“The creature has resolved it,” Noh agreed ruefully. “Inquire next riddle.”
This was no easy Game! Stile felt nervous sweat cooling on his body. He feared he was overmatched in spatial relationships. He had invoked the third dimension, and the alien had in turn invoked something like the fourth dimension. Better to move it into another region. “Using no other figures, convert four eights to three ones,” Stile said. Probably child’s play for this creature, but worth a try.
“Permissible to add, subtract, multiply, divide, powers, roots, tangents?” Noh asked.
“Permissible—so long as only eights are used,” Stile agreed. But of course simple addings of eights would never do it.
“Permissible to form shapes from numbers?”
“You mean like calling three ones a triangle or four eights a double row of circles? No. This is straight math.” Noh was on the wrong trail.
But then the alien brightened. His skin assumed a lighter hue. “Permissible to divide 888 by 8 to achieve 111?”
“Permissible,” Stile said. That really had not balked Noh long—and now the return shot was coming. Oh, dread!
“Human entity has apparent affinity for spheres, as witness contours of she-feminine of species,” Noh said. “Appreciate geography on sphere?”
“I fear not,” Stile said. “But out with it, alien.”
“In human parlance, planetary bodies have designated north and south poles, apex and nadir of rotation, geo-graphical locators?”
“Correct.” What was this thing leading up to?
“So happenstance one entity perambulates, slithers, or otherwise removes from initiation of north pole, south one unit, then east one unit, and right angle north one unit, discover self at point of initiation.”
“Back at the place he started, the north pole, yes,” Stile agreed. “That’s the one place on a planet that such a walk is possible. Walk south, east, north and be home. That’s really a variant of the triangle paradox, since two right angle turns don’t—“
“Discover another location for similar perambulation.”
“To walk south a unit, east a unit, and north a unit, and be at the starting point—without starting at the north pole?”
“Explicitly.”
The creature had done it again. Stile would have sworn there was no such place. Well, he would have to find one! Not the north pole. Yet the only other place where polar effects occurred was the south pole—and how could a person travel south from that? By definition, it was the southernmost region of a planet.
“All units are equal in length and all are straight?” Stile inquired, just in case.
“Indelicately.”
“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.”
So Stile had won after all! Yet he wished he could have done it by solving the riddle. The west pole—where could it be? He might never know, and that was an aggravation. Stile had been late in the Round Three roster, and now was early in the Round Four roster, as the Game Computer shifted things about to ensure fairness. Hence he had less than a day to wait. He spent much of that time sleeping, recovering from his excursion with the tanks and resting for possibly grueling forthcoming Games. He had been lucky so far; he could readily have lost the Football Game and the Riddle Game, and there was always the spectre of CHANCE to wash him out randomly. Most duffer players were fascinated by CHANCE; it was the great equalizer. So he hoped he would encounter a reasonably experienced player, one who preferred to fight it out honestly. One who figured he had a chance to win by skill or a fortuitous event in a skill contest, like the referee’s miscall in the Football Game. But a true skill contest could be arduous, exhausting both participants so that the winner was at a disadvantage for the next Game.