An excursion through mathematics and its history

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: A. It was invented by Non Euclid. B. It negates Euclid’s parallel postulate. C. It negates all of Euclid’s postulates. D. Euclid did not care for it. E. Nobody really knows why it is so called..

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

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

HQ0-Warm Up, no points  Non Euclidean Geometry is so called because: A. It was invented by Non Euclid. B. It negates Euclid’s parallel postulate. C. It negates all of Euclid’s postulates. D. Euclid did not care for it. E. 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

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. 4

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. 3

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. 2

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. 1

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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? A. In Egypt. B. In Greece. C. In Iraq. D. In Turkey. E. 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? A. In Egypt. B. In Greece. C. In Iraq. D. In Turkey. E. 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? A. In Egypt. B. In Greece. C. In Iraq. D. In Turkey. E. In Florida. Time's Up!

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? A. In Egypt. B. In Greece. C. In Iraq. D. In Turkey. E. 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 n 3 +n 2 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 n 3 +n 2 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

MQ1-Babylonian Tables A Babylonian tablet has a table listing n 3 +n 2 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. 2

MQ1-Babylonian Tables A Babylonian tablet has a table listing n 3 +n 2 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. 1

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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 A.Palm leaves B.Cotton C.The stems of a water plant. D.The leaves of a desert lily. E.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 A.Palm leaves. B.Cotton. C.The stems of a water plant. D.The leaves of a desert lily. E.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 A.Palm leaves. B.Cotton. C.The stems of a water plant. D.The leaves of a desert lily. E.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 A.Palm leaves. B.Cotton. C.The stems of a water plant. D.The leaves of a desert lily. E.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

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? 2

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? 1

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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: A. That there was an infinity of numbers. B. That some numbers could not be expressed as a quotient of integers. C. That every number could be expressed as a product of primes. D. That some perfect squares are sums of perfect squares. E. 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: A. That there was an infinity of numbers. B. That some numbers could not be expressed as a quotient of integers. C. That every number could be expressed as a product of primes. D. That some perfect squares are sums of perfect squares. E. 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: A. That there was an infinity of numbers. B. That some numbers could not be expressed as a quotient of integers. C. That every number could be expressed as a product of primes. D. That some perfect squares are sums of perfect squares. E. That numbers will get you, if you don’t watch out. Time's Up!

HQ3. Pythagoreans  Hippasus the Pythagorean is said to have drowned at sea, or suffered some other punishment for revealing: A. That there was an infinity of numbers. B. That some numbers could not be expressed as a quotient of integers. C. That every number could be expressed as a product of primes. D. That some perfect squares are sums of perfect squares. E. 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 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, = 284. The proper divisors of 284 are: 1, 2, 4, 71, = 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

MQ3. Looking for Friends It is known that 1184 is a member of an amicable pair. It is feeling lonely. Find its friend. 3

MQ3. Looking for Friends It is known that 1184 is a member of an amicable pair. It is feeling lonely. Find its friend. 2

MQ3. Looking for Friends It is known that 1184 is a member of an amicable pair. It is feeling lonely. Find its friend. 1

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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 = 2 5 ● 37. The divisors are all of the form 2 p and 2 p ● 37, 0 ≤ p ≤ 5 : = 1210

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

HQ4. The invention of nothing The number 0 in its modern form, and the concept of a negative number first appears in: A. India. B. Greece. C. Rome. D. Egypt. E. China. 20 seconds

HQ. The invention of nothing The number 0 in its modern form, and the concept of a negative number first appears in: A. India. B. Greece. C. Rome. D. Egypt. E. China. Time’s Up!

HQ. The invention of nothing The number 0 in its modern form, and the concept of a negative number first appears in: A. India. B. Greece. C. Rome. D. Egypt. E. China. Time’s Up!

