MATH DAY 2012—Team Competition

An excursion through mathematics and its history MATH DAY 2012—Team Competition

A quick review of the rules • History (or trivia) questions alternate with math questions • Math questions are numbered by MQ1, MQ2, etc. History questions by HQ1, HQ2, etc. • Math answers should be written on the appropriate sheet of the math answers booklet. • History questions are multiple choice, answered using the clicker. • Math questions are worth the number of points shown on the screen when the runner gets your answer sheet. That equals the number of minutes left to answer the question. • Have one team member control the clicker, another one the math answers booklet

Rules--Continued • All history/trivia questions are worth 1 point. • The team with the highest math score is considered first. Next comes the team with the highest history score, from a school different from the school of the winning math team. Finally, the team with the highest overall score from the remaining schools.

HQ0-Warm Up, no points • Non Euclidean Geometry is so called because: • It was invented by Non Euclid. • It negates Euclid’s parallel postulate. • It negates all of Euclid’s postulates. • Euclid did not care for it. • Nobody really knows why it is so called..

HQ0-Warm Up, no points • Non Euclidean Geometry is so called because: • It was invented by Non Euclid. • It negates Euclid’s parallel postulate. • It negates all of Euclid’s postulates. • Euclid did not care for it. • Nobody really knows why it is so called.. 20 seconds

HQ0-Warm Up, no points • Non Euclidean Geometry is so called because: • It was invented by Non Euclid. • It negates Euclid’s parallel postulate. • It negates all of Euclid’s postulates. • Euclid did not care for it. • Nobody really knows why it is so called.. Time's Up!

Demonstrating the points system • For math questions there will be a number in the lower right corner. It will change every minute. Here I am illustrating with numbers changing every 10 seconds. Try to imagine 10 seconds is a minute. The first number tells you the maximum number of points you can get for the question. Assume a question is on the screen.

Demonstrating the point system • For math questions there will be a number in the lower right corner. It will change every minute. Here I am illustrating with numbers changing every 10 seconds. Try to imagine 10 seconds is a minute. The first number tells you the maximum number of points you can get for the question. Assume a question is on the screen. 5

TIME’s UP!

THE CHALLENGE BEGINS VERY IMPORTANT! Put away all electronic devices; including calculators. Mechanical devices invented more than a hundred years ago, are OK.

HQ1. Babylonians One of the oldest of all known civilizations is that of the Babylonians, with capital in Babylon. Where was the city of Babylon located? • In Egypt. • In Greece. • In Iraq. • In Turkey. • In Florida.

HQ1. Babylonians One of the oldest of all known civilizations is that of the Babylonians, with capital in Babylon. Where was the city of Babylon located? • In Egypt. • In Greece. • In Iraq. • In Turkey. • In Florida. 20 seconds

HQ1. Babylonians One of the oldest of all known civilizations is that of the Babylonians, with capital in Babylon. Where was the city of Babylon located? • In Egypt. • In Greece. • In Iraq. • In Turkey. • In Florida. Time's Up!

The Babylonians The name Babylonians is given to the people living in the ancient Mesopotamia, the region between the rivers Tigris and Euphrates, modern day Iraq. They wrote on clay tablets like the one in the picture. The civilization lasted a millennium and a half, from about 2000 BCE to 500 BCE.

MQ1-Babylonian Tables A Babylonian tablet has a table listing n3 +n2 for n = 1 to 30. Here are the first 10 entries of such a table. Tables as these seem to have been used to solve cubic equations. Solve (using the table or otherwise) for an integer solution. Hint: Set x = 2n.

MQ1-Babylonian Tables A Babylonian tablet has a table listing n3 +n2 for n = 1 to 30. Here are the first 10 entries of such a table. Tables as these seem to have been used to solve cubic equations. Solve (using the table or otherwise) for an integer solution. Hint: Set x = 2n. 3

TIME’s UP!

HQ2. Papyrus Writing The Rhind or Ahmes papyrus, dated to 1650 BCE is one of the oldest remaining mathematical documents. Papyrus is a paper like material made from Palm leaves Cotton The stems of a water plant. The leaves of a desert lily. Apple peels.

HQ2. Papyrus Writing The Rhind or Ahmes papyrus, dated to 1650 BCE is one of the oldest remaining mathematical documents. Papyrus is a paper like material made from Palm leaves. Cotton. The stems of a water plant. The leaves of a desert lily. Apple peels. 20 seconds

HQ2. Papyrus Writing The Rhind or Ahmes papyrus, dated to 1650 BCE is one of the oldest remaining mathematical documents. Papyrus is a paper like material made from Palm leaves. Cotton. The stems of a water plant. The leaves of a desert lily. Apple peels. Time's Up!

