“Ah, I believe you mean indubitably?”
“Indecisively,” the alien agreed.
Just so. “Can’t move the planet to a black hole?”
“Correct. Cannot. Would squash out of shape.” So it had to be settled right here; no four-dimensional stunts. Yet where in the world could it be? Not the north pole, not the south pole—
Wait! He was assuming too much. He did not have to go south from the south pole. He had to go south to it. Or almost to it....
“Picture a circle around the south pole,” Stile said. “A line of latitude at such distance north of the pole that its circuit is precisely one unit. Now commence journey one unit north of that latitude. Walk south, then east around the pole, and north, retracing route to starting point.”
“Accursed, foiled additionally,” Noh said. “This creature is formidability.”
Stile’s sentiments exactly, about his opposition. He was afraid he was going to lose this match, but he struck gamely for new territory, seeking some intellectual weakness in his opponent. “The formula X2 plus Y2 equals Z2, when graphed represents a perfect circle with a radius of Z, as described in what we call the Pythagorean Theorem,” he said. “Are you familiar with the mechanism?”
“Concurrently. We term it the Snakegrowltime Equation.”
Stile suspected there was a bit of alien humor there, but he had to concentrate on the job at hand instead of figuring out the reference. He was glad he had not gotten into a punning contest! “What variation of this formula represents a square?”
“No variation!” Noh protested. “That formula generates only a curve; a variation must remain curvaceous. No straight lines from this.”
“I will settle for an approximate square,” Stile said helpfully. “One that curves no more than the width of the lines used to draw it.”
“How thickness lines?”
“Same thickness as those used to make the circle.”
“Extraordinarily unuseful,” Noh grumped, pacing the floor with the little feet tramping in threes. “Geometric curves do not transformation so. It is a fact of math.”
“Math is capable of funny things.” Stile was regaining confidence. Had he found a weakness?
Noh paced and questioned and did the alien equivalent of sweating, and finally gave it up. “Incapability of sensiblizing this. Demand refutation.”
“Try Xoo plus Yoo equals Zoo,” Stile said.
“Party of the first part raised to the infinitive power, plus party of the second part raised similarly? This is mean-ingful-less!”
“Well, try it partway, to get the drift. X3 plus—“
“Partway?” Noh demanded querulously. “Cannot split infinity!”
Stile thought of the infinities of the scientific and magical universes, split by the curtain. But that was not relevant here. “X8 plus Y8 equals Z8 yields a distorted loop, no longer a perfect circle. Raise the powers again. X4 plus Y4 equals Z*, and it distorts further towards the comers. By the time it goes to the tenth or twelfth power, it is beginning to resemble a square. By the time it reaches the millionth power—“
Noh did some internal calculating. “It approaches a square! Never a perfect one, for it yet is a curve, but within any desired tolerance—remarkable! I never realized a curve could fashion into—amaztonishing!”
“Now I must answer yours,” Stile reminded the alien. He knew he hadn’t won yet; he had only gained a temporary advantage, thanks to splitting an infinity.
“Appreciably so. Where is the west pole?”
“West pole?”
“North pole, south pole, east pole, west pole. Where?”
“But a planet can only have one axis of rotation! There can’t be two sets of poles!”
“There cannot be a square generated from a curve, alternatively.”
“Urn, yes.” Stile lapsed into thought. If he could get this one, he won the Round. But it baffled him as much as his square had baffled Noh. Was the west pole simply a matter of semantics, a new name for the north or south poles? That seemed too simple. There really had to be a pole in addition to the conventional ones, to make it make sense. Yet unless a planet could have two axes of rotation—
In the end, Stile had to give it up. He did not know where the west pole was. His advantage had disappeared. They were even again. “Where?”
“Anticipation-hope you would solution it,” Noh said. “Solve has eluded me for quantity of time.”
“You mean you don’t know the answer yourself?” Stile asked incredulously.
“Affirmation. I have inexplicably defaulted competition. Negative expedience. Remorse.”
