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Chapter 1: The Art of Problem Solving

Chapter 1: The Art of Problem Solving. Letters Game. A BCD EF G HI J KL M N OPQRS T U VW X YZ B C D EF G HI J KLMN OPQ R S T U VWXYZ Where could C, M, R and X belong?. Letters Game. Possible solution: A BCD EF G HI J KL M N OPQRS T U VW X YZ B C D EF G HI J KLMN OPQ R S T U VWXYZ.

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Chapter 1: The Art of Problem Solving

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  1. Chapter 1: The Art of Problem Solving

  2. Letters Game ABCDEFGHIJKLMNOPQRSTUVWXYZ BCDEFGHIJKLMNOPQRSTUVWXYZ Where could C, M, R and X belong?

  3. Letters Game Possible solution: ABCDEFGHIJKLMNOPQRSTUVWXYZ BCDEFGHIJKLMNOPQRSTUVWXYZ

  4. “Die Hard” Problem • Riddle to defuse a bomb, from the movie “Die Hard”: On the fountain there should be two jugs: 5 gallon and 3 gallon. Fill one of the jugs with exactly 4 gallons of water. You must be precise.

  5. What is the next number? For the following sequence, what do you think the next number is? Why? 1, 4, 7, 10, 13, …

  6. What is the next number? For the following sequence, what do you think the next number is? Why? 1, 1, 2, 3, 5, 8, …

  7. Inductive ReasoningSpecific  General Inductive reasoning is characterized by drawing a general conclusion/conjecture from repeated observations of specific examples. The conjecture may or may not be true. You are looking for patterns to make predictions. Examples: letters game, number sequences, predicting next equation

  8. Deductive ReasoningGeneral  Specific Deductive reasoning is characterized by applying general principles to specific examples. Examples: Pythagoras’ theorem Pythagoras and Flatiron

  9. Flatiron BuildingPythagoras and FlatironFlatiron

  10. 1.2 Number Patterns • An ordered list of numbers such as 1, 1, 2, 3, 5, 8, … is called a sequence • Arithmetic sequence: each term is obtained by adding the same number to the previous term: 1, 3, 5, 7, … • Geometric sequence: each term is obtained by multiplying the previous term by the same number (called the common ratio): 1, 2, 4, 8, 16, …

  11. Successive Differences • 1, 3, 6, 10, … • 3, 14, 31, 54, 83, 118, … • 1, 11, 35, 79, 149, 251, …

  12. Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics. Sometimes referred to as "the Prince of Mathematicians", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians. He referred to mathematics as "the queen of sciences". Gauss 1.2

  13. How to find the sums? 1 + 2 + 3 + 4 + 5 + 6 = 1 + 2 + 3 + … 20 = 1 + 2 + 3 + … 100 =

  14. Triangular Numbers

  15. Geometric argument for Gauss’s formula

  16. Frog Climbing a Well 1.3 #50, page 29 A frog is at the bottom of a 20 foot well. Each day it crawls up 4 feet but each night it slips back 3 feet. After how many days will the frog reach the top of the well?

  17. Polya’s 4 Steps • Understand the problem • Devise a plan • Carry out the plan • Look back and check

  18. Units Digit of a Power of 7 #51, page 29 What is the units digit in 7 raised to 491?

  19. Matching Socks #55, page 29 A drawer contains 20 black socks and 20 white socks. If the light is off and you reach into the drawer to get your socks, what is the minimum number of socks you must pull out in order to be sure that you have a matching pair?

  20. Growth of a Lily Pad #59, page 27 A lily pad grows so that each day it doubles in size. On the twentieth day of its life, it completely covers a pond. On what day was the pond half-covered?

  21. Crossing a River A person must take a wolf, a goat, and some cabbage across a river. The rowboat to be used has room for one person plus either the wolf, the goat or the cabbage. If the person takes the cabbage in the boat, the wolf will eat the goat. If the wolf goes in the boat, the goat will eat the cabbage. The goat and cabbage are safe only when the person is present. Even so, the person gets everything across the river. How?

  22. 1623-1662 French mathematician, physicist, inventor, writer and Catholic philosopher Was told not to study math until 15 but rebelled Invented first digital calculator Blaise Pascal “We arrive at truth, not by reason only, but also by the heart.”

  23. Pascal’s Triangle

  24. Patterns in Pascal’s Triangle • Symmetry • Sums of Rows • Prime Numbers • Diagonals and Fibonacci • Hockey Stick Pattern

  25. Pascal and Pizza If there are 8 different toppings to choose from, how many possibilities are there? (assuming no doubles) • No toppings • Just one • Two • Three • Pattern?

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