**From:** Owen Jones (*oj@maths.anu.edu.au*)

**Date:** Wed 14 May 1997 - 06:59:50 EEST

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MEASURING THE BLOCK

A story from Prax of the man who counted*

The man who counted had been sent to Moonbroth, commissioned by his

emperor to count the number of herd beasts in Prax. While travelling

the wastes South of Moonbroth, he was accosted by Furious Dust Cloud, a

storm khan of fearsome reputation.

Furious Dust Cloud bellowed his greeting, "So this is the man sent to

count tails! Does your emperor think to conquer warriors of Urox with

paper lists?"

"Thankfully, I am not privvy to the emperor's thoughts on such

matters", the man who counted replied softly. "My emperor commands me

to count, and so I do. But in so doing I am fortunate to encounter many

of the strange and beautiful trysts that numbers make.

"For instance, I have already determined that our meeting is marked by

a most fortuitous omen; a sign of friendship."

"How can this be?" shouted Furious Dust Cloud. "What sign of friendship

can you see, that will not soon be trampled in the dust?"

The man who counted smiled reassuringly as he explained; "While we have

been talking, I have counted the ears of you and your companions.

"You have 64 llama's between you, each with 2 ears, and on each llama a

fearsome warrior, each also having 2 ears. The total number of ears is

thus 4 times 64, or 256. My 41 friends and I have with us 43 mules,

thus giving a count of 2 times 42, plus 2 times 43, or

170 ears. However, I must subtract 1 from this total, for as you can

see, my own mule has but one ear, giving a total of 169 ears.

"The numbers 256 and 169 share a curious bond, which I call a quadratic

friendship. The digits of 256 add up to 13. The square of 13 is 169.

The digits of 169 add up to 16, and the square of 16 is 256. In this

way 256 pays it's respects to 169, which 169 amply returns in kind.

Thus our meeting is marked by this omen of friendship."

The khan was taken aback by this omen, for he had imagined no

friendship when he spied the dust of this caravan that morning. After

pondering the meaning of these numbers, he declared, "You must come

with us to the block, and count the ears of those present. Then we will

see if that number, too, is a friendly one." Of course, the khan

planned that his clan would take as slaves the man who counted and his

companions, for such is often the thinking of innumerate men.

When the man who counted and Furious Dust Cloud arrived at the block, a

noisy argument was in place. Three warriors had returned from a raid

with 35 fine beasts, but could not determine how they should be

distributed between them.

"What nonsense is this?" thundered Furious Dust Cloud. "If you butcher

2 beasts and distribute the meat fairly, there will then be 33 beasts

remaining, or 11 each."

"Unfortunately, the division can not be so", replied the first of the

three warriors. "We saught an augery before the raid, and were told the

following by a spirit of Waha; 'You will each meet a single warrior.

One of you will cut down your enemy with just 2 blows. To him will go

one half of the beasts taken. The next will cut down his enemy in 3

blows, and to him will go one third. The last will require 9 blows to

fell his opponent, and to him will go only one ninth of the beasts.' We

are loath to butcher the beasts unnecessarily, but whatever solution

one of us suggests, the other two dispute."

At this the man who counted stepped forward. "I can make this division

for you fairly. Though to do so, I must add to your catch my own humble

mule."

* "How can this be?" yelled the first.
** "Who is this man?" thundered the second.
*

"I want no mule!" insisted the third.

"Quiet!" roared Furious Dust Cloud, in a voice that silenced the

others. "We will hear his solution, and then judge the speaker by its

fairness." For, though he felt the man who counted would fail, his

curiosity had been roused.

The man who counted proceeded as follows; "Together with my mule, you

will agree that the number of beasts is now 36. One half of 35 is 17

and 1/2, but one half of 36 is 18, which is the share I give to the

first warrior, who thus gains by my division and has nothing to

complain about.

"One third of 36 is 12, which I give to the second warrior, who thus

also gains."

Turning to the third warrior, he said, "And one ninth of 36 is 4, which

is your share, by which you also gain.

"By this division, you have all gained, and no beasts need be

butchered. The first of you has 18 beasts, the second 12 and the third

4, which in total comes to 34. Thus of the 36 beasts, 2 remain. One is

my mule, which I take back, while the other is surely due to Furious

Dust Cloud, my sponsor at the block."

Furious Dust Cloud did not like being called the sponsor of the man who

counted, but appreciated the fairness of his solution, so said nothing.

