The Man Who Counted
W. W. Norton & Company, 1993; ISBN 0-393-30934-7. [ See this book at Amazon.com]
or Canongate Books, 1993; ISBN 0-86241-883-6.
compiled by Tom Verhoeff in June 2005.
The actual stories are much nicer, and include other mathematical allusions.
Introduces Hanak Tade Maia (HTM), who has experienced numerous adventures with ``The Man Who Counted'', in and around Baghdad. HTM ``transcribes'' the latter's life story in the following chapters.
Introduces Beremiz Samir (``The Man Who Counted'') born in Khoi (Persia). As a boy shepherd, Beremiz used to count his master's flock of sheep, many times a day, for fear of losing an animal. He developed the skill to count many things at a glance.
While traveling, HTM and Beremiz meet 3 men who inherited 35 camels. The will prescribes that the oldest son gets 1/2, the middle son 1/3, and the youngest 1/9. The men quarrel because they don't want to cut up animals.
Beremiz solves this by offering HTM's camel (not without complaints). The 36 animals can be divided according to the will, without cutting an animal: 18+12+4=34, each getting more than his share, and still leaving 2 camels. Thus, HTM gets back his own animal, and Beremiz is offered a camel for himself as reward.
HTM has 3 loaves of bread; Beremiz has 5. They meet a hungry man named Salem Nasair (SN), who offers to pay them 8 pieces of gold if they share their bread with him. Later, SN indeed offers 3 pieces of gold to HTM and 5 to Beremiz. Beremiz objects, arguing that he should get 7 pieces of gold and HTM only 1.
His argument is as follows. Each loaf was divided evenly among the 3 men. Thus, HTM brought in 3x3=9 pieces and Beremiz 5x3=15; making 24 pieces altogether. Each ate 8 pieces; hence, HTM contributed 9-8=1 piece to SN's meals and Beremiz 15-8=7.
A jeweler promised to pay old Salim 20 dinars for lodging if he sold all of his jewels for 100 dinars, and 35 if he sold them for 200. He actually sold them all for 140 dinars. The jeweler argues that he owes old Salim 24.5 dinars: 35 dinars for lodging per 200 sales is the same as 3.5 per 20. Thus, 140 sales translates into 7x3.5=24.5 for lodging.
Old Salim wants to get 28 dinars, because 20 dinars for lodging per 100 sales is the same as 2 per 10. Thus, 140 sales translates into 14x2=28 for lodging.
Beremiz reasons that 200-100=100 dinars extra in sales would have resulted in 35-20=15 dinars extra for lodging, or 1.5 dinars extra for lodging per 10 dinars extra sales. The sales was actually 40 extra (above 100), hence the lodging costs 4x1.5=6 extra (above 20). Therefore, 26 dinars for lodging is the fair price.
Beremiz argues that 256 camels is a much nicer wedding present for Astir, than 257 camels.
257 is prime. Astir is 16 years old. 256 is a power of 2, and it is the square of 16. Furthermore, the sum of the digits of 256 is 13. The square of 13 is 169, whose digit sum is again 16. This is called quadratic friendship.
Beremiz shows how to make each of the numbers from zero to ten by combining exactly four 4s.
|0||44 - 44|
|2||4/4 + 4/4|
|4||4 + (4-4)/4|
|6||(4+4)/4 + 4|
|7||44/4 - 4|
|8||4 + 4 + 4 - 4|
|9||4 + 4 + 4/4|
A shopkeeper has lent 50 dinars to a sheik, and another 50 to a merchant. The sheik paid the debt in four installments of 20, 15, 10, and 5 dinars:
The merchant paid the debt in four installments of 20, 18, 3, and 9 dinars:
How to explain the difference of 1 dinar?
Beremiz points out that summing the amounts still owed has no relevance. For example, if the debt would be paid in three installments of 10, 5, and 35 dinars, the table would be:
21 identical casks of wine, of which 7 full, 7 half full, and 7 empty, are to be divided among three men. Each should receive the same amount of wine and the same number of casks, without opening them. How to do this?
The first receives 3 full casks, 1 half full, and 3 empty. The second and third each receive 2 full casks, 3 half full, and 2 empty. (There is also another solution.)
A bill of 30 dinars is paid by the three men, each putting up 10 dinars. It turns out that there was an error; the bill was only 25 dinars. They get back 5 dinars; each takes 1 dinar, and the remaining 2 dinars are given to the slave who served them. However, someone finds this strange, because now each has paid 9 dinars making a total of 27 dinars. Adding in the 2 dinars given to the slave only makes 29 dinars. How did 1 dinar disappear?
One should account as follows: 27=25+2 (rather than 27+2).
Where Beremiz promises to teach geometry to Telassim, the daughter of Iezid, aged 17.
Where Beremiz explains why he prefers 496 over 499, because the former is a perfect number, whose true divisors add up to the number itself. Other perfect numbers are 6 and 28.
Where Beremiz teaches his first lesson to Telassim. He tells about Pythagoras, mathematics, arithmetic, algebra, geometry, astronomy.
Harim and Hamed each bring 30 melons to be sold in the market. Harim's melons are to be sold at a price of 3 for 1 dinar, whereas Hamed's are to be sold at the higher price of 2 melons for 1 dinar. The total revenue would be 10+15=25 dinars. The seller fears that selling them as instructed will not work out, and decides to sell the 60 melons at the same time for the price of 3+2=5 melons for 1+1=2 dinars. However, after selling all melons this way, he only made (60/5)x2=24 dinars.
Beremiz explains the discrepancy, by pointing out that only 10 batches of 5 melons can be made that should go for 2 dinars; the remaining 10 melons should all have been sold at the higher price (yielding 5 dinars, instead of 4).
Where Beremiz tells about friendly numbers, such as 220 and 284. The true divisors of 220 (i.e. 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110) add up to 284, and those of 284 (i.e. 1, 2, 4, 71, 142) add up to 220.
Three daughters of a peasant, who bragged about their intelligence, are challenged by a qadi (judge). The qadi instructs them to sell 90 apples on the market: Fatima, the oldest, must sell 50, Cunda 30, and Shia, the youngest, 10. If Fatima sells her apples at a price of 7 to a dinar, then the others must sell them at the same price. If Fatima sells her apples for 3 dinars each, then the others must do so as well. Furthermore, they must end up with the same amount of money. They cannot give their apples away for nothing.
Beremiz explains a solution:
Fatima Cunda Shia Phase Price apples dinar apples dinar apples dinar One 7 to a dinar 7x7=49 7 4x7=28 4 1x7= 7 1 Two 3 dinar a piece 1 3 2 6 3 9 Total 50 10 30 10 10 10
Beremiz shows that a few examples do not make a (true) mathematical theorem. He considers the three 4-digit numbers 2025, 3025, and 9801. Their square roots are 45, 55, and 99, because 45x45=2025, 55x55=3025, and 99x99=9801. He also observes that 20+25=45, 30+25=55, and 98+01=99. However, it is not a theorem that the square root of a 4-digit number can always be obtained by adding the left half to the right half.
Where Beremiz wishes to marry young Telassim. His wish is granted provided he solves a strange problem....
HTM tells that in 1258, Baghdad was destroyed by Tartars and Mongols. He and Beremiz had fled 3 years earlier to Constantinople.