psilon Camp 2015 - Welcome to Mathematics and Statistics |...
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ϵ psilon Camp 2015 The Inclusion-Exclusion Principle Consider a large cube made from unit cubes. Suppose our cube is n×n×n. Look at the cube from a corner so that you can see three faces. How many unit cubes are in your vision? Build a table that shows how many cubes are visible from one corner. n n 3 number visible 1 1 1 2 8 7 3 27 19 4 64 37 How does the table continue? Make some guesses and then try to prove your answer. Let’s name the number we’re looking for. Let G(n) denote the number of cubes visible from a corner of the n × n × n cube. Notice that the sequence of differences G(2) − G(1) = 6; G(3) − G(2) = 19 − 7= 12; G(4) − G(3) = 37 − 19 = 18 has an interesting property. The differences are all multiples of 6. When we explore such a sequence in which the sequence of successive differences is eventually constant, we can build a polynomial that produces the sequence. Since the second order differences are constant, we propose that G(n) is a quadratic polynomial, G(n)= an 2 + bn + c. We can solve this without great difficulty to get G(n)=3n 2 − 3n + 1. But what do these coefficients have to do with the problem? One way to see this is to extend the chart by one more column that shows the cubes that are not visible. n n 3 number not visible number visible 1 1 0 1 2 8 1 7 3 27 8 19 4 64 27 37 Now you can see that we can count the number of visible cubes by counting the invisible ones first. So, in general, G(n)= n 3 − (n − 1) 3 = n 3 − (n 3 − 3n 2 +3n − 1) = 3n 2 − 3n + 1. This doesn’t completely answer the question however. What are the coefficients telling us? How does the algebra help us reason geometrically? The answer is below. But before we leave this rich area, we have one more question to pursue. If we paint the entire outside, how many of the n 3 unit cubes receive some paint? Can you write this as a polynomial in n in standard form. What do the coefficients tell you? 1
Transcript of psilon Camp 2015 - Welcome to Mathematics and Statistics |...
ϵpsilon Camp 2015 The Inclusion-Exclusion Principle
Consider a large cube made from unit cubes. Suppose our cube is n×n×n.Look at the cube from a corner so that you can see three faces. How manyunit cubes are in your vision? Build a table that shows how many cubes arevisible from one corner.
n n3 number visible1 1 12 8 73 27 194 64 37
How does the table continue? Make some guesses and then try to proveyour answer. Let’s name the number we’re looking for. Let G(n) denotethe number of cubes visible from a corner of the n × n × n cube. Noticethat the sequence of differences G(2) − G(1) = 6;G(3) − G(2) = 19 − 7 =12;G(4)−G(3) = 37− 19 = 18 has an interesting property. The differencesare all multiples of 6. When we explore such a sequence in which the sequenceof successive differences is eventually constant, we can build a polynomialthat produces the sequence. Since the second order differences are constant,we propose that G(n) is a quadratic polynomial, G(n) = an2 + bn + c. Wecan solve this without great difficulty to get G(n) = 3n2 − 3n+1. But whatdo these coefficients have to do with the problem? One way to see this isto extend the chart by one more column that shows the cubes that are notvisible.
n n3 number not visible number visible1 1 0 12 8 1 73 27 8 194 64 27 37
Now you can see that we can count the number of visible cubes bycounting the invisible ones first. So, in general, G(n) = n3 − (n − 1)3 =n3 − (n3 − 3n2 + 3n − 1) = 3n2 − 3n + 1. This doesn’t completely answerthe question however. What are the coefficients telling us? How does thealgebra help us reason geometrically? The answer is below. But before weleave this rich area, we have one more question to pursue. If we paint theentire outside, how many of the n3 unit cubes receive some paint? Can youwrite this as a polynomial in n in standard form. What do the coefficientstell you?
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ϵpsilon Camp 2015 The Inclusion-Exclusion Principle
These counting problems succumb to a combinatorial technique called thePrinciple of Inclusion-Exclusion, aka, PIE. The following exercises use PIE.
1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. LetA = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7},and C = {1, 5, 8}. Recall that × denotes Cartesian product and Xdenotes the complement of X with respect to U . Find each of thefollowing. Recall that A ⊕ B = A\B ∪ B\A denotes the symmetricdifference of A and B and that |X| denotes the number of elements ofthe finite set X.
