to do the addition, I make a list of all the negative terms and all the positive terms. If my finite sum is currently more than one, I choose the first available term on the negative list and tack it on the end of my sum. If the sum is currently less than one, I choose the first positive term on that list and tack it on the end. If that's not clear, just go through the series above, computing the partial sums after each term, and notice how it's oscillating from above one to below one and back again. Every time the sum dips below one, I switch from subtracting terms to adding them, and it eventually goes back up above one again.
Every term in the original definition of S is also in the second version. Notice that I never skip a term. The negative terms all appear in order relative to each other, as do the positive terms.
Also, since both series (the series of positive terms and the series of negative terms) individually diverge, I can never get "stuck", for example at 1.1 and unable to bring myself all the way back to below 1.0 by adding more and more negative terms. Whichever side of 1.0 I'm on, I can always flipflop to the other side by adding enough terms of the appropriate sign.
Finally, notice that I flipflop infinitely many times, and therefore I use infinitely many terms of each sequence. That is, I use all of them.
Conclusion: all the terms in the original definition of S are also in the secondary definition of S, and so they are equivalent to each other.

Now I make a third definition:
S = 1 - 1/2 + 1/3 - 1/4 - 1/6 + 1/5 - 1/8 + 1/9 - 1/10 - 1/12 + ... = 1/2

This is done in just the same way as before, except this time I oscillate back and forth around 1/2 instead of one. But all the same terms are there. But if you take a bunch of numbers and add them all up, you get the same thing no matter the order, by the associative law of addition.
Evidently
1/2 = 1
Subtracting 1/2 from both sides
0 = 1/2
multiplying by A*2
0 = A
or instead multiplying by B*2
0 = B
And by the transitive property
A = B

there you have it. All numbers are equal to each other.