“You courtesy explained situation,” Noh said. “Appreciation I yield initial.”
It really did not make any difference who went first, since only an unanswered or misanswered riddle followed by a successful defense counted. But Stile was glad to get into it, for psychological reasons. He had a number of tricky questions stored in the back of his mind. Now he would find out just what the alien was made of, intellectually.
“Picture three equal-length sticks,” Stile said carefully. “Each quite straight, without blemish. Form them into a triangle. This is not difficult. With two additional sticks of the same size and kind a second triangle can be formed against a face of the first. Now can you fashion four congruent triangles from only six such sticks?”
Noh considered. “Enjoyability this example! Permissible to employ one stick to bisect double-triangle figure formed from segments of sticks?”
“No. Each stick must represent exactly one side of each triangle; no projecting points.” But Stile felt a tingle of apprehension; such a device would indeed have formed four congruent triangles, and similar overlapping could make up to six of them. This creature was no patsy.
“Permissible to cross sticks to form pattern of star with each point a triangle?”
“No crossing.” So quick to explore the possibilities!
The head stalks bobbled thoughtfully for almost a minute. Then: “Permissible to employ a third dimension?”
The alien had it! “Permissible,” Stile agreed gamely.
“Then to elevate one stick from each angle of first triangle, touching at apex to form four-sided pyramid, each side triangle.”
“You’ve got it,” Stile admitted. “Your turn.”
“Agreeable game. Triangles amenable to my pleasure. Agree sum of angles is half-circle?”
“One hundred eighty degrees,” Stile agreed.
“Present triangle totaling three-quarter circle.”
“That’s—” Stile began, but choked off the word “impossible.” Obviously the alien had something in mind. Yet how could any triangle have a total of 270°? He had understood 180° was part of the definition of any triangle. Each angle could vary, but another angle always varied inversely to compensate. If one angle was 179°, the other two would total 1°. Otherwise there would be no triangle.
Unless there could be an overlay of triangles, one angle counting as part of another triangle, adding to the total. That didn’t seem sensible, yet—
“Permissible to overlap triangles?” Stile inquired.
“Never.”
So much for that. Stile paced the floor, visualizing triangles of all shapes and sizes. No matter how he made them, none had more than 180°.
Could they have differing definitions of triangles? “Permissible to have more than three angles in the figure?”
“Never.”
Down again. Damn it, it wasn’t possible! Yet somehow, by some logic, it had to be, or the alien would not have proposed it. Stile had encountered situations in which the supposedly impossible had turned out to be possible, like turning a torus inside out through a hole in its side. Topology—there was a fertile field for intellectual riddles! Shapes that were infinitely distortable without sacrificing their fundamental qualities. Bend it, twist it, stretch it, tie it in knots, it did not really change. Now if he could do that with a triangle, bowing out the sides so as to widen the angles—but then its sides would be curved, no good. Maybe if it were painted on a rubberite sheet, which sheet was stretched—aha! A curved surface! Noh had not specified a flat surface. A triangle drawn on a sphere—
“Permissible to employ a curved surface?” Stile inquired triumphantly.
“Never. Triangle must be rigid frame, as were your own.”
Ouch! He had been so sure! On the surface of a sphere he could have made eight triangles each with three right angles, or even four triangles with two right angles and one 180° straight angle each—a quarter section of the whole. The curvature of the surface permitted straight lines, in effect, to bow. He had often carved the skin of a pseudo-orange that way. But the alien forbade it.
Still, perhaps he was getting warmer. Noh’s antennae were flexing nervously, which could be a good sign. Suppose the surface were not curved, but space itself was? That could similarly distort a rigid triangle, by changing the laws of its environment. In theory the space of the universe was curved; suppose the triangle were of truly cosmic proportion, so that it reflected the very surface of the cosmos? “Okay to make a very large triangle?”
“Nokay,” Noh responded. “Standard triangle held in tentacles readily.”
Brother! Stile was getting so inventive, stretching his imagination, to no avail. If he could not draw on the curvature of space—
But he could! “How about taking it to another location?”
The stalks wobbled ruefully. “Permissible.”
“Like maybe the vicinity of a black hole in space, where intense gravity distorts space itself. Normal geometric figures become distorted, despite no change in themselves, as though mounted on a curved surface. Down near the center of that black hole, space could even be deformed into the likeness of a sphere, just before singularity, and a triangle there could have two hundred and seventy degrees, or even more.”