MQ4. Brahmagupta’s legacy A quadrilateral of sides of lenghts 4, 5, 7, and 10, is inscribed in a circle. The radius of the circle can be expressed in the form where m, n, s are positive integers, the greatest common divisor of m and n is 1, and s is square free. Determine m + n + s

MQ4. Brahmagupta’s legacy A quadrilateral of sides of lenghts 4, 5, 7, and 10, is inscribed in a circle. The radius of the circle can be expressed in the form where m, n, s are positive integers, the greatest common divisor of m and n is 1, and s is square free. Determine m + n + s 3

MQ4. Brahmagupta’s legacy A quadrilateral of sides of lenghts 4, 5, 7, and 10, is inscribed in a circle. The radius of the circle can be expressed in the form where m, n, s are positive integers, the greatest common divisor of m and n is 1, and s is square free. Determine m + n + s 2

MQ4. Brahmagupta’s legacy A quadrilateral of sides of lenghts 4, 5, 7, and 10, is inscribed in a circle. The radius of the circle can be expressed in the form where m, n, s are positive integers, the greatest common divisor of m and n is 1, and s is square free. Determine m + n + s 1

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MQ4. Brahmagupta’s legacy A quadrilateral of sides of lenghts 4, 5, 7, and 10, is inscribed in a circle. The radius of the circle can be expressed in the form where m, n, s are positive integers, the greatest common divisor of m and n is 1, and s is square free. Determine m + n + s The answer is 78

HQ5. Fibonacci’s rabbits The title of the book in which Fibonacci has the famous rabbit problem, solved by the famous Fibonacci sequence is A.The book of numbers. B.The book of the golden mean. C.The book of calculations. D.The book of sums. E.Peter Rabbit and family.

HQ5. Fibonacci’s rabbits The title of the book in which Fibonacci has the famous rabbit problem, solved by the famous Fibonacci sequence is A.The book of numbers. B.The book of the golden mean. C.The book of calculations. D.The book of sums. E.Peter Rabbit and family. 20 seconds

HQ5. Fibonacci’s rabbits The title of the book in which Fibonacci has the famous rabbit problem, solved by the famous Fibonacci sequence is A.The book of numbers. B.The book of the golden mean. C.The book of calculations. D.The book of sums. E.Peter Rabbit and family. Time’s Up!

HQ5. Fibonacci’s rabbits The title of the book in which Fibonacci has the famous rabbit problem, solved by the famous Fibonacci sequence is A.The book of numbers. B.The book of the golden mean. C.The book of calculations. D.The book of sums. E.Peter Rabbit and family.

MQ5. Recurrence The Fibonacci sequence is an example of a sequence defined recursively. Here is another example. The sequence of numbers a 0, a 1, a 2, a 3, a 4,... satisfies: if n ≥ 2. If a 0 = 3 and a 5 = 307, what is a 1 ?

MQ5. Recurrence The Fibonacci sequence is an example of a sequence defined recursively. Here is another example. The sequence of numbers a 0, a 1, a 2, a 3, a 4,... satisfies: if n ≥ 2. If a 0 = 3 and a 5 = 307, what is a 1 ? 4

MQ5. Recurrence The Fibonacci sequence is an example of a sequence defined recursively. Here is another example. The sequence of numbers a 0, a 1, a 2, a 3, a 4,... satisfies: if n ≥ 2. If a 0 = 3 and a 5 = 307, what is a 1 ? 3

MQ5. Recurrence The Fibonacci sequence is an example of a sequence defined recursively. Here is another example. The sequence of numbers a 0, a 1, a 2, a 3, a 4,... satisfies: if n ≥ 2. If a 0 = 3 and a 5 = 307, what is a 1 ? 2

MQ5. Recurrence The Fibonacci sequence is an example of a sequence defined recursively. Here is another example. The sequence of numbers a 0, a 1, a 2, a 3, a 4,... satisfies: if n ≥ 2. If a 0 = 3 and a 5 = 307, what is a 1 ? 1

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MQ5. Recurrence The Fibonacci sequence is an example of a sequence defined recursively. Here is another example. The sequence of numbers a 0, a 1, a 2, a 3, a 4,... satisfies: if n ≥ 2. If a 0 = 3 and a 5 = 307, what is a 1 ? The answer is 7

MQ5. Recurrence The sequence of numbers a 0, a 1, a 2, a 3, a 4,... satisfies: if n ≥ 2. If a 0 = 3 and a 5 = 307, what is a 1 ?

HQ6. Talking of Fibonacci Leonardo Pisano ( ) One of the main purposes of Fibonacci’s book was A.To discuss sequences and show how they can be used. B.To introduce the Arabic numerals to Europe. C.To introduce new techniques for solving equations. D.To discuss the logarithmic spiral and its applications. E.To warn people about the danger of breeding rabbits.

