The Rite of Spring: The Student

The Rite of Spring: Fragment 16g

The Student
in which Taynuic sits an examination, and how he fairs therein

There are two ways of getting your surveyor's rule. Officially there's only one. You pay a fee, and sit an examination in numbering. The examination is always very hard; the Hand think up some new trick every year, so it's no good asking someone who's tried before. But if you pay the Hand a lot more, you can have special coaching, in which they show you the trick for the year before the exam. There's never been any hope of us finding the money for the special coaching, so my father has just tried to teach me every numbering trick there is.

Anyway, the surveying exam is held in the spring, at the beginning of the month that we don't name; and on the day of the exam, my father sent me off to the House of the Hand with his blessing, and his prayers. I was feeling fairly confident. We'd spent a lot of time in the weeks leading up to it calculating all sorts of strange things; I thought I was ready for any trick.

I wasn't.

I sat down in a quiet room on the River side of the House of the Hand. There will have been thirty or so other people there sitting the examination. I had six clean wax tablets, and my stylus ready, and I took down what the examiner said word for word. It was this:

"Today's problem is a simple surveying problem, concerning a field with five corners, which we will label with the glyphs of the foot, the nose, the hand, the mouth, and the ear. The distance between foot and nose is thirty six manheights. The distance between nose and hand is fourty-two manheights. The distance between hand and mouth is twenty-three manheights. The distance between mouth and ear is thirty-one manheights. The distance between ear and nose is thirty-seven manheights. The distance between ear and hand is also thirty-seven manheights. The distance between foot and hand is fifty-five manheights.

So far, simple triangulation. Any fool could solve that in five minutes. So where is the trap, I thought? The examiner went on.

"This field is in a land in which the people use a number base of twelve, not ten. Thus they count one, two, three, four, five, six, seven, a numeral we shall call eye, which you will denote by showing a box with the glyph of the eye within it, and a numeral we shall call foot, which you will denote by showing a box with the glyph of the foot within it, and then ten. Thus the number one hundred in this base twelve system is equal to one hundred and fourty four in our own system.

"Now imagine also that they use a unit of linear measure which we will call a feather, which is exactly half a manheight. This they divide into fourty four (or, in their numbering system, thirty six) smaller units which we will call peas. Their unit of area, which we will call a rock, is of four thousand eye hundred and fourty square feathers in their numbering system. Express the area of the field I have described to you in terms which the owner of the field will understand.

When he had finished, I sat and stared at my wax in disbelief. None of it made any sense. I suppose a really primitive people might use a base twelve, if they counted on their thumbs as well as their fingers, but as soon as they began to do any sort of sophisticated numbering they would quickly abandon it. It is clearly impossible! Why, half of ten in this numbering system would be five, and half of five is two and a half. Equally, two cubed would be this number eye, not ten; to the fourth power it would be sixteen, not twenty. In fact, you could never get ten in this system by raising any whole number - or even, I worked out later, any rational number - to any whole power! It is obvious that anyone who used such a system could never learn either to multiply or divide.

And then this business of the units. Fourty four peas to a feather would make sense if you used a base six number system, but no-one who counted in base twelve would use it, because fourty four won't divide by twelve, and niether will any multiple of fourty four that isn't a multiple of five. Then this ridiculous unit of area! I quickly calculated that four thousand eye hundred and fourty in base twelve meant eleven thousand three hundred and fifty; then I worked out the square root; it is a bit more than one hundred and five and eleven twentieths. I don't think it's even a rational number. Surely to goodness no-one ever used a unit of area which wasn't the square of some sensible number of units? Intrigued, I tried dividing it by the circular ratio, and thought at first that the answer was three thousand, which would have made some sort of sense (but not in base twelve). When I did the sum more carefully, though, I found it came out to a fraction more than three thousand and four.

At first I was angry. I was furious! But then I thought, all I can do is my best. All the information I need is here; all I need to do is work out how to tackle it. Obviously the thing to do was to work out the size of the field in cloths, and then work out how many cloths there are to a rock (two thousand two hundred and seventy two). That way of tackling it turned out to be very easy, ignoring all the silly nonsense about peas and feathers.

That gave me one thousand five hundred and thirty three and three quarter cloths, or a tiny bit more than five thousand, five hundred and thirty-six ten-thousandths of an acre. All that remained was to convert my answer into base twelve; although indeed that is not easy to do. I made it just over seventy-one hundredths in base twelve. Very pleased with myself, for few others seemed to be finished, I made a tidy copy of my workings on two fresh waxes, handed them to the examiner, and walked out into the hall.

I was not so pleased so pleased when I returned the next day. On a large clay standing by the entrance to the House was written 'fifty five sixty fourths of an acre', followed by a list of the names of those who had passed - and mine was not on it. I went into the House, found a Priest, and demanded to see the Examiner. The priest, a sour looking man in the black robe of the House, looked me up and down as one might a stray dog.

"What is your business?"

he asked. I explained that I had sat the examination the previous day, and had arrived at an answer that was not only more accurate than that given, but also stated in fractions in orders of magnitude of the number base, yet I had not passed.

"The answer is the answer", said the priest. I explained, with some heat, that my answer was a better answer, and said why.

"It was not the answer that was wanted", he said. "If you had paid for tuition, you would have known how to find the answer that was wanted.

"That's unfair!"

I said.

"That's a cheat. My father is a poor man, and cannot afford to pay for tuition. Yet I worked out the answer, more precisely than that given."

The priest shook his head.

"The question was what the owner would have understood. The owner would not have understood your precision, or your fractions. They are mathematicians precision, mathematicians fractions. The answer is the answer. You failed."

I called out again, more loudly,

"that's unfair!"

All around the great hall of the House, people turned to stare at us. "Hear me, lad", the priest said, sternly.

"This is the House of the God. It is an offence to disturb the peace of this House. Furthermore, the God is outside time, and sees everything. Not only what you have done, but what you will do. The God has judged you. And as the God judges, so the God rewards. Some are rewarded with wealth; some are rewarded with luck; and some are rewarded by knowledge. Now this", he said again,

"is the House of the God; and we admit into this House, and into the crafts governed by this House, only those who can show the God's favour. Some show that favour in money, and buy their craft; some in luck, and win their craft by guessing; and some by ability, and win their craft by knowing. You did not. Therefore we know that the God has judged you, and found you wanting. Therefore, you are not admitted to the craft, and you are not welcome within this House. Do you understand?"

.npÿ

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