the problem with hats

If you have read Chaos Theory, you may not have been aware that this piece was originally the introduction to an (unpublished) comic fantasy novel. More about Gelgins introduced the three principal characters in that novel, and the following story is a rather silly reworking of a well-known logic puzzle from the novel.

For those of you who have wondered how well I’ve been convalescing after my recent accident, the good news is that the plaster on my leg comes off in two days, and I should then be able to sit comfortably long enough to be able to get back to more serious writing. Meanwhile, I hope that you find the following piece of ‘daft crack’ amusing.

*  *  *

The regular morning debate as to who was going to cook the porridge was interrupted by a sharp knock on the door. There might easily have been another debate as to who should answer the door, but there was only the briefest of pauses before the door was opened by the previously unseen knocker, who marched confidently into the parlour and stood rigidly to attention before the trio with an air of supreme self-importance.

“A minor functionary,” thought Sneedl’bodja.

“Grandmaster’s orders! Report to the training halls in ten minutes! Do not be late!” announced their unexpected visitor in a loud staccato bark.

He turned on his heel and marched smartly out of the door.

Qumfl’quelunx was too hungry to even think of taking out his pocket watch. “His idea of timekeeping is to keep time waiting!” is Sneedl’bodja’s latest patronizing comment on the subject.

However, Shunshelstinx, as eager as ever to impress, calculated that they would have to leave immediately in order to ensure that they were on time, even though the venue was next door. It wasn’t an idea that found favour, even with Shunshelstinx himself, because they were all hungry, but viable alternatives were as scarce as lions in Antarctica.

And thus the three gelgins found themselves awaiting the arrival of Dweebl’gulja in the central hall of his famous pranking academy. They did not have long to wait. As each of the three looked around nervously, the Grandmaster appeared from a long tunnel at one end of the hall and walked slowly towards them.

“Ah! Qunzl’bowza’s challenge!” he said softly.

“Right! Remedial training, lesson one. Deductive reasoning!” he continued more loudly.

Shunshelstinx and his friends were directed to fetch three wooden stools from a corner of the hall, which they did.

“Now! Place them in a line here, like this, about two paces apart,” ordered Dweebl’gulja.

He left to the three gelgins’ collective ingenuity the insignificant matter of whose paces were to be employed, well knowing that this level of accuracy would not be needed for the lesson he had in mind. However, what he did not know was the degree of confusion that this apparently simple instruction would cause. Sneedl’bodja, because he is always first, placed his stool firmly on the ground. Shunshelstinx, because he is in charge, took two paces forward and did likewise, if a little less firmly. Qumfl’quelunx was in unknown territory as regards both counting and measuring, so he simply stood there, motionless, unaware that he was still holding the stool.

“Mmm!” mused the Grandmaster, his face betraying nothing of his dismay. “This could turn out to be even more difficult than I thought. Surely, it won’t be necessary to enrol these fools in a foundation course before they can be admitted to remedial training? Why, I’ve seen better teamwork between two boxers in the same ring. I’ll give them just one more chance.”

With the Grandmaster’s close supervision, Shunshelstinx and his friends were able to arrange the three stools precisely according to the original instruction, and as expected, whose paces were used to measure the intervening distances really was irrelevant.

“Now! Which of you is the cleverest?” enquired Dweebl’gulja softly.

Shunshelstinx thought that a rather obvious question but at the risk of having two thoughts in quick succession thought it better not to say so.

“I am!” he announced confidently, pointing with both index fingers towards his chest.

“Right! Sit on the stool at this end so that you are facing the other two stools,” instructed the Grandmaster, sensibly pointing to the stool in question to reinforce the point, not wishing to see a repeat of the earlier confusion. “Now! Who’s next?”

Sneedl’bodja was not sure how Shunshelstinx had beaten him to the previous question, but he was quite sure that he did not want it on his training record that he, Sneedl’bodja, was not as clever as Qumfl’quelunx. Surely, he thought, nobody is not as clever as Qumfl’quelunx.

“I am!” he spoke up.

As it happens, he had no need to worry. Qumfl’quelunx was still pondering the first question.

“Okay!” said Dweebl’gulja. “You sit on the middle stool with your back to your friend so that you can see only the empty stool.”

“Now!” he continued, turning to Qumfl’quelunx. “Sit on the last empty stool, facing away from the other two.”

“I’m not sure I like this academy,” thought Qumfl’quelunx as he tried desperately to settle comfortably on to the stool and quickly decided that he wasn’t prepared to settle for the meagre level of comfort that it is possible to achieve on a hard wooden stool with only three legs and no padding (and not even a cushion). “This is a hard lesson.”

“Now, stay there until I return. I won’t be long. And on no account turn around to look at the gelgin or gelgins behind you,” said Dweebl’gulja.

Shunshelstinx was the only one to turn around, a waste of time in his position, although you can never be sure, as he might have said in justification.

The Grandmaster returned clutching five almost identical hats. He positioned himself in front of Qumfl’quelunx and a little to the side, so that Sneedl’bodja and Shunshelstinx could also see Dweebl’gulja and his hats. As hats, they were unremarkable. The only difference between them, it seemed to each of the three, was that three were black. The other two were white. That much was obvious.

“What horrible headgear!” thought Qumfl’quelunx, quietly outraged. “I wouldn’t be seen dead wearing such a boring bonnet, such a calamitous cap, such a hideous homburg, such a horrid hat, such a passé panama, such a terrible titfer, such a….”

“Shut up you fool!” hissed Sneedl’bodja. “You’re thinking aloud again!”

