Survey of Mathematical IdeasMath 100Chapter 1

John Rosson

Tuesday January 23, 2007

Chapter 1The Art of Problem Solving

1. Solving Problems by Inductive Reasoning

2. Number Patterns

3. Strategies for Problem Solving

4. Calculating, Estimating and Reading Graphs

Number Patterns: Successive Differences

The method of successive difference tries to find a pattern in a sequence by taking successive differences until a pattern is found and then working backwards.

2 57 220 575 1230 2317 ….55 163 355 655 1087 ….108 192 300 432 ….

84 108 132 ….24 24 ….

Number Patterns: Successive Differences

Fill in the obvious pattern and work backwards by adding.

2 57 220 575 1230 2317 399255 163 355 655 1087 1675

108 192 300 432 58884 108 132 156

24 24 24

The method of successive differences predicts 3992 to be the next number in the sequence.

Number Patterns: Successive Differences

The method of successive differences is not always helpful. Consider

1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 ….1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ….

0 1 1 2 3 5 8 13 21 34 55 89 144 233 ….1 0 1 1 2 3 5 8 13 21 34 55 89 ….

Since the sequence reproduces itself after applying successive differences, the method can give us no simplification.

Number Patterns; SumsWe can use patterns to conjecture formula for extended sums.

Conjecture: The sum of the first n numbers

2

)1(...321

nnn

102

544321

62

43321

32

3221

12

211

red is n

blue is n+1

True

Number Patterns; SumsA little more interesting is the sum of squares…

Conjecture: The sum of the squares of the first n numbers

6

)21)(1(...321 2222 nnn

n

306

)421()41(4432130

146

)321()31(332114

56

)221()21(2215

16

)121()11(111

2222

222

22

2

True

Number Patterns; SumsConsider the patterns that we have seen….

Conjecture: The sum of the cubes of the first n numbers

13 23 33 ... n3 n

11 n

21 2n

31 3n

4

10 20 30 ... n0 n

1

11 21 31 ... n1 n

11 n

2

12 22 32 ... n2 n

11 n

21 2n

3

Number Patterns; Sums

So 2 is a counterexample (as is 3) and the conjecture is false. In this case close is not enough

Check some cases.

113 1111

21 21

31 31

41

9 13 23 2

11 2

21 22

31 32

4

35

4

36 13 23 33 31

1 32

1 233

1 334

35

Number Patterns; SumsLooking at the sums of cubes, the true conjecture is ….

Conjecture: The sum of the cubes of the first n numbers

13 23 33 ... n3 n

11 n

2

2

True

113 1

111

2

2

1

9 13 23 21

1 22

2

9

36 13 23 33 3

11 3

2

2

36

100 13 23 33 43 4

11 4

2

2

100

The sums of higher powers have no simple formula. They are formulated in terms of a new idea: the Bernoulli polynomials.

Number Patterns; Figurate Numbers

Number Patterns; Figurate Numbers

The figurate numbers are a classical source of number sequences.

Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55…Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100…

Pentagonal numbers: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145…Hexagonal numbers: 1, 6, 15, 28, 45, 66, 91, 120, 153, 190…

Heptagonal numbers: 1, 7, 18, 34, 55, 81, 112, 148, 189, 235…Octagonal numbers: 1, 8, 21, 40, 65, 96, 133, 176, 225, 280…

Nonagonal numbers: 1, 9, 24, 46, 75, 111, 154, 204, 261, 325…

Number Patterns; Figurate Numbers

We can calculate the figurate numbers using successive differences.Consider the nonagonal numbers.

1 9 24 46 75 111 154 204 261 325 396

8 15 22 29 36 43 50 57 64 717 7 7 7 7 7 7 7 7

Number Patterns; Figurate Numbers

Formulas for the figurate numbers:

2

)57(2

)46(2

)35(2

)24(2

)13(

2

)1(

2

nnNo

nnO

nnHp

nnH

nnP

nS

nnT

n

n

n

n

n

n

n Considering these formulas leads us to conjecture a formula for a general N-agonal number:

2

))4()2((

NnNnNn

Note that this formula works for N=3 and N=4. It even works for N=2 (biagonal numbers).

True

Strategies;Polya’s Four Step Process

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Look back and check

Other Strategies

• Make a table or chart.• Look for a pattern.• Solve a similar

simpler problem.• Draw a sketch.• Use inductive

reasoning• Solve an equation.

• Use a formula.• Work backward.• Guess and check• Common sense (?)• Look for a “catch” if

the problem seems too easy or impossible.

ExampleHow must one place the integers from 1 to 15 in each of the spaces below in such a way that no number is repeated and the sum of the numbers in any two consecutive squares is a perfect square?

ExampleMaybe the problem is easy. Let us just try to fill in the squares following the “consecutives sum” to square rule. Start, say, with 1. This “playing with the problem” is part of understanding it.

97214115412133610151Nothing left to add to 9.

1510631Nothing left to add to 15.

Let us construct a table to see how things “add up”.

Example1+3=4

1+8=9

1+15=16

6+3=9

6+10=16

11+5=16

11+14=25

2+7=9

2+14=16

7+2=9

7+9=16

12+4=16

12+13=25

3+6=9

3+13=16

8+1=9 13+3=16

13+12=25

4+5=9

4+12=16

9+7=16 14+2=16

14+11=25

5+4=9

5+11=16

10+6=16

10+15=25

15+1=16

15+10=25

The table tells us that we have to start with either an 8 or a 9 since these two numbers can only be paired with one other number.

ExampleThe plan is now to start with either 8 or 9 see if we can fill in the table. Most numbers have only two choices for neighbors and one choice will eliminate the next.

811510631312451114279

916251691625169162516916

Answer

Sum Check

Assignments 2.1, 2.2, 2.3, 2.4Read Section 2.1 Due January 25

Exercises p. 541-8, 21-28, 33-40, 41-50, and 67-76

Read Section 2.2 Due January 25

Exercises p. 61 1-6, 23-42, 44, 49-54

Read Section 2.3 Due January 30

Exercises p. 731-6, 7-27 odd, 47, 51, 52, 71, 75, 97, 115, 127, 129, 131, 133.

Read Section 2.4 Due February 1

Exercises p. 79 1, 3, 5, 7, 9, 17, 19, 25, and 27