HQ6. Talking of Fibonacci Leonardo Pisano ( ) One of the main purposes of Fibonacci’s book was A.To discuss sequences and show how they can be used. B.To introduce the Arabic numerals to Europe. C.To introduce new techniques for solving equations. D.To discuss the logarithmic spiral and its applications. E.To warn people about the danger of breeding rabbits. 20 seconds

HQ6. Talking of Fibonacci Leonardo Pisano ( ) One of the main purposes of Fibonacci’s book was A.To discuss sequences and show how they can be used. B.To introduce the Arabic numerals to Europe. C.To introduce new techniques for solving equations. D.To discuss the logarithmic spiral and its applications. E.To warn people about the danger of breeding rabbits. Time’s Up

HQ6. Talking of Fibonacci Leonardo Pisano ( ) One of the main purposes of Fibonacci’s book was A.To discuss sequences and show how they can be used. B.To introduce the Arabic numerals to Europe. C.To introduce new techniques for solving equations. Liber Abaci begins with: ``The nine Indian figures are: : With these nine figures and with sign 0 which the Arabs call zephir any number whatsoever is written, as is demonstrated below.’’

HQ7. Donkeys Crossing The fifth proposition in the first volume of Euclid’s elements was known as the ``pons asinorum,’’ ``bridge of asses.’’ It states A.If in a triangle two angles equal each other, then the triangle is isosceles. B.In an isosceles triangle, the base angles are equal. C.In any triangle the sum of any two sides is greater then the remaining side D.In any triangle the sum of the three interior angles of the triangle equals two right angles. E.In any triangle, the side opposite the greater angle is greater.

HQ7. Donkeys Crossing The fifth proposition in the first volume of Euclid’s elements was known as the ``pons asinorum,’’ ``bridge of asses.’’ It states A.If in a triangle two angles equal each other, then the triangle is isosceles. B.In an isosceles triangle, the base angles are equal. C.In any triangle the sum of any two sides is greater then the remaining side D.In any triangle the sum of the three interior angles of the triangle equals two right angles. E.In any triangle, the side opposite the greater angle is greater. 20 seconds

HQ7. Donkeys Crossing The fifth proposition in the first volume of Euclid’s elements was known as the ``pons asinorum,’’ ``bridge of asses.’’ It states A.If in a triangle two angles equal each other, then the triangle is isosceles. B.In an isosceles triangle, the base angles are equal. C.In any triangle the sum of any two sides is greater then the remaining side D.In any triangle the sum of the three interior angles of the triangle equals two right angles. E.In any triangle, the side opposite the greater angle is greater. Time's Up!

HQ7. Donkeys Crossing The fifth proposition in the first volume of Euclid’s elements was known as the ``pons asinorum,’’ ``bridge of asses.’’ It states A.If in a triangle two angles equal each other, then the triangle is isosceles. B.In an isosceles triangle, the base angles are equal. C.In any triangle the sum of any two sides is greater then the remaining side D.In any triangle the sum of the three interior angles of the triangle equals two right angles. E.In any triangle, the side opposite the greater angle is greater.

MQ6. Circles  Let R be the radius of the circumscribed, r the radius of the inscribed circle of a triangle of sides 21, 20, 13. The ratio R/r can be expressed as a ratio of two positive integers a, b with no common divisors other than 1. Find a + b.: R/r = a/b, gcd(a,b)=1. What is a+b ? Possible hint: 21 = 5+16

MQ6. Circles  Let R be the radius of the circumscribed, r the radius of the inscribed circle of a triangle of sides 21, 20, 13. The ratio R/r can be expressed as a ratio of two positive integers a, b with no common divisors other than 1: R/r = a/b, gcd(a,b)=1. What is a+b ? Possible hint: 21 =

MQ6. Circles  Let R be the radius of the circumscribed, r the radius of the inscribed circle of a triangle of sides 21, 20, 13. The ratio R/r can be expressed as a ratio of two positive integers a, b with no common divisors other than 1: R/r = a/b, gcd(a,b)=1. What is a+b ? Possible hint: 21 =

MQ6. Circles  Let R be the radius of the circumscribed, r the radius of the inscribed circle of a triangle of sides 21, 20, 13. The ratio R/r can be expressed as a ratio of two positive integers a, b with no common divisors other than 1: R/r = a/b, gcd(a,b)=1. What is a+b ? Possible hint: 21 =