“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, geographical 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 starring 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 indubita
bly?”
“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 X plus Y equals Z,” Stile said.
“Party of the first part raised to the infinitive power, plus party of the second part raised similarly? This is meaningful-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. “X3 plus Y3 equals Z3 yields a distorted loop, no longer a perfect circle. Raise the powers again. X4 plus Y4 equals Z4, and it distorts further towards the corners. 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.”
“Um, 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.
If Stile’s employer discovered anything in the course of her investigation into the matter of the forged message-address, she did not confide it to him. That was the way of Citizens. They often treated serfs with superficial courtesy, but no followthrough. The Citizen he had encountered in Round One, the Rifleman, was an example; there had been no further communication from him. There was no such thing, in Proton law and custom, as a binding commitment by Citizen to serf. Everything went the other way.
Stile remained troubled by the continuing campaign against him by his anonymous enemy. At first he had thought this person had sent Sheen and then lasered his knees as a warning to get out of horseracing. But he had gotten out—and the threat continued. There had been a Citizen who was after him, but that had been effectively neutralized, and Sheen’s self-willed robot friends had verified that he was not the present offender. They had not been able to trace the source of the substituted address, because it had not been handled through any computer circuits; it had been a “mechanical” act. But they had watched that particular Citizen, and knew that he was relatively innocent.
Someone wanted Stile dead—in Proton and in Phaze. Perhaps it was the same person—a frame-traveler. There were a number of people who crossed the curtain regularly, as Stile himself did. Maybe that same one had killed Stile’s other self, the Blue Adept, and tried for Stile with less success. Probably another Adept; no lesser person could have done it. But who? The maker of golems—or the maker of amulets? Stile was becoming most eager to know.
If he beat the long odds and won the Tourney, he would have at his disposal the resources of a Citizen. Then he would be in a position to find out—and to take remedial action. That was the real imperative of his present drive. He could not wash out of the Tourney and simply return to Phaze to court the Lady Blue—not when a curtain-crossing Adept was laying constant deathtraps for him. He had life-and-death riddles to solve first!
His Round Four opponent was a woman his own age: Hella, first Rung on the Age 35 ladder. Stile had qualified for the Tourney by being fifth on the male 35 ladder, but he was actually the top player of his age. Many top players remained deliberately low on their Game ladders, so as to avoid the annual Tourney draft of the top five. Hella, however, really was the top female player, whose tenure ended this year; she had been eager to enter the Tourney.
Nevertheless, she
was not in Stile’s class. He could outperform her in most of the physical Games and match her in the mental ones. If he got the numbers he would have no gallantry at all. He would choose PHYSICAL. If he had the letters he would have to go for TOOL, to put it into the boardgames block, where he retained the advantage over her.
Hella was a fit, statuesque woman, taller and heavier than Stile. She had half-length dark-blonde hair, slightly curly, and lips a little too thin. She looked like what she was: a healthy, cynical, hard-driving woman, nevertheless possessed of a not inconsiderable sex appeal. Larger men found her quite attractive, and she was said to be proficient at private games, the kind that men and women played off record. Stile had played her often, in random Games, but had never socialized with her. Most women did not get romantic about men who were smaller than themselves, and she was no exception. Stile himself had always been diffident about women, and remained so. Sheen was special, and not really a woman. The Lady Blue was special too—and Stile really found himself unable to forward his suit with her. She was his other self’s widow…
“I would rather not have come up against you,” Hella told him in the waiting room. “I’m already half out; a duffer caught me in CHANCE.”
“That’s the way it goes,” Stile said. “I have nothing against you, but I intend to put you away.”
“Of course,” she said. “If you get the numbers.”
“If I get the numbers,” he agreed.
Their summons came. Stile did not get the numbers. Thus they landed in 2B, Tool-Assisted MENTAL GAMES. They played the sixteen choices of the subgrid and came to MAZE.
They adjourned to the Maze-section of the Game premises. The Game Computer formed new mazes for each contest by sliding walls and panels along set channels; there was an extraordinary number of combinations, and it was not possible to anticipate the correct route through. One person started from each end, and the first to complete the route won.