06 Induction and Recusion 2019 2.pdf - Induction and RecursionPart 2 1 Covering a Board with L-shaped Trominoes Prove that for any integer ≥ 1 if one

06 Induction and Recusion 2019 2.pdf - Induction and...

This preview shows page 1 - 12 out of 72 pages.

Induction and RecursionPart 2 1
Image of page 1
Covering a Board with L-shaped Trominoes Prove that for any integer if one corner square is removed from a checkerboard, the remaining squares can be completely covered by L-shaped trominoes. ࠵?   ≥ 1 2 ࠵?   ×   2 ࠵?
Image of page 2
Proof by induction on n Basis case: Show that a checkerboard minus a corner square can be covered by a trominoe. Inductive Hypothesis: Assume that a checkerboard minus a corner can be completely covered by l-shaped trominoes. 2 1   ×   2 1 2 ࠵?   ×   2 ࠵?
Image of page 3
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Image of page 4
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. Divide into four quadrants…use I.H. on each quadrant…. 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Image of page 5
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. use I.H. 4 times, different corners 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Image of page 6
Inductive Step: Show that a checkerboard minus a corner can be completely covered by trominoes. Consider an checkerboard minus a corner. 4 times plus one 2 ࠵? +1   ×   2 ࠵? +1 2 ࠵? +1   ×   2 ࠵? +1
Image of page 7
Combinations Define as the number of sets of size that can be chosen from a set of size . Like the sum of the first numbers, the argument seems better than the proof. Let’s recall the argument for a formula for . ( n k ) k n n ( n k )
Image of page 8
Combinations - the argument If we are looking to pick a subset of elements from a set of size , we have choices to write down a first element, choices for the second element, and so on down to choices for the last element, giving us permutations. But, we are talking combinations. (i.e., choosing a set of committee members, not ways of arranging of them around a table). Any set of members can be written in how many different ways? k n n n 1 n ( k 1) n ( n 1) ( n 2) ( n ( k 1)) k
Image of page 9
Combinations - the argument If we are looking to pick a subset of elements from a set of size , we have choices to write down a first element, choices for the second element, and so on down to choices for the last element, giving us permutations. But, we are talking combinations. (i.e., choosing a set of committee members, not ways of arranging of them around a table). Any set of members can be written in how many different ways? k n n n 1 n ( k 1) n ( n 1) ( n 2) ( n ( k 1)) k We can write any of members in the first position, any of the remaining in the second position, and so on, giving us k k 1 k ( k 1) ( k 2) 3 2 1
Image of page 10
Combinations - the argument If we are looking to pick a subset of elements from a set of size , we have choices to write down a first element, choices for the second element, and so on down to choices for the last element, giving us permutations. But, we are talking combinations. (i.e., choosing a set
Image of page 11
Image of page 12

You've reached the end of your free preview.

Want to read all 72 pages?

  • Spring '14

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors