Chapter 1: The Art of Problem Solving

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

Letters Game Possible solution: ABCDEFGHIJKLMNOPQRSTUVWXYZ

“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.

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

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

Inductive Reasoning Specific  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

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

Flatiron Building Pythagoras and Flatiron Flatiron

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, …

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

Gauss 1.2 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".

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

Triangular Numbers

Geometric argument for Gauss’s formula

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?

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

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

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?

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?

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?

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

Pascal’s Triangle

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

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?