Monday, March 4, 2013

Solution to Cryptogram

And the answer [1]  is…

"Within your lifetime will, perhaps,
As souvenirs from distant suns
Be carried back to earth some maps
Of planets and you'll find that one's
So hard to color that you've got
To use five crayons. Maybe, not."

The poet was Marlow Sholander. He was my freshman calculus professor at Western Reserve University (before it united with Case Institute of Technology). I don't know when he wrote it, or why. He was known for chain smoking and for phrases like, "There are no Gausses in this class"---proved by lofting epsilons and deltas over our heads. But he said not a word about the four-color-map theorem. It was only a conjecture and not a theorem when I knew Sholander.  The proof would come in 1976 and be published in 1977 [2,3]. Even then, the proof was questioned because it required a computer. In fact, it was the first major theorem that was proved using a computer.

For new initiates: The four-color map theorem says that, no matter how you carve up a plane into connected (contiguous) areas, to assure that no two abutting regions have the same color, you don’t need more than four colors. “Abutting” means sharing a boundary of at least two points, so, e.g., Arizona and Colorado (which share only one point) could have the same color on a U.S. map.

You won’t find the theorem bandied about by geographers. The maps are entirely in the minds of mathematicians, e.g. the following from Wikipedia (

Why spend a career trying to prove (or disprove) something about four-color maps? To put it abstractly, I think it allows you to hold (and maybe control) certainty in the palm of your hand.  The intoxication of knowing exactly what could not have come from a distant planet---no matter how far away---is the essence of Sholander’s poem.

I would not have guessed he had it in him, and it was not he who got me excited about what was then the four-color conjecture.  Years before, my tiny sixth-grade class trooped across the soccer field to Brentwood High School, invited to partake in a flight of fancy led by a 12th-grade prodigy. This prodigy inundated us with fun and challenges from constructing flexagons and polyhedra to reading Fantasia Mathematica---which contains a story about an impossible five-color map. We made three visits after school, as I recall.
The others in my class returned and made flexagons. I spent every boring class moment for the next six years trying to disprove various “easy” truths like the four-color-map problem. And math classes didn’t have boring moments anymore.

The name of the prodigy? Jef Raskin, who started the MacIntosh project at Apple Computer. The rest is history, as Wikipedia will attest (in a different article). Some years after our visits, I chanced to meet him again, and he said he’d outgrown the childish pursuits he had started me on.

How strange that it was the dry, hierarchy-obsessed professor who carried the wonder to distant planets through his poem!

Michael H. Brill

1. The only solver was Paul Centore.
2. Appel, Kenneth; Haken, Wolfgang (1977), "Every Planar Map is Four Colorable Part I. Discharging", Illinois Journal of Mathematics 21: 429–490
3. Appel, Kenneth; Haken, Wolfgang; Koch, John (1977), "Every Planar Map is Four Colorable Part II. Reducibility", Illinois Journal of Mathematics 21: 491–567