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MQ6. Circles  Let R be the radius of the circumscribed, r the radius of the inscribed circle of a triangle of sides 21, 20, 13. The ratio R/r can be expressed as a ratio of two positive integers a, b with no common divisors other than 1: R/r = a/b, gcd(a,b)=1. What is a+b ? Possible hint: 21 = 5+16 The answer is 93

HQ8. The Word Became Number The inventor of logarithms was A. An Italian priest. B. A Greek mathematician. C. A Scottish nobleman. D. A German philosopher. E. A Chinese mathematician

HQ8. The Word Became Number The inventor of logarithms was A. An Italian priest. B. A Greek mathematician. C. A Scottish nobleman. D. A German philosopher. E. A Chinese mathematician 20 seconds

HQ8. The Word Became Number The inventor of logarithms was A. An Italian priest. B. A Greek mathematician. C. A Scottish nobleman. D. A German philosopher. E. A Chinese mathematician Time's Up!

HQ8. The Word Became Number The inventor of logarithms was A. An Italian priest. B. A Greek mathematician. C. A Scottish nobleman. D. A German philosopher. E. A Chinese mathematician Time's Up! John Napier ( )

MQ7. Logarithms  With log being logarithm in base 10, compute log 2 + log 4 + log 8 + log 25 + log 625

MQ7. Logarithms  With log being logarithm in base 10, compute log 2 + log 4 + log 8 + log 25 + log 625 2

MQ7. Logarithms  With log being logarithm in base 10, compute log 2 + log 4 + log 8 + log 25 + log 625 1

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MQ7. Logarithms  With log being logarithm in base 10, compute log 2 + log 4 + log 8 + log 25 + log 625 = log (2 ×4×8×25×625) = log(10 6 ) = 6. The answer is 6.

HQ9.The Renaissance Rafael Bombelli was one of the leading mathematicians of the 16 th century. He was born in the city whose university, founded 1088, may be the oldest in the world. That city is A.Rome B.Venice C.Pisa D.Florence E.Bologna

HQ9.The Renaissance Rafael Bombelli was one of the leading mathematicians of the 16 th century. He was born in the city whose university, founded 1088, may be the oldest in the world. That city is A.Rome B.Venice C.Pisa D.Florence E.Bologna 20 seconds

HQ9.The Renaissance Rafael Bombelli was one of the leading mathematicians of the 16 th century. He was born in the city whose university, founded 1088, may be the oldest in the world. That city is A.Rome B.Venice C.Pisa D.Florence E.Bologna Time’s Up

HQ9.The Renaissance Rafael Bombelli was one of the leading mathematicians of the 16 th century. He was born in the city whose university, founded 1088, may be the oldest in the world. That city is A.Rome B.Venice C.Pisa D.Florence E.Bologna Time’s Up

MQ8. From Bombelli’s Algebra A square is inscribed in triangle ABC with one side on BC. If |AB| = 13, |BC| = 14, |CA| = 15, the length of the side of the square has the form a/b, where a, b are positive integers with no common divisor other than 1. Find a + b.

MQ8. From Bombelli’s Algebra A square is inscribed in triangle ABC with one side on BC. If |AB| = 13, |BC| = 14, |CA| = 15, the length of the side of the square has the form a/b, where a, b are positive integers with no common divisor other than 1. Find a + b. 5

MQ8. From Bombelli’s Algebra A square is inscribed in triangle ABC with one side on BC. If |AB| = 13, |BC| = 14, |CA| = 15, the length of the side of the square has the form a/b, where a, b are positive integers with no common divisor other than 1. Find a + b. 4

MQ8. From Bombelli’s Algebra A square is inscribed in triangle ABC with one side on BC. If |AB| = 13, |BC| = 14, |CA| = 15, the length of the side of the square has the form a/b, where a, b are positive integers with no common divisor other than 1. Find a + b. 3

MQ8. From Bombelli’s Algebra A square is inscribed in triangle ABC with one side on BC. If |AB| = 13, |BC| = 14, |CA| = 15, the length of the side of the square has the form a/b, where a, b are positive integers with no common divisor other than 1. Find a + b. 2

MQ8. From Bombelli’s Algebra A square is inscribed in triangle ABC with one side on BC. If |AB| = 13, |BC| = 14, |CA| = 15, the length of the side of the square has the form a/b, where a, b are positive integers with no common divisor other than 1. Find a + b. 1

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MQ8. From Bombelli’s Algebra A square is inscribed in triangle ABC with one side on BC. If |AB| = 13, |BC| = 14, |CA| = 15, the length of the side of the square has the form a/b, where a, b are positive integers with no common divisor other than 1. Find a + b. The answer is 97

HQ10. Mathematical Advances Who said “Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals” A. Woody Allen. B. Jon Stewart. C. Stephen Colbert. D. Lord Bertrand Russell. E. Stephen Hawking.

