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Problem Solving in mathematics and computer science<br /> Team 4<br /> Chapter 13 <br /> Reading in Mathematics<br />Team members:<br />Lishu Liu<br />Yun Zhang<br />ProsunjitBiswas<br />Tahmina Ahmed<br />

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

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Thought pattern in mathematics<br />The three patterns are:<br /> Process<br />Problem solving<br />Comparison and contrast<br />

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The kissing problem<br />

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Problem solving in mathematics- step1<br />Define the problem<br />Everyone must kiss each other.<br />We define one kiss as occurring when two people kiss each other, any number of times.<br />Q: How many kisses will occur if we have n people? <br />

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Problem solving in mathematics- step2<br />Try to solvethe problem for simple cases<br />

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Problem solving in mathematics- step3<br />Look for a pattern or clue<br />n=2: 1 kiss <br /> n=1: 0 kiss <br />3<br />1<br />2<br />3<br />1<br />4<br />2<br />n=4: 3+2+1 kisses<br />n=3: 2+1 kisses <br />

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Problem solving in mathematics- step3 (continuing…)<br />n = 5, kisses = 4 + 3 + 2 + 1<br />n = 6, kisses = 5 + 4 + 3 + 2 +1<br />n= 7, kisses = 6 + 5 + 4 + 3 + 2 + 1<br />So, what can be the pattern for the kissing problem?<br />

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Problem solving in mathematics- step4<br />Guess and check<br />What if we have n people?<br />?<br />

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Problem solving in mathematics- step5<br />Use knowledge to solve the problem and extend the solution <br />Kisses = (n-1) + (n-2) + … + 1<br />

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Problem solving in mathematics- step6<br />Try to find a better solution <br />Kisses = (n-1) + (n-2) + … + 1 <br /> = (n + n + … + n) – (1 + 2 + … + (n-1))<br /> = n(n-1) – n(n-1)/2<br /> = n(n-1)/2<br />(n-(n-1))<br />(n-1) factors<br />(n-1) factors<br />(n-1) factors<br />

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Summarizing steps in mathematics<br />Flow chart<br />

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Problem solving in computer science - step1<br />Define the problem<br />Everyone must kiss each other.<br />We define one kiss as occurring when two people kiss each other, any number of times.<br />Q: For N number of People how many kisses there will be ?<br />

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Problem solving in computer science - step2<br />Get sample input and output for simple cases<br />Define input 1, 2, 3, 4 …<br /> | | | |<br />Define output 0, 1, 3, 6 …<br />

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Problem solving in computer science - step3<br />Construct logic with the pattern<br />For first two input<br />So, We can assume that for n people n -1 kisses.<br />n=1: 0 kiss n=2 : 1 kiss<br />

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Problem solving in computer science - step3<br />We consider more input & output to justify our guess(kiss=n-1).<br /> n=1: 0 kiss n=2: 1 kiss<br /> n=3: 2 kisses n=4: 3 kisses<br />But this time we are indeed wrong. So, need to improve our guess.<br />

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Problem solving in computer science - step4<br />We will improve our solution through trial & error.<br />what will happen if we cannot reach the exact solution ?<br />

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Problem solving in computer science – step4<br />So, we will construct & justify our logic using mathematical formula. <br />Kisses = n (n-1) / 2 <br />

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Problem solving in computer science - step5<br />Now we have the solution<br />Now its time to code !<br />FIND NO OF KISSES( N) <br /> KISS=N(N-1)/2<br />RETURN KISS<br />

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Problem solving in computer science - step6<br />Problem solved!<br />Why do we really need a computer to solve the problem?<br />

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Summarizing steps in computer science<br />Flow chart<br />

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Mathematics vs. Computer Science<br />

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Historical relation between mathematics & computer science<br />Donald Knuth (Professor of Stanford)<br />-- Father of Analysis of Algorithm<br />Graduated in Mathematics (Case Institute of Tech.)<br />PhD in Mathematics (Caltech)<br />

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Acknowledgement<br />

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Questions?<br />