He selected the finest of the beasts for himself, and declared the

issue settled.

Word of the arrival of the man who counted, and his wise division of

the 35 beasts, soon spread. Shortly, he found himself before a mighty

assemblage of khans and their shamans, with Furious Dust Cloud beside

him. The storm khan stepped forward and related the story of the

numbers 256 and 169, and then demanded the following. "This man who

counts has claimed ties of friendship by grace of the number of ears

present at our meeting. Such friendship by numbers was not known to

Waha, and is strange to me. I ask you all to devise for me a test, by

which this friendship can be truely measured.

"If he should pass, then I shall call him friend. But should he fail,

then I claim him and his companions as slaves, to serve me and my

clan."

The khans and their shamans retired to consider the problem Furious

Dust Cloud had put to them. Many spirits were summoned and questioned,

and their guidance heeded. Finally a wizened shaman called Nodder came

forth and spoke thus. "At first was suggested a feat of arms, but this

was discounted, as this man who counts bears no arms. Next was

suggested a test of strength, but he is clearly a puny man, and would

surely fail. If you truely wish to test his friendship, then you must

have a test for his strengths and not his weaknesses, for it is with

his strengths that he will aid you, should he be a friend. The test we

have for you was given to us by a spirit of Urox that dwells at the

very top of the block, and is a proper test for a man skilled with

numbers."

Turning to the man who counted, Nodder continued; "You must measure the

height of the block! Be warned, however, that as one not initiated into

the following of Urox, you will not be allowed to touch the block, nor

even come close to it."

The man who counted replied, "By the grace of my emperor, and the

forbearance of your gods, I think I can solve this problem. I will,

however, need two javelins."

The next morning a great crowd had gathered to watch the testing of the

man who counted. Not even his companions believed he could succeed. The

test was begun by the shaman Nodder, who handed the man who counted two

javelins, and declared; "The height of the block has been given to us

by a spirit that dwells at its very peak. You have until sundown to

tell us what this height is, or your freedom is forfeit."

The man who counted took one of the javelins, and walked towards the

block with it held before him, point down. When still some distance from the

block he stopped, and thrust the javelin into the ground point first. He

then turned to Furious Dust Cloud, and asked, "I must now ask of you a

service. Will you carry the second javelin to the block for me? In doing

so, you must count the number of strides it takes you to get there."

Furious Dust Cloud agreed, for he was not an unjust man, and left for

the block. After a quarter of an hour he returned. "I took 880 strides

to reach the base of the block, where your second javelin now lies. But

how does this tell you its height?"

The man who counted replied, "The answer lies in the kinship of

triangles, which I explain as follows. The ratios between the sides of

a triangle are fixed by their shape, and not their size. Thus for two

triangles of the same shape but different size, the ratio of the

shortest side to the longest side remains the same.

"When I place my head on the ground at this point, the height of the

first javelin appears to me to be the same as the height of the block.

Thus, I create two triangles of the same shape, which I call kin. The

first has as its corners, my eye and the top and bottom of the javelin.

The second shares my eye as its first corner, and uses the top and

bottom of the block for its others. The distance between my eye and the

base of the javelin is one stride, and its height is two strides,

exactly twice the distance to its base. So, by the kinship of these two

triangles, the height of the block must be twice the distance to its

base. That is, 2 times 880, or 1760 strides."**

A great roar went up, though Furious Dust Cloud was unusually quiet,

for Nodder had been forced to agree that the man who counts was

correct. Finally, as if waking from a dream, Furious Dust Cloud spoke.

"Your skill with numbers honours Urox, and in return I grant you

freedom to return to Moonbroth. Should we meet in the wastes, I will

honour my pledge and share my water with you. However, you must tell me

one thing. Why did you require the second javelin?"

At this the man who counted smiled, and said, "For this you must

forgive me. I did not need the second javelin placed by the block, only

that you carry it there. And I neglected to ask you to bring it back

with you."

- ---------------------------------------------------------------

*The man who counted is the creation of Malba Tahan, also known

as Helio Marcial de Faria Pereira, a Brazilian mathematician.

**Judging a stride to be approximately 3 feet, this would make

the block approximately one mile, or 5280 feet, high.

Owen Jones

Centre for Maths and its Applications, School of Math. Sciences

Australian National University, ACT 0200

Ph +61 6 249 2897 (office) 249 4552 (direct) Fax +61 6 249 4675

Web page http://wwwmaths.anu.edu.au/~oj/

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