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.”
“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.”
