Title: Problem Solving, Searching for a Pattern and Math Induction
1Problem Solving, Searching for a Pattern and Math
Induction
- JUMELA F. SARMIENTO
- Mathematics Department
- Ateneo de Manila University
2Outline
- Introduction to Problem Solving
- Searching for a Pattern
- Using Direct Proofs
- Working Backward
- Using Indirect Proofs By Contradiction
- Using Mathematical Induction
3Introduction to Problem Solving
- What is a Problem?
- A situation, quantitative or otherwise, that
confronts an individual, that requires resolution
and for which the individual sees no apparent
path to the solution. - It is open-ended, paradoxical and sometimes
unsolvable and requires investigation before one
can come close to a solution.
4- Problem requires analysis and synthesis of
previously learned knowledge to resolve. - A problem is different from a question and an
exercise. - A question can be resolved by mere recall and
memory - An exercise involves drill and practice to
reinforce a previously learned skill or algorithm
5What is Problem Solving?
- In the book, How to solve it, G. Polya stated
that to solve a problem is
to find a way, where no way is known
off-hand, to attain a desired end that is not
immediately attainable, by appropriate means. - Problem solving is a process.
- It is the process by which one uses a previously
acquired knowledge, skills and understanding to
satisfy the demands of an unfamiliar situation. - It begins with some initial investigation and
concludes when an answer is obtained.
6Who is a good problem solver?
- A good problem solver has a desire to solve
problems. - He/She is perseverant when solving problems.
- He/She is not afraid to guess.
- He/She is confident, concentrated and courageous.
7Polyas four-step problem solving process
8Problem solving employs certain strategies called
Heuristics.
- Searching for a patterns
- Working backwards
- Drawing pictures and diagrams
- Exploiting symmetry
- Using direct and indirect arguments
- Arguing by contradiction
- Using calculator/computer as a problem solving
tool - Using Mathematical Induction
- Using the Pigeonhole Principle
9Searching for a Pattern
- Gathering initial data is important in a
mathematical investigation. - It is best to begin problem solving by examining
specific instances of the problem. - If exploration is done systematically, patterns
may emerge that will suggest a solution to the
entire problem.
10Example. Suppose there are n lines in the
plane, no two are parallel and no three intersect
at a point. Into how many regions do these lines
divide the plane?
11Division of a plane by n lines
n 1
n 2
1
2
1
2
3
4
n 3
n 4
5
2
5
2
10
1
6
1
11
6
9
3
4
3
4
7
7
8
Figure 1
12Let f (n) be the number of regions into which
the plane is divided by n lines.
n f (n) 1 2 2 4 3 7 4
11 5 6
f (n) - f (n-1)
2
3
4
Conjecture
?
16
f (n) f ( n 1) n for n gt 1
?
22
13Show f (n) f ( n 1) n for n gt 1 or
f (n) f ( n 1) n for n gt 1
- When we have n 1 lines and add an nth line,
the nth line must intersect all the others.
- The nth line is divided into n segments by the
other lines. - Since each of these segments divides an old
region into new regions, the addition of the nth
line increases the number by n. -
14To obtain a more explicit formula for f (n), we
compute for f (n) as follows
f (n) f ( n 1) n f (
n 2) ( n 1) n f ( n 3)
( n 2) ( n 1) n .
. . f (
1) ( 2 3 4 n ) 2
n(n1)/2 ? 1
Therefore, f (n) (n2 n
2) / 2 Finally, check if it fits the data.
15Using Direct Proofs
- A direct proof begins with the given information
and by a succession of logical steps leads to the
desired (and required) conclusion.
- In particular, we must come up with a sequence of
statements - P1
- P2
- .
- .
- .
- Pn (Conclusion)
16For example, a typical sequence of proofs is as
follows (1) P1 definition (2) P1 ?
P2 proved theorem (3) P2 follows from (1) and
(2) by MP (4) P2 ? P3 proved theorem (5)
P3 follows from (3) and (4) by MP (6) P3 ?
P4 proved theorem (7) P4 follows from (5) and
(6) by MP
17Example. Consider the axioms for the four-point
geometry. Axiom 1. There are exactly four
points. Axiom 2. Each pair of points has
exactly one line in common. Axiom 3. Each line is
on exactly two points.
Prove If P is a point there exists at least
one line not containing P.
18Solution
Let P be a point. From Axiom 1, such a
point exists.
Axiom 1 also guarantees that there exists a
second point Q and a third point R.
From Axiom 2, there exists a line l containing
P and Q. Point R is not on l from Axiom 3.
Moreover, Axiom 2 guarantees that there exists
a line m containing Q and R. P does not
belong to m from Axiom 3. Thus, m is a
line not containing P
19Using Indirect Proof By
Contradiction
When do we use proof by contradiction ?
- If a certain conclusion can be easily negated
and it further gives some useful information - If a set of hypotheses do not have much to offer
- When the direct method fails, that is, we get
stuck at a stage and do not know how to proceed.
20How do we prove by contradiction?
21Example. Given that a, b, c are odd integers,
prove that the equation ax2 bx c 0
cannot have a rational root.
22Using Mathematical Induction
Suppose we wish to know the sum of the first n
positive odd integers. We first look at some
instances and search for a pattern. 1 12
1 3 22
1 3 5 32
1 3 5 7 42 1 3 5
7 9 52 It is reasonable to guess that the
sum of the first n positive odd integers is n2 .
That is, 1 3 5 (2n 1) n2
23- We have to show that the formula is true for any
positive integer n. - The method of proof known as Mathematical
Induction gives an alternative to direct proof in
the case where an infinite number of cases is
involved. - The idea is that we start with some statement
that depends on a positive integer n. Call this
statement S(n).
For example, S(n) 1 3 5 (2n 1)
n2 where n is a positive integer
24- THE PRINCIPLE OF MATHEMATICAL INDUCTION
- Let S(n) be a statement for each positive
integer n. Suppose that the following two
conditions hold - S(1) is true.
- If S(k) is true then S(k 1) is true, where k
is any positive integer. - Then S(n) is a true statement for all positive
integers n.
25Consider a long (endless) row of dominoes
How can we make them all fall down?
- Push down the first domino towards the second.
- We are sure that the 1st knocks down the 2nd,
the 2nd knocks down the 3rd, and in general, the
kth domino knocks down the (k 1)st. - Eventually, all will fall down.
26The following guarantees this chain
reaction 1. The first domino will fall. 2.
If any domino falls, then so is the next These
two conditions were the guidelines in forming the
Principle of Mathematical Induction.
How do we prove by Mathematical Induction?
27Example. Prove that S(n) is true for all
positive integers n, where S(n) is the
statement 1 3 5 . . .(2n 1) n2
28Example Let S(n) 13 23 33 . . . n3
(1/4) n2 (n 1)2 Prove that it is true for
all positive integers n.
29Example. Prove that 72n - 48n -1 is divisible by
2304 for every positive integer.
30Thank you very much for listening...