Some Surprising Patterns!

Transcription

1 Kasman MSM 715 Some Surprising Patterns! Part I The What 1. A landscape architect is going to lay a border of bricks around a garden. The bricks are each 1 unit wide by 2 units long, and the border will be 2 units high. She wants to know how many different arrangements she could use to make a border that has a specified length. In the picture below, you can see the different ways that she can lay a border that is 1, 2, or 3 units long. Find the number of arrangements that give borders which are 4, 5 and 6 units long. 2. A group of students will be studying abroad in Spain next month. Some of the students are native Spanish speakers, while others are native English speakers. In order to make sure students practice their Spanish, she insists that whenever students are sitting in a row there can never be two native English speakers next to each other. Below you can see the number of ways that students can sit in 1 or 2 chairs. Find the number of ways that they can sit in 3,4 or 5 chairs. 1 chair S or E 2 chairs SS or SE or ES

2 3. A set of identical square boxes need to be mailed. They are going to be taped together in a single layer so that the edges of one box are always completely aligned with the edges of any adjacent boxes. The tape will then go around the perimeter of the boxes. Since the parcels will stand on their own, no box should have blank space below it unless it s on the bottom. number of sides on perimeter parcel shapes number of parcels Find all of the arrangements which have 10 sides on its perimeter. 4. In each of the previous 3 problems, can you predict how the number of arrangements grows?

3 Part II The Why Each of your patterns likely had something to do with the Fibonacci Numbers. You may recall that the Fibonacci Numbers usually start with the pattern 1,1,2,3,5,8,13,, where each number (starting with 2) is the sum of the previous two numbers in the sequence. So the question for us is why did these numbers show up in three completely different situations? The Brick Problem Let s let B n represent the number of arrangements of length n. That is, Our goal is to justify the recursive equation. B 1 =1, B 2 =2 B 3 =3, B 4 =, B 5 =, B 6 =, and B n =B n-1 + B n-2, for n>2. 1. (a) Let s examine all of your brick borders of length 5. Divide your borders into categories based on what they have at their rightmost ends. (b) How many categories did you need in part (a)? How many borders were in each category? (c) What will happen if you divide your borders of length 6 into the same categories? (d) Can you use this categorization to explain the recursive equation B n =B n-1 + B n-2?

4 The Chair Problem 2. Can you come up with a recursive sequence for the chair problem, and then justify your recursion in a similar way to the way we analyzed the Brick Problem? The Parcel Problem The parcel problem is a bit trickier. Not every term of the Fibonacci sequence is showing up only every other one is. There are different ways to analyze this problem, but we ll take advantage of a particular identity about these terms. P 1 represents the number of parcels of perimeter 4. P 2 represents the number of parcels of perimeter 6. P 3 represents the number of parcels of perimeter 8. P 4 represents the number of parcels of perimeter Based on the pattern above, what will be the perimeter of the parcels counted in P n? 2. From your earlier work, fill in the blanks below: P 1 =, P 2 =, P 3 =, P 4 =,

5 3. Choose a parcel with perimeter 10 that protrudes on the left in the bottom row. If you remove one box on that left end of the bottom row, what is the resulting perimeter of the parcel? 4. Choose a parcel with perimeter 10 that protrudes on the right in the bottom row. If you remove one box on that right end of the bottom row, what is the resulting perimeter of the parcel? 5. Choose a parcel with perimeter 10 that doesn t protrudes on either the left end or the right end in the bottom row. If you remove the entire bottom row, what is the resulting perimeter of the parcel? 6. Can you divide your parcels with perimeter 10 into 3 disjoint categories ones that protrude on the left of the bottom row, ones that protrude on the right of the bottom row, and ones that don t protrude at all on the bottom row? Why or why not? 7. Use the ideas in Problems 3-6 of this section to justify the following recursive formula for P n for n>2: P n =3P n-1 - P n-2 Source for problems: