This section is from the book "Feeling Better? Amusements and Occupations for Convalescents", by Cornelia R. Trowbridge. Also available from Amazon: Feeling Better.
ACCORDING to Aristotle the science of Mathe-Amatics developed in Egypt because there the priestly class had the leisure necessary for the study of it. The oldest mathematical work known is a papyrus by the Egyptian Ahmes, the Moon-Born, written before 1700 b. c, which contains problems in arithmetic and geometry. It is appropriately entitled "Directions for Obtaining a Knowledge of All Dark Things."
While you are a member of the Leisure Class, here are some Dark Things to study out.
1. Euclid himself is said to have propounded this riddle: An ass and a donkey were plodding along with burdens of corn. The ass said to the donkey, "If you gave me one bushel of corn, I should carry twice as much as you. If I gave you one bushel, our loads would be equal." How much was each carrying?
2 An Arab who owned 17 camels left half of them to his oldest son, a third to his second son and a ninth to his youngest. As they could not agree on the division of their inheritance, they appealed to the chief of their tribe. How did he settle the question so as to satisfy them all?
3 If a hen and a half lay an egg and a half in a day and a half, how many eggs can seven hens lay in six days?
4 A frog in a well 42 feet deep jumps three feet each day and falls back two feet. In how many days will he be out of the well?
5 I sold a house for $5000. I bought it back for $4500, and sold it again for $5500. How much profit did I make?
6 Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann?
7 What integer multiplied by the next higher equals 600?
8 What part of half a square yard is half a yard square?
9 In three days A can build a boat which it takes
B six days to build. How long will it take them working together to build it?
10 If the dividend is multiplied by 4 and the divisor is divided by 2, the quotient will be 40. What was the original quotient?
11 Prove that one equals two.
12 Someone asked a farmer how many cows he had and got this answer. "If I should give you 1/4,1/5 and 1/6 of my herd, you would have 37 cows." How many cows did the farmer have?
13 With two measures containing five pints and three pints how can you measure half a gallon of water?
14 Write 100 with four 9's; with five i's; with four 5's; with five 3's.
15. Add together the figures from 1 to 7 so that they make 100.
16 When do 2 and 2 not make 4?
17 A grocer has but four weights, yet by combining them in various ways he can weigh all quantities up to forty pounds. What are his four weights?
18. Work out this example in addition by substituting for each letter the digit or zero for which it stands. Each letter stands for a different figure.
19. Divide 100 into 4 parts such that if one is increased by 4, one decreased by four, one multiplied by
4, and one divided by 4, the result is in each case the same.
20. A man at a post office asked for some 2c stamps, 10 times as many 1 c stamps and the balance from a dollar in 5c stamps. How many did he get of each?
In letter divisions, the letters of a word are substituted in their order for the digits of an example in division. The problem is to work out the key word. Suppose you are asked to find the key word for this
Only six letters occur, which means that only the first six digits are used. The first multiplication shows that S x R = R or 10 + R or a multiple of 10 plus R. The multiplication tables prove that to get this result, S must equal 6 and R either 2 or 4. £ must be j. T must be 1 or 2. The first subtraction shows that D = 2R and must be 4. Then R is 2, T is 1, and as A = T + R, A is 3. The key word is TRADES. Replacing the letters by figures proves that this answer is correct.
Only words of ten letters, using each letter but once, like PRECAUTION, can supply ail the digits and zero.
i. EAS) URDNPGI(EGRDU
PI S I
If you wish to make up additional Letter Divisions, these are good key words: importance, blacksmith, subtrahend. Figure out how your own Letter Divisions can be solved and then try them on your friends.
These hieroglyphics, found on the walls of one of the Egyptian pyramids, are supposed to represent a long division example such as the governesses of the young Pharaohs used to give their royal pupils. Substitute the correct Arabic numerals.
Test these statements for yourself:
1 In whatever order the digits 123456789 are arranged the number which they make is always divisible evenly by 9.
2 Any number containing an even number of figures which read the same backward and forward (for example 49266294) is divisible by 11. Therefore to any row of figures add or prefix the same figures in reverse order and the result will be divisible by 11. For example take 376 and by prefixing or adding 673 you will get 673376 or 376673, both of them divisible by 11.
3 Any prime number which, if divided by four, leaves a remainder of one, is the sum of two perfect squares. For example, since 13,37 and 113 are all prime numbers and all leave a remainder of one if they are divided by four, they are each equal to the sum of two square numbers. 13 =4 + 9, the squares of 2 and 3. 37 = 36 + 1, the squares of 6 and 1. 113 = 64 + 49, the squares of 8 and 7. Make a list of all the prime numbers to one hundred and as much further as you wish, and you will find this rule always holds.
4 Any number is divisible by nine when its digits added together make nine or a multiple of nine. For example: 45 (4 + 5=9), 5*3 (5 + 1 + 3 = 9). ***7 (2 + 1+8 + 7=18), 52884 (5+2 + 8 + 8+4 = 27) are all divisible by nine. As a consequence of this rule, you can take any number, add its digits, find what other digit will make their sum equal nine or a multiple of nine, insert this digit anywhere in the original number and the resulting number will be divisible by nine. For example, take 7213. Its digits add up to 13 and 13 + 5 = 18, a multiple of nine. Insert 5 where you will. Whether you write 57213 or 75213 or 72513 or 72153 or 72135, the result is divisible by nine.
5 To multiply any figure by nine, add a cipher to it and from the result subtract the original number. For example, if you want to multiply 3 5 by nine, subtract 35 from 350. The answer, 315, is the product of 35 x 9.
6 Try this out: Write the digits from 1 to 9, omitting 8. Take any digit and multiply it by nine and with the result multiply the original number. What curious result will you have? For example:
12345679 x (2x9) =? 12345679 x (8x9)=?
7 Try this also: Take any ten consecutive numbers, the first of which ends with 1 (e. g., 361, 362, 363, etc.
-370). Multiply the first by nine. If the last digit of this product is decreased by one each time and the digit in front of it is increased by one each time, a series will be obtained which is the series of products resulting from multiplying each of the original numbers by nine. (e. g. 3249, 3258, 3267, etc.)
Fun with Figures, A. Frederick Collins. D. Appleton-Century Company, New York. 1928. And very good fun it is. Mathematical Wrinkles, Samuel I. Jones. Published by the author, Nashville, Tennessee. 1929. A teacher's handbook. Includes puzzles, ingenious problems and mathematical short-cuts.
Mathematical Nuts, Samuel I. Jones. Published by the author,
Nashville, Tennessee. 1932. A companion book to Mathematical Wrinkles. Mathematical Excursions, Helen Abbot Merrill. Bruce Humphries,
Conducted by the Emeritus Professor of Mathematics of Welles-ley College.
Mathemagic, Royal V. Heath. Simon and Schuster, New York. 1933.
The Boy's Own Arithmetic, Raymond Weeks. E. P. Dutton and Company, New York. 1924. At once an amusing satire and a book of knotty problems.