HQ2. Papyrus Writing The Rhind or Ahmes papyrus, dated to 1650 BCE is one of the oldest remaining mathematical documents. Papyrus is a paper like material made from Palm leaves. Cotton. The stems of a water plant. The leaves of a desert lily. Apple peels. Papyrus is made from the stems of the papyrus plant, a reed like plant growing on the shores of the Nile.

MQ2. Solve like an Egyptian This problem appears in the Rhind papyrus: Divide 100 loaves among 5 men in such a way that the shares received shall be in arithmetic progression and that one seventh of the sum of the largest 3 shares shall be equal to the sum of the smallest two. All shares but the third (middle) one are fractions. What is the middle share?

MQ2. Solve like an Egyptian This problem appears in the Rhind papyrus: Divide 100 loaves among 5 men in such a way that the shares received shall be in arithmetic progression and that one seventh of the sum of the largest 3 shares shall be equal to the sum of the smallest two. All shares but the third (middle) one are fractions. What is the middle share? 3

TIME’s UP!

MQ2. Solve like an Egyptian This problem appears in the Rhind papyrus: Divide 100 loaves among 5 men in such a way that the shares received shall be in arithmetic progression and that one seventh of the sum of the largest 3 shares shall be equal to the sum of the smallest two. All shares but the third (middle) one are fractions. What is the middle share? Answer: 20

HQ3. Pythagoreans • Hippasus the Pythagorean is said to have drowned at sea, or suffered some other punishment for revealing: • That there was an infinity of numbers. • That some numbers could not be expressed as a quotient of integers. • That every number could be expressed as a product of primes. • That some perfect squares are sums of perfect squares. • That numbers will get you, if you don’t watch out.

HQ3. Pythagoreans • Hippasus the Pythagorean is said to have drowned at sea, or suffered some other punishment for revealing: • That there was an infinity of numbers. • That some numbers could not be expressed as a quotient of integers. • That every number could be expressed as a product of primes. • That some perfect squares are sums of perfect squares. • That numbers will get you, if you don’t watch out. 20 seconds

HQ3. Pythagoreans • Hippasus the Pythagorean is said to have drowned at sea, or suffered some other punishment for revealing: • That there was an infinity of numbers. • That some numbers could not be expressed as a quotient of integers. • That every number could be expressed as a product of primes. • That some perfect squares are sums of perfect squares. • That numbers will get you, if you don’t watch out. Time's Up!

More on Pythagoras and his friends The Pythagoreans were a secret society flourishing ca. 600-400 BC They called themselves followers of Pythagoras of Samos, who may or may not have existed. The most important achievement of the Pythagoreans was the discovery of irrational numbers, specifically: The hypotenuse of a right triangle of legs of length 1 is incommensurable with the legs. Or, as we say now, the square root of 2 is irrational, cannot be expressed as a ratio of two integers. The Pythagoreans loved numbers, they adored numbers, they said “Everything is number.” A property that amazed them was the existence of what they called amicable or friendly numbers. A pair of numbers m, n is an amicable pair if each is the sum of the proper divisors of the other one.

Pythagoras and his friends Their only example was the pair 220, 284. The proper divisors of 220 are: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110. 1+2+4+5+10+11+20+22+44+55+110 = 284. The proper divisors of 284 are: 1, 2, 4, 71, 142. 1+2+4+71+142 = 220.

MQ3. Looking for Friends It is known that 1184 is a member of an amicable pair. It is feeling lonely. Find its friend.

MQ3. Looking for Friends It is known that 1184 is a member of an amicable pair. It is feeling lonely. Find its friend. 4

TIME’s UP!

The answer is 1210 • To find the divisors of 1184 we can start seeing that it is even, see how many powers of 2 divide it. 1184/2 = 592, 592/2 = 296, 296/2 = 148, 148/2 = 74, 74/2 = 37, and 37 is prime. That is: 1184 = 25●37. The divisors are all of the form 2p and 2p ●37, 0 ≤ p ≤ 5: 1+2+4+8+16+32+37+74+148+296+592 = 1210

HQ4. The invention of nothing The number 0 in its modern form, and the concept of a negative number first appears in: • India. • Greece. • Rome. • Egypt. • China.

MATH DAY 2012—Team Competition