"No, Snickchoo, I'm not crazy. I mean the gate told me. Look."

Zelki swung open the door and pointed to the gate before Snickchoo's cabin. The gate was now closed to pen in Galatea.

"See the way I braced that gate for you with the cross-piece? See the 'zee' the brace makes with the top and bottom of the gate? As with any good gate.

"Oh, forgive me. I guess you don't see what I mean -- the connection with your problem. Let's sit down by the fire again, Snickchoo and Atchoo, and drink some more of that delicious cider. There. Umm. Delicious!

"Well, when I was a boy, I was apprenticed to a master joiner, Bolthar, to learn to build doors, and windows, and window sashes, and wall paneling. And, especially, wooden inlays. A good carpenter is a good craftsman. But a joiner must be more than that. A joiner must be an artist! And Bolthar was a wonderful master! So creative. So skillful. So brilliant. Yet so kind and understanding. And so full of stories! For he had traveled far and wide as a journeyman.

"Well, it was Bolthar who told me -- about the gate. He'd set me to building a gate for an old widow -- when I really wanted to get my hands on some paneling. But I worked willingly. And, afterwards, to soothe me, Bolthar told me of this mystery -- discovered a long time ago. In Greece, I think.

"Bolthar said, 'Look, Zelki, at the square frame of this gate: the bar at the top; the bar at the bottom; and the bars on either side.'"

Zelki drew the outline of a gate in the ashes of the hearth.

"'See', Bolthar said to me. 'The bars form a square. And, now, to brace the square -- to keep it from sagging, under its weight, on its hinges -- we brace it with a diagonal cross-bar, which divides the square into two triangles. As you know, the triangle is the strongest structure we can build with rigid parts.' He meant, Snickchoo, that the rigid triangle will break somewhere along its sides before it sags at its corners.

"'Now, Zelki', said Bolthar, gripping my shoulder, 'it's a great mystery. But the side and the diagonal of the square, which fit so well together (structurally) -- the side and diagonal are, in the sense of measurement -- are im --.' What was the word Bolthar used? Incompatible. No! That's Wopsi the Tinker and his wife Bekan. They're incompatible -- always fighting. No, the word was 'incommensurable'. That's it.

"Commensurable" means sort of 'measurable'. If two sticks are commensurable, either one can be used to MEASURE OUT the other. But, for two INCOMMENSURABLE sticks or boards, neither one can be used to MEASURE OUT THE OTHER. To say the same thing, the physical ratio of the diagonal length to the side length does not equal a whole number ratio or a fractional ratio. That means this physical ratio is not numerically rational -- but irrational.

"And it means that -- listen! Snickchoo -- that no matter how many times you lay down a line of rods cut to the length of the diagonal and a line of rods cut to the length of the side, the two lines can never never match! Oh, be close, I dare say. So that the eye may miss the difference. But, geometrically, they can never be the same! You can try to get closer. But the Candy Miser will trot out a doorsill spider to test how close it is. Or perhaps something even finer."

Snickchoo nodded. "Who knows? He may have found a waterspider to write up conditions for an even finer test -- in such fine print that even Nulf can't read it!"

Zelki looked puzzled. "A waterspider? Why a waterspider? Well, anyway, that's what I think the Candy Miser is doing to you. Whenever you take a 'path', I mean, choose a length for your licorice laces, he has only to build a square with your length as the sides. From this, the Candy Miser obtains the square's diagonal -- showing the length that the diagonal makes. And then he constructs a mold the length of that diagonal -- to turn out bundles of licorice laces you can't possibly match in length by the laces you've made -- which are the incomensurable sides of his square. By this trick, copies of your ('side') laces and copies of his ('diagonal') laces can never match!

"It's just like the Candy Miser bidding or bargaining up and down. He uses the knowledge of what the other person is going to do to construct a move that will thwart him. That's what had bothered me! I knew it could be done with prices and such. I'd forgotten it could be done with lengths of rods -- or lengths of licorice laces -- until Atchoo repeated to me the Candy Miser's "Riddle Song' and I looked at my own gate."

When Snickchoo continued to look puzzled, Zelki drew a deep breath.

"I'll explain another way. You have played Tic-Tac-Toe. Yes? And you remember that, if I allow you to start first, and you put an 'X' in the center-box, then there is no way I can put in a line of 'O's to beat you, so you always win with that strategy. But if I can trick you into allowing me to start first, and if I put an 'X' in the center-box, then there is no way you can put in a line of 'O's to beat me, so I win by that strategy. First person can always win."

Snickchoo nodded. "Yes, I understand that."

"Good!", smiled Zelki. "Then the Miser's trick is just the reverse of that. He allows you to make the first move: choosing the length of licorice stick. Oh, you think you have tricked him by discovering his length of licorice stick and duplicating it. But the Miser cleverly allows you to think that. Then, when you have begun to make sticks of that length, he secretly confirms the length. He uses this as a side stick for a square of licorice sticks. Understand me? The Miser tricks you into choosing the s-stick for a licorice stick square. Using duplicates of your s-stick, the Miser makes a licorice-square and, placing a long stick across it, he cuts it off so it exactly fits the diagonal of that square to make a d-stick. Then, in contest, the Miser lays down a line of d-sticks which you cannot exactly duplicate with any number of your s-sticks!"

Snickchoo shuddered. "Yes. Now I see that. But what can we do?"

Zelki rubbed his hands and laughed. "We can reverse that game. We must find a way to trick the Miser into making his d-sticks as the first move in this reverse game. Secretly, with the help of dear Atchoo, we discover their lengths. You secretly make sticks of that length, the d-length, while allowing the Miser to think he is secretly discovering you at making some (false!) s-sticks. And then, in contest, you can easily duplicate a line of d-sticks!"

Snickchoo slowly nodded an understanding agreement.

Zelki laughed again. "I'm certain of it, Old Chap. 'The Licorice Puzzle' is 'The Mystery of the Gate'. Now that we know, here's what we can do!"

Zelki leaned forward, clasping one arm around Snickchoo's neck, the other around Atchoo's, and brought them together, so that three heads huddled in the flickering firelight.

(Ready for Chapter 10?)