HQ10. Mathematical Advances Who said “Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals” A. Woody Allen. B. Jon Stewart. C. Stephen Colbert. D. Lord Bertrand Russell. E. Stephen Hawking. 20 seconds

HQ10. Mathematical Advances Who said “Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals” A. Woody Allen. B. Jon Stewart. C. Stephen Colbert. D. Lord Bertrand Russell. E. Stephen Hawking. Time's Up!

HQ10. Mathematical Advances Who said “Standard mathematics has recently been rendered obsolete by the discovery that for years we have been writing the numeral five backward. This has led to reevaluation of counting as a method of getting from one to ten. Students are taught advanced concepts of Boolean algebra, and formerly unsolvable equations are dealt with by threats of reprisals” A. Woody Allen. B. Jon Stewart. C. Stephen Colbert. D. Lord Bertrand Russell. E. Stephen Hawking. Time's Up!

HQ11. Back to the Pythagoreans Which of the following was NOT a rule the Pythagoreans had to follow: A.Don’t eat beans. B.Never stir the fire with a knife. C.Touch the earth when it thunders. D.Spit on your nail pairings and hair trimmings. E.No women are allowed at the meetings.

HQ11. Back to the Pythagoreans Which of the following was NOT a rule the Pythagoreans had to follow: A.Don’t eat beans. B.Never stir the fire with a knife. C.Touch the earth when it thunders. D.Spit on your nail pairings and hair trimmings. E.No women are allowed at the meetings. 20 seconds

HQ11. Back to the Pythagoreans Which of the following was NOT a rule the Pythagoreans had to follow: A.Don’t eat beans. B.Never stir the fire with a knife. C.Touch the earth when it thunders. D.Spit on your nail pairings and hair trimmings. E.No women are allowed at the meetings. Time's Up!

HQ11. Back to the Pythagoreans Which of the following was NOT a rule the Pythagoreans had to follow: A.Don’t eat beans. B.Never stir the fire with a knife. C.Touch the earth when it thunders. D.Spit on your nail pairings and hair trimmings. E.No women are allowed at the meetings. Time's Up!

HQ12. Euler  Leonhard Euler ( ) was the greatest mathematician of the 18 th century and one of the greatest of all times. He was born in A. Basel, Switzerland. B. Berlin, Germany. C. Bonn, Germany. D. Salzburg, Austria. E. Vienna, Austria.

HQ12. Euler  Leonhard Euler ( ) was the greatest mathematician of the 18 th century and one of the greatest of all times. He was born in A. Basel, Switzerland. B. Berlin, Germany. C. Bonn, Germany. D. Salzburg, Austria. E. Vienna, Austria. 20 seconds

HQ12. Euler  Leonhard Euler ( ) was the greatest mathematician of the 18 th century and one of the greatest of all times. He was born in A. Basel, Switzerland. B. Berlin, Germany. C. Bonn, Germany. D. Salzburg, Austria. E. Vienna, Austria. Time’s Up!

HQ12. Euler  Leonhard Euler ( ) was the greatest mathematician of the 18 th century and one of the greatest of all times. He was born in A. Basel, Switzerland. B. Berlin, Germany. C. Bonn, Germany. D. Salzburg, Austria. E. Vienna, Austria. Time’s Up!

HQ13. Etymologies  The word algebra is A. Of Greek origin. B. Of Arabic origin. C. Of Indian origin. D. Of Latin origin. E. Of Italian origin.

HQ13. Etymologies  The word algebra is A. Of Greek origin. B. Of Arabic origin. C. Of Indian origin. D. Of Latin origin. E. Of Italian origin. 10 seconds

HQ13. Etymologies  The word algebra is A. Of Greek origin. B. Of Arabic origin. C. Of Indian origin. D. Of Latin origin. E. Of Italian origin. Times Up

HQ13. Etymologies  The word algebra is A. Of Greek origin. B. Of Arabic origin. C. Of Indian origin. D. Of Latin origin. E. Of Italian origin.

