Problem Solving in mathematics and computer science

Team members:Lishu Liu

Yun ZhangProsunjit BiswasTahmina Ahmed

Team 4Chapter 13 Reading in Mathematics

How ideas and relationships are expressed in mathematics?

Ideas and relationships are expressed through notations, symbols, numbers and formulas.Symbol: =,<,>,+,-,*,/ etc.Number: 1,2,3,0.1,1/2,logn etc.Formula: ex. y = x+2

Thought pattern in mathematics

The three patterns are: ProcessProblem solvingComparison and contrast

The kissing problem

Problem solving in mathematics- step1

Define the problemEveryone must kiss each other.We define one kiss as occurring when two people kiss each other, any number of times.

Q: How many kisses will occur if we have n people?

Problem solving in mathematics- step2

Try to solve the problem for simple cases

Problem solving in mathematics- step3

Look for a pattern or clue

n=1: 0 kiss

n=4: 3+2+1 kisses n=3: 2+1 kisses

n=2: 1 kiss

1 3

2

1

2

3

4

Problem solving in mathematics- step3 (continuing…)

n = 5, kisses = 4 + 3 + 2 + 1n = 6, kisses = 5 + 4 + 3 + 2 +1n = 7, kisses = 6 + 5 + 4 + 3 + 2 + 1

So, what can be the pattern for the kissing problem?

Problem solving in mathematics- step4

Guess and checkWhat if we have n people?

?

Problem solving in mathematics- step5

Use knowledge to solve the problem and extend the solution Kisses = (n-1) + (n-2) + … + 1

Problem solving in mathematics- step6

Try to find a better solution Kisses = (n-1) + (n-2) + … + 1

= (n + n + … + n) – (1 + 2 + … + (n-1))

= n(n-1) – n(n-1)/2 = n(n-1)/2

(n-1) factors

(n-1) factors (n-1) factors

(n-(n-1))

Summarizing steps in mathematics

Flow chart

Step 1

•Define the problem

Step 2

•Try some solution to the problem for simple cases

Step 3

•Look for a pattern or clue

Step 4

•Guess and check

Step 5

•Use knowledge to solve the problem and extend the solution

Step 6

•Try to find a better solution

Problem solving in computer science - step1

Define the problemEveryone must kiss each other.We define one kiss as occurring when two people kiss each other, any number of times.

Q: For N number of People how many kisses there will be ?

Problem solving in computer science - step2

Get sample input and output for simple casesDefine input 1, 2, 3, 4 … | | | |Define output 0, 1, 3, 6 …

Problem solving in computer science - step3

Construct logic with the patternFor first two input

So, We can assume that for n people n -1 kisses.

n=1: 0 kiss n=2 : 1 kiss

Problem solving in computer science - step3

We consider more input & output to justify our guess(kiss=n-1).

n=1: 0 kiss n=2: 1 kiss

n=3: 2 kisses n=4: 3 kisses

But this time we are indeed wrong. So, need to improve our guess.

Problem solving in computer science - step4

We will improve our solution through trial & error.

what will happen if we cannot reach the exact solution ?

Problem solving in computer science – step4

So, we will construct & justify our logic using mathematical formula.

Kisses = n (n-1) / 2

Problem solving in computer science - step5

Now we have the solutionNow its time to code !

FIND NO OF KISSES( N) KISS=N(N-1)/2RETURN KISS

Problem solving in computer science - step6

Problem solved!

Why do we really need a computer to solve the problem?

Summarizing steps in computer science

Flow chart

Step 1

•Problem Understanding

Step 2

•Get sample Input and Output

Step 3

•Construct logic with the pattern

Step 4

•Improve logic through Trial & Error

Step 5

•Finally solve the problem with mathematics

Step 6

•Fitting the mathematical solution in computer programming.

Mathematics vs. Computer Science

Mathematics Computer ScienceStep 1 Define the problem Problem understandingStep 2 Try to solve the problem for

some simple cases Get sample input and output

Step 3 Look for a pattern or clue Construct logic with the pattern

Step 4 Guess and check Improve logic through trail & error

Step 5 Use knowledge to solve the problem and extend the solution

Finally get the solution

Step 6 Try to find a better solution Problem solved

Historical relation between mathematics & computer

science

Donald Knuth (Professor of Stanford)-- Father of Analysis of Algorithm

Graduated in Mathematics (Case Institute of Tech.)PhD in Mathematics (Caltech)

Acknowledgement

Contact Person Ahmed, TahminaHandout preparation Ahmed, Tahmina & Lishu LiuProof reading Zhang, YunSlide Customization Biswas, ProsunjitContent Preparation AllSlide Preparation All

Questions?