Chapter 1 The Art of Problem Solving Letters

Chapter 1: The Art of Problem Solving

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

Letters Game Possible solution: ABCDEFGHIJKLMNOPQRSTUVWXYZ

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

Triangular Numbers

Gauss • 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".

Geometric argument for Gauss’s formula

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

Buckets of Water #17, page 27 You have brought two unmarked buckets to a stream. The buckets hold 7 gallons and 3 gallons, respectively. How can you obtain exactly 5 gallons of water to take home?

Frog Climbing a Wall #30, page 28 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?

Matching Socks #31, page 28 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?

Crossing a River #53, page 29 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 “We arrive at truth, not by reason only, but also by the heart. ” • 1623 -1662 • French mathematician, physicist, inventor, writer and Catholic philosopher • Was told not to study math until 15 but rebelled • Invented first digital calculator

Pascal’s Triangle

Patterns in Pascal’s Triangle • • Symmetry Sums of Rows Prime Numbers Diagonals Hockey Stick Pattern Magic 11 s Fibonacci

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?

Pascal and Sierpinski Try this at home: print off a blank Pascal’s triangle, fill in at least 16 rows of numbers. What happens when you fill in all the hexagons that have odd numbers and leave the other ones blank? ? ?