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

4. 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

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

6. Triangular Numbers

8. 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".

9. Geometric argument for Gauss’s formula

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

11. 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?

12. 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?

13. 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?

14. 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?

15. Blaise Pascal
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.”

16. Pascal’s Triangle

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

18. 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?

19. 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???