HQ14. Women in mathematics She was one of the very important mathematicians of the 20 th century, making essential contributions to both abstract algebra and physics. She was recently the subject of a long article in the New York Times. Her name is A. Frances Langford. B. Matilda Neuberger. C. Lise Meitner. D. Anne Dubreil. E. Emmy Noether.

HQ14. Women in mathematics She was one of the very important mathematicians of the 20 th century, making essential contributions to both abstract algebra and physics. She was recently the subject of a long article in the New York Times. Her name is A. Frances Langford. B. Matilda Neuberger. C. Lise Meitner. D. Anne Dubreil. E. Emmy Noether. 10 seconds

HQ14. Women in mathematics She was one of the very important mathematicians of the 20 th century, making essential contributions to both abstract algebra and physics. She was recently the subject of a long article in the New York Times. Her name is A. Frances Langford. B. Matilda Neuberger. C. Lise Meitner. D. Anne Dubreil. E. Emmy Noether. Time’s Up!

HQ14. Women in mathematics She was one of the very important mathematicians of the 20 th century, making essential contributions to both abstract algebra and physics. She was recently the subject of a long article in the New York Times. Her name is A. Frances Langford. B. Matilda Neuberger. C. Lise Meitner. D. Anne Dubreil. E. Emmy Noether. Time’s Up!

HQ15. A French nobleman.  The Marquis of L’Hôpital ( ) was the author of the very first Calculus textbook. His famous rule appears in it. But it seems the book was really written by: A. Isaak Newton. B. Johan Bernoulli. C. Gottfried Leibniz. D. Isaak Barrow. E. Abraham DeMoivre.

HQ15. A French nobleman.  The Marquis of L’Hôpital ( ) was the author of the very first Calculus textbook. His famous rule appears in it. But it seems the book was really written by: A. Isaak Newton. B. Johan Bernoulli. C. Gottfried Leibniz. D. Isaak Barrow. E. Abraham DeMoivre. 10 seconds

HQ15. A French nobleman.  The Marquis of L’Hôpital ( ) was the author of the very first Calculus textbook. His famous rule appears in it. But it seems the book was really written by: A. Isaak Newton. B. Johan Bernoulli. C. Gottfried Leibniz. D. Isaak Barrow. E. Abraham DeMoivre. Time’s Up!

HQ15. A French nobleman.  The Marquis of L’Hôpital ( ) was the author of the very first Calculus textbook. His famous rule appears in it. But it seems the book was really written by: A. Isaak Newton. B. Johan Bernoulli. C. Gottfried Leibniz. D. Isaak Barrow. E. Abraham DeMoivre. Time’s Up!

HQ16. Conjectures.  The Goldbach conjecture states: A. Every even integer greater than 4 is a sum of two primes. B. There exists an infinity of primes p such that p+2 is also prime. C. There exists an infinity of primes of the form 5k+7, where k is a positive integer. D. If 2 p – 1 is prime, then p is prime. E. An infinite number of Fibonacci numbers are prime.

HQ16. Conjectures.  The Goldbach conjecture states: A. Every even integer greater than 4 is a sum of two primes. B. There exists an infinity of primes p such that p+2 is also prime. C. There exists an infinity of primes of the form 5k+7, where k is a positive integer. D. If 2 p – 1 is prime, then p is prime. E. An infinite number of Fibonacci numbers are prime. 10 seconds

HQ16. Conjectures.  The Goldbach conjecture states: A. Every even integer greater than 4 is a sum of two primes. B. There exists an infinity of primes p such that p+2 is also prime. C. There exists an infinity of primes of the form 5k+7, where k is a positive integer. D. If 2 p – 1 is prime, then p is prime. E. An infinite number of Fibonacci numbers are prime. Time’s Up!

HQ16. Conjectures.  The Goldbach conjecture states: A. Every even integer greater than 4 is a sum of two primes. B. There exists an infinity of primes p such that p+2 is also prime. C. There exists an infinity of primes of the form 5k+7, where k is a positive integer. D. If 2 p – 1 is prime, then p is prime. E. An infinite number of Fibonacci numbers are prime.