In times of severe stress, we often find out who our real friends are. The Winter of 1991 was the most stressful time in my life so far. I would not have gotten through it without my friends Basho and Axel Thu. Axel Thu was a Chinese mathematician who died in the year 400 AD. Basho was my cat.

January 14 was my preliminary examination in Mathematics. The preliminary examination is the biggest hurdle that lies between the hopeful graduate student, and the Ph. D. that is his or her heart's desire. Picture a two hour oral exam consisting of two professors hurling questions at you, where you only have a piece of chalk for self defense. At the University of Michigan, the rule was that if you failed, you got one more chance. If you failed a second time, that's it. You're out. Thanks for playing. You leave college with nothing but debt, misery, and a Master's degree. I started studying six weeks before the date, telling my family that Christmas was canceled for me that year.

This is the story of a friendship. Basho the Cat, Axel Thu and I became very close during those six weeks. Unfortunately, this is also the story of a betrayal, and the end of a friendship.

On the first day of my ordeal, I sat down and began making a list of every possible question Professors Hanlon and Storer could possibly ask me. I figured that I would then merely spend six weeks figuring out, and then memorizing all the answers. It only took me a day to make the list. It was long. I lived in a tiny efficiency without a television. This was doable.

After one week, I became depressed. As I learned more and more about my subject, I kept thinking of more and more possible questions. For every question I could cross off my list, I could add two or three more. I would spend long periods of time sitting at my desk, with my head in my hands. When I sat still for a long time, however, Basho would jump in my lap and this would cheer me up, at least for a while.

During week two, I decided that I would pick one question, and not only memorize the answer, but practice saying it with grace and flair. That way, I'd have at least one thing I could feel secure about. My notebook was opened to Lagrange's Four Squares Theorem, a very nice theorem that states that any number can be written as the sum of four squares. Seven, for example, is 2 squared plus 1 squared plus 1 squared plus 1 squared. Every night, just before closing the books, I'd recite the proof of Lagrange's Four Squares Theorem to Basho the Cat. He always listened.

After three weeks, things were finally beginning to come under control. I began to see relationships between all the things I was studying. It turned out that what I thought were different questions were really all asking the same types of things. I progressed from memorizing to understanding. Ironically, the one thing that was still causing me stress was Lagrange's Four Squares Theorem. There was a part of the proof where I had to say, "Now we apply Axel Thu's Theorem." Try to say it out loud. I mean it - do it. I am not going to go on until you try. Okay... now... Isn't that impossible? Whenever I did it, it kept coming out "Akthel Thuth Theorem." I would go to bed moaning, "I know I'm gonna blow it I know I'm gonna blow it." I wanted the Lagrange Proof to be the Platonic Ideal - the quintessence of the Perfect Answer, and I was going to lisp.

If you've never done it, you don't know what it's like to spend a month and a half studying for an exam. You stop caring after a while; your soul just can't keep up that level of worry for that long. But you start caring about other things. You run out of oatmeal when it is too late to buy more and you either cry, or start thinking that someone stole it to sabotage you. You stare at the Quaker Oats guy on the box, and start getting sentimental about how happy he was, and wonder if his wife was hot.

Week number four, two weeks left: An amazing revelation took place. I was practicing the proof of Lagrange's Four Squares Theorem with Basho, and I complained, "I'm going to flunk out of Grad School because of that Theorem of Axel Thu!!" Did you notice it? "The Theorem of Axel Thu." It had style! It had Panache. What a moment that would be when I would apply, nay when I would Invoke The Theorem of Axel Thu.

After that moment, Axel Thu became my pal! I would study all day, looking forward to the evening, when I'd get to prove the Lagrange Four Squares Theorem and get to Invoke The Theorem of Axel Thu.

When there were just seven days left, I was under so much stress, I could barely think. Sometimes I'd be at my desk, stuck on some point, and I would just quiver. During those times I would say, "Okay, Basho. Time to prove Lagrange's Four Squares Theorem." And I would prove the theorem, Invoking The Theorem of Axel Thu, and feel better. Then I could sit back down and usually solve my problem.

When there were two days left, I didn't have the energy to learn any more new things. I just went over my notes, and rehearsed answering questions out loud to Basho. When I was too wound up to speak any more, I would sit down with a box of crayons and... draw pictures of Axel Thu! Sometimes I would just draw him, in China, with happy villagers in the background, and sometimes I would draw him, Basho, and me, all smiling because Doug had passed the Prelim. The three pals, triumphant!

