The final grid was about as simple as they came: four squares. The top was 1. COMPUTER-GENERATED 2. SELF-GENERATED. The side was A. DUAL RESPONSE B. INDIVIDUAL RESPONSE. Just four alternatives. Stile had the numbers.
Noh’s antennae wavered in agitation. “Nonetheless is this naked mental? How justified computer involvement?”
“These categories are fundamentally arbitrary,” Stile explained. “Too many Games are in fact mixed types. The Game Computer assumes for the sake of convenience that it, itself, has no Game significance. The riddles could come from a book or a third person, but it is most convenient and random to draw on the computer memory banks. There are all kinds of little anomalies like this in the Games; I had to play Football using androids termed animals, with robots for referees.”
“This is delightfully mistrustful. Expedient to avoid?”
“Well, I happen to have the numbered facet, so you cannot control that. Myself, I’d rather avoid the dual response; that’s timed, with the first one to answer being the winner. I’m more of a power thinker; I get there, but not always in a hurry.” This was true, but perhaps misleading;
Stile was stronger as a power thinker in proportion to his other skills, but still by no means a slow thinker. “That is—“
“Can we collude? Choose 2B for mutual accommodationality?”
“We could—but how could we know one of us won’t cheat? Deals are permissible but not legally enforceable in 1 the Game. An expert liar makes an excellent grid-player. The computer accepts nothing but the signal-buttons.”
“Chance it must be risked,” Noh said. “Some trust exists in the galaxy, likewise on little planets.”
“Agreed,” Stile said, smiling at the alien’s phrasing. He touched 2, and sure enough, it came up 2B. They had each kept faith. Game players normally did; it greatly facilitated things on occasion.
Now they adjourned to a bare private chamber. “Select a recipient for the first riddle,” the Game Computer said from a wall speaker. “Recipient must answer within ten minutes, then propose a counter-riddle for the other. In the event of failure to answer, proponent must answer his own riddle, then answer opponent’s riddle within the time limit. The first contestant to achieve such success is the victor.
Computer will arbitrate the technical points.”
“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 hobbled 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 tri-angles 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. Sup-pose 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.”