Last time, on Patman: the inanimate series:
Patman: You'll never get away with this, Puzzler...
Puzzler: Hahaha, try this one! [writes something on the chalkboard]
Noggin: Holy Hadamard matrices Patman! It's a quadruple integral over a Klein bottle embedded in 4-dimensional hyperbolic space!
And now, the conclusion of "Double Induction":
Patman: ...my...Pat Futility Belt (TM) (not available in stores! order now!) ...and...get...my...Table...of...Rainy-day...Integrals...
Noggin: Wait, Patman! It simplifies down to exp(-gamma) by symmetry! Intuitively obvious...You're no match for the Dynamical Systems Duo, Puzzl-- He's gone! What do we do now, Patman?
Patman: Quick, Noggin, to the PatMobile!
Noggin: But wait, neither of us can drive! And my learner's licence is only class Pi, no trucks, motorcycles, or ultra-high-tech-armor-plated-voice-controlled-fuzzy-dark-blue-cars-with-positive- Schnirelman-density.
Patman: Wait a minute, then how did we get here?
Noggin: Don't you remember, our grad student Derfla drove us here...But he's working on his PhD thesis right now...
[Meanwhile, at SFU, in the Pub]
Derfla: Argh, I lost again! Oh well, here goes another quarter...Hmmm, wasn't there something I was supposed to be doing? Nah...
[Back to our heroes]
Patman: Hey, I have an idea...I'll use my patented PatTopologyModifier and change the Euclidean metric to a discrete one! Then all distances will be the same, and we can walk there in one step.
Noggin: Leaping limits, Patman, that's brilliant!
Noggin: Wow, Patman, that was one non-differentiable ride!
Patman: Non-differentiable? Felt more like non-Lebesgue integrable!
Noggin: Definitely not Lebesgue, but maybe Generalized Riemann...Hmmm...
Patman: Now where's that dastardly villain Puzzler?
Voice of Puzzler: You solved my last problem, now try this one! Find a 2-regular subgraph with the same vertex set!
Patman: What? This graph has 1000 vertices, and finding Hamilton cycles is NP-complete!
Noggin: Then there's no hope...
Patman: But wait! He didn't say it had to be connected! That means we only need to find a 2-factor! Easy...
[The Puzzler suddenly leaps down from the ceiling and scratches his head]
Puzzler: What? That's impossible! Oh wait, I didn't say connected! No problem, I still have some tricks up--Hey, what?!?
Patman: We've got you now, Puzzler...Quick, Noggin, tie him up with a Conway knot.
Noggin: All done...Hey, who's that?
Patman: It's the Pencilguin!
Pencilguin: That's right, and I have some invalid proofs...Like this:
0*1 = 0*2, therefore 1 = 2.
1 = sqrt(1) = sqrt((-1)^2) = (sqrt(-1))^2 = i^2 = -1.
[Noggin slumps to the floor, unable to withstand the horrible fallacies]
Patman: Hey, wait, you can't divide by 0! And square roots do not extend to the complex numbers while preserving multiplicativity!
[Noggin recovers, partially...]
Noggin: Let's see, where's the Patman Official Archenemy Handbook...A... G... Q...P...Pencilguin, here it is! It says that his weakness is number theory! Alright, Pencilguin, take this:
Prove Euler's theorem, Gauss' Law of Quadratic Reciprocity, Dirichlet's theorem, the Rosser-Schoenfeld inequalities, the Prime Number Theorem, Mordell's conjecture, the Extended Riemann hypothesis, Goldbach's conjecture, the ABC conjecture, the Prime k-tuple conjecture, and Fermat's Last Theorem!
Pencilguin: Arggggh! I can't!
Patman: Good work, Noggin! Let's go back to the PatCave. I'll get the PatTopo--
Noggin: On second thought, I think I'll take the bus...
Next time, on Patman: the inanimate series: The Return of Jacobier
Burnaby, B. C., Canada
August and November 1993
(P.S. Can you spot the factual error in one of the episodes?)