The morning of my prelim, I had intended to review my notes one last time. Instead I found myself drawing a big picture of Axel Thu. There was a voice balloon which said, "Let's do it. You and me, Doug."

As I walked to the Math building, I was confident. After all, with Basho and Axel Thu behind me, how could I fail?

The first half hour of my test went well. I knew the answers to most of the questions immediately, and I was able to figure out the ones I didn't know. Then Professor Hanlon asked, "Are you familiar with Lagrange's Four Squares Theorem?" HO HO HO! ARE YOU FAMILIAR WITH THE SKY? BET YOUR GENIUS MATHEMATICAL ASS I AM FAMILIAR WITH -

I said, "Yes. Would you like to see a proof?"
"Just the outline please."

"Right!" I said. "First we prove that for any prime p there exists an m such that mp is the sum of four squares. We do this by-" He interrupted me. "No need to go into detail. I just want the outline. What is the next step?"

Just an outline? He wasn't going to let me Invoke The Theorem of Axel Thu! But I was going to go too fast for him to stop me. "Step 2: We show that m is odd. We do this by Invoking-"

"Okay, you obviously know La Grange's four squares theorem, lets move on."

Lets move on? Lets move on? My dear web-surfers, you have to understand something. By this point, Axel Thu was more than a dead mathematician to me. Axel Thu was friend. By not invoking his theorem, I was betraying him. I pictured my drawing of him back home. He looked so happy, "You and me, Doug." And I didn't get to invoke his theorem. After another hour or so, Professor Storer asked me to leave the room. Then Professors Hanlon and Storer came out and shook my hand. "You passed." "Yeah, thanks." And I walked home, wondering how I could face Basho? I'd ruined everything.

When I got home, I decided I would prove Lagrange's Four Squares Theorem one last time. And I would do it better than ever. This was going to be the grand finale of six weeks of hell. I put Basho in his place on my desk, and I proceeded to prove it for him, clearly, wonderfully. My gestures were crisp and on-the-money. My mathematics was concise. My voice was confident and proud.

"Step 2: We show that m is odd. We do this by Invoking the Theorem of AXEL THU!" At this he twitched his ears. Now for the last six weeks, when I was rehearsing, I had always pretended that Basho twitching his ears meant that he was asking a question. So I said, "Well, Basho, we can invoke the theorem of Axel Thu because... because..." And then I had to think... and then I had to get out a sheet of paper... You guessed it: I had been horribly, horribly, wrong! Axel Thu did not apply! To use the theorem, I had to assume that m was odd, and that was what I was trying to prove! If Professor Hanlon hadn't cut me off, I would have made a complete idiot out of myself. I may not have even passed!

Axel Thu was never my friend. He was my enemy! A traitor, a time-bomb, lurking in my notes just waiting to make me fail my prelim!

In times of severe stress, we learn who our real friends are. In the Winter of 1991, Basho the Cat was my true friend. And if Axel Thu had not been dead for over a thousand years, I would have been forced to kill him. The bastard.

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© Douglas J. Shaw,  2002


Historical Note: His name was actually Axel Thue, he was Norwegian, and he died in 1922. But I didn't know that at the time. And it's a much better story if he was from the year 400 A.D. Actually, as long as we are talking like this, I never thought for a moment that I would need his theorem to prove the Lagrange Four Squares Theorem, but, again, I lied to make it a better tale. I DID get cut off while trying to invoke his theorem, and I DID have to prove the La Grange Four Squares theorem, so those parts are true... well except for the fact that I didn't really get cut off while trying to invoke the theorem, I said that just now but I didn't mean it. And the biggest hurdle to getting a Ph.D. is the doctoral thesis, not the prelims... So I guess I'm trying to say I made a lot of this essay up. But I've told this story to every member of my family, so I kind of remember it that way now.

The real Axel Thue said some wonderful things about mathematics: "The further removed from usefulness or practical application, the more important." and "For the development of the logical sciences it will be important to find wide fields for the speculative treatment of difficult problems, without regard to eventual applications." He looked like this:

And Basho really did listen to me patiently, every day, as I practiced answering questions for my prelims.