(a) |(U × U)\(A× A)|(b) |A× A|(c) |(A×B) ∪ (B × A)|(d) |A ∪B ∪ C|(e) | (A× A) ∩ (B ×B) ∩ (C × C)|(f) | (A× A) ∪ (B ×B) ∪ (C × C)|(g) | (A×B) ∪ (B × C) ∪ (C × A)|(h) | (A× A) ∪ (A×B) ∪ (A× C)|(i) | (A×B)⊕ (B × A)|(j) | (A⊕B)× (B ⊕ A)|(k) | (A ∪B)⊕ C|
The problem below appeared at the KenKen website during the weekof October 7 to 11. As counting problems often do, this one provokedsome discussion about the solution.The purpose of this essay is to solvethis and some related problems.
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ϵpsilon Camp 2015 The Inclusion-Exclusion Principle
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•A
•B
•C
• •
• •
• •♡
The figure above is built with nine line segments, each with either Aor B as an endpoint. Triangles and quadrilaterals of various sizes areformed with parts of these segments as boundary.
(a) How many triangles are there?
(b) How many quadrilaterals are there?
(c) How many regions (triangles and quadrilaterals) have a ♡?
(d) Generalizing, suppose there are n − 2 points inserted on the seg-ment AC and m − 2 points inserted on the segment BC. Howmany triangular regions are there.
2. Among 18 students in a room, 7 study mathematics, 10 study science,and 10 study computer programming. Also, 3 study mathematics andscience, 4 study mathematics and computer programming, and 5 studyscience and computer programming. We know that 1 student studiesall three subjects. How many of these students study none of the threesubjects?
3. Let A,B, and C be sets with the following properties:
• |A| = 100, |B| = 50, and |C| = 48
• The number of elements that belong to exactly one of the threesets is twice the number that belong to exactly two of the sets.
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ϵpsilon Camp 2015 The Inclusion-Exclusion Principle
• The number of elements that belong to exactly one of the threesets is three times the number that belong to all of the sets.
How many elements belong to all three sets?
4. Three setsA,B, and C have the following properties: N(A) = 63, N(B) =91, N(C) = 44, N(A
∩B) = 25, N(A
∩C) = 13, N(C
∩B) = 11.
Also, N(A∪
B∪C) = 153. What is N(A
∩B∩
C)?
5. Two distinct circles and a triangle are given in a plane. What is thelargest number of points that can belong to at least two of the threefigures? By a circle we mean just the circumference, by a triangle wemean just the union of the edges.
6. How many equilateral triangles in the same plane as the lattice shownhave at least two vertices in the hexagonal lattice?
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7. How many integers in the set {1, 2, 3, 4, . . . , 360} have at least one primedivisor in common with 360?
8. Let U = {1, 2, 3, . . . , 1000} and let A2, A3, and A5 denote the subsetsof U defined as follows:A2 = {n | 1 ≤ n ≤ 1000 and n is even },A3 = {n | 1 ≤ n ≤ 1000 and n is a multiple of 3},A5 = {n | 1 ≤ n ≤ 1000 and n is a multiple of 5},All complements are taken with respect to U . Find the number of ele-ments of each of the sets listed below. a. A2
∩A3
∩A5; b. A2
∩A3
∩A5;
c. A2
∩A3
∩A5; d. A2
∩A3
∩A5; e. A2
∩A3
∩A5; f. A2
∩A3
∩A5;
g. A2
∩A3
∩A5; and h. A2
∩A3
∩A5.
9. In a survey of the chewing gum tastes of a group of baseball players, itwas found that: 22 liked juicy fruit25 liked spearmint
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ϵpsilon Camp 2015 The Inclusion-Exclusion Principle
39 like bubble gum9 like both spearmint and juicy fruit17 liked juicy fruit and bubble gum20 liked spearmint and bubble gum6 liked all three4 liked none of theseHow many baseball players were surveyed?
10. Mr. Brown raises chickens. Each can be described as thin or fat, brownor red, hen or rooster. Four are thin brown hens, 17 are hens, 14 arethin chickens, 4 are thin hens, 11 are thin brown chickens, 5 are brownhens, 3 are fat red roosters, 17 are thin or brown chickens. How manychickens does Mr. Brown have?
11. Consider the following information regarding three sets A, B, and C allof which are subsets of a set U . If N(S) denotes the number of membersof S, suppose that N(A) = 14, N(B) = 10, N(A ∪ B ∪ C) = 24 andN(A ∩B) = 6.Consider the following assertions:
1. C has at most 24 members
2. C has at least 6 members
3. A ∪B has exactly 18 members
Which ones are true?
12. Cam Newton and Steve Smith attend a press conference at the 2012Pro Bowl with 2 other pairs of NFL teammates. Since the playersspend so much time with their teammates during the season, playerson the same team agree not to sit next to each other. If the six playerssit in a row occupying six consecutive seats, how many ways can theplayers sit down for the press conference?
13. How many squares in the plane have two or more vertices in the set(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)?
14. How many circles in the plane contain at least three of the points(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)?
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