The Grandmaster walked along the line until he was standing behind Shunshelstinx. When he reappeared a moment later, he was holding only one hat, which he placed on Qumfl’quelunx’s head from behind, with the fat one’s not so silent disapproval. He produced a second hat, which he placed on Sneedl’bodja’s head, also from behind. A third hat was soon in place on the head of Shunshelstinx, who could see that both Qumfl’quelunx and Sneedl’bodja were also wearing hats. Sneedl’bodja could clearly see that Qumfl’quelunx was wearing a hat, but he couldn’t see Shunshelstinx, let alone his hat. Qumfl’quelunx could see nothing, including the point of the exercise. None of the three could see his own hat.

“Now think carefully,” said Dweebl’gulja.

He paused for a moment before addressing Shunshelstinx.

“Now! O cleverest one,” he said in a voice dripping with enough irony to keep a medium-sized foundry going for a week. “What is the colour of the hat you are wearing?”

“How could I possibly be expected to know that? I can’t see it,” thought Shunshelstinx. “Still, perhaps I should think about it for a while, to create a good impression, you know. And who knows? Perhaps an answer will turn up. You never know. They often do.”

He continued to deliberate in this preposterously optimistic fashion for what he deemed to be an appropriately polite interval, by which time he had forgotten the original question. That did it. He had only one answer to offer.

“I have no idea…,” he said.

“Excellent!” interrupted the Grandmaster, a smile beginning to break out over his face, no doubt in relief that some progress was being made. “That is the right answer.”

You may feel that the Grandmaster’s interruption was more than a little fortuitous. After all, what Shunshelstinx had intended to say was this: “I have no idea what you’re talking about!” Fool’s luck. It’s the only plausible explanation.

Still, Shunshelstinx was pleased to have made such a worthwhile contribution to the exercise, even if the value of judgement in the luck/judgement equation was zero, and everyone knows that you can’t divide by zero. Everyone but Shunshelstinx, that is. However, he had no time to shower himself with self-congratulations. Dweebl’gulja turned to Sneedl’bodja.

“What is the colour of the hat you are wearing?” he asked.

“Well, that’s easy,” thought Shunshelstinx. “I can see the answer from here.”

“That’s a stupid question!” thought Sneedl’bodja contemptuously. “How does he expect me to answer that? I’m not a clairvoyant. I’ve as much chance of finding the right answer as that fat fool in front of me will have when he’s asked the same question.”

He allowed barely enough time to elapse to avoid the impression that he was being too hasty before speaking up.

“I don’t know…,” he said.

“Excellent!” interrupted the Grandmaster, smiling broadly. “That is the right answer.”

You can be sure that the anonymous author of the proverb ‘lightning never strikes twice in the same place’ never had any contact with gelgins. What Sneedl’bodja had meant to say was this: “I don’t know why I bother to put up with this!” Double fool’s luck. And they do say that things always happen in threes. But therein lies the problem. That would require Qumfl’quelunx to be correct too, and you would get long odds at your local turf accountant against that happening in the remotely near future.

“Now!” said the Grandmaster, turning to Qumfl’quelunx. “Tell me, what is the colour of the hat you are wearing?”

Dead silence, as you might have expected. What else were you expecting? Miracles? Try the shop down the road. And then, just as hope was fading, Qumfl’quelunx spoke up.

“That’s easy!” he announced confidently.

When Sneedl’bodja heard these two words, he was unable to restrain a loud, heartfelt groan. But if you thought that this sounds like dismay, then the only way to describe the feeling occasioned by Qumfl’quelunx’s next words is dismay multiplied by itself an absurd number of times. It was like being thwocked over the head with a large, wet and presumably dead fish (also an absurd number of times).

“I’m wearing a black hat,” he continued in a tone that you would swear was a vocal swagger.

Sneedl’bodja stared blankly at the black hat perched jauntily atop his friend’s head and was unable to contain his astonishment.

“How could you possibly know that?” he gasped.

“That’s easy,” thought Shunshelstinx, warmly congratulating himself for getting three questions right in a row. “I can see that his hat is black from here.”

“That’s easy!” said Qumfl’quelunx for the second time. “If me and Bodge both had white hats, Stinky would know that he must be wearing a black hat, because there are only two white hats….”

“Now why didn’t I think of that?” thought Shunshelstinx.

“…But he couldn’t say what colour of hat he was wearing,” continued Qumfl’quelunx, in the process ascribing a quality of reasoning to Shunshelstinx that was, well, unreasonable. “That means that Stinky can see at least one black hat. Now, I assume that Bodge also worked this out….”

Sneedl’bodja failed to notice the unjustified assumption on which Qumfl’quelunx was basing his argument and silently congratulated the latter for his astute insight.

“…So he would know that if the hat I’m wearing is white, his hat must be black,” concluded Qumfl’quelunx. “But he didn’t know, so it must be my hat that is black.”

“Excellent! Excellent!” said the Grandmaster, chuckling to himself as he spoke. “But perhaps Qunzl’bowza has underestimated your special talents. It would be just like him. By the book. Follow the rules. He never learns. That is all for now. Return here at the same time tomorrow for lesson two.”

There is a mystery to clear up here. Where did Qumfl’quelunx find the wherewithal to answer the Grandmaster’s question, let alone answer it correctly? And that reasoning. Surely he must have deducted when he should have deduced, but you can’t take anything away from him this time. One highly implausible explanation for this unexpected display of mental acuity that is currently doing the rounds is that his brain has been taken over by some awesome alien force, unknown to science or anyone else, that transforms dullards, dunces, dummies, dolts, dopes and dimwits, all of which terms adequately describe Qumfl’quelunx, into scholars, savants and sages, none of which describe, adequately or otherwise, our fat friend. But Qumfl’quelunx is unlikely to be able to offer any definitive guidance on the subject, so you must draw your own conclusions.