• Home
  • Textbooks
  • Introductory Combinatorics
  • What is Combinatorics?

Introductory Combinatorics

Richard A. Brualdi

Chapter 1

What is Combinatorics? - all with Video Answers

Educators

Chapter Questions

14:41

Problem 1

Show that an $m$ -by-n chessboard has a perfect cover by dominoes if and only if at least one of $m$ and $n$ is even.

Chris Trentman
Chris Trentman
Numerade Educator
02:31

Problem 2

Consider an $m$ -by-n chessboard with $m$ and $n$ both odd. To fix the notation, suppose that the square in the upper left-hand corner is colored white. Show that if a white square is cut out anywhere on the board, the resulting pruned board has a perfect cover by dominoes.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 3

Imagine a prison consisting of 64 cells arranged like the squares of an 8 -by-8 chessboard. There are doors between all adjoining cells. A prisoner in one of the corner cells is told that he will be released, provided he can get into the diagonally opposite corner cell after passing through every other cell exactly once. Can the prisoner obtain his freedom?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
07:54

Problem 4

(a) Let $f(n)$ count the number of different perfect covers of a 2 -by- $n$ chessboard by dominoes. Evaluate $f(1), f(2), f(3), f(4)$, and $f(5) .$ Try to find (and verify) a simple relation that the counting function $f$ satisfies. Use this relation to compute $f(12)$.
(b) * Let $g(n)$ be the number of different perfect covers of a 3 -by-n chessboard by dominoes. Evaluate $g(1), g(2), \ldots, g(6)$.

Bryan Lynn
Bryan Lynn
Numerade Educator
02:30

Problem 5

Find the number of different perfect covers of a 3 -by- 4 chessboard by dominoes.

Foster Wisusik
Foster Wisusik
Numerade Educator
01:33

Problem 6

Consider the following three-dimensional version of the chessboard problem: $\mathrm{A}$ three-dimensional domino is defined to be the geometric figure that results when two cubes, one unit on an edge, are joined along a face. Show that it is possible to construct a cube $n$ units on an edge from dominoes if and only if $n$ is even. If $n$ is odd, is it possible to construct a cube $n$ units on an edge with a 1 -by- 1 hole in the middle? (Hint: Think of a cube $n$ units on an edge as being composed of $n^{3}$ cubes, one unit on an edge. Color the cubes alternately black and white.)

James Chok
James Chok
Numerade Educator
01:15

Problem 7

Let $a$ and $b$ be positive integers with $a$ a factor of $b .$ Show that an $m$ -by- $n$ board has a perfect cover by $a$ -by-b pieces if and only if $a$ is a factor of both $m$ and $n$ and $b$ is a factor of either $m$ or $n .$ (Hint: Partition the $a$ -by-b pieces into a 1-by-b pieces.)

James Chok
James Chok
Numerade Educator
01:44

Problem 8

Use Exercise 7 to conclude that when $a$ is a factor of $b$, an $m$ -by-n board has a perfect cover by $a$ -by-b pieces if and only if it has a trivial perfect cover in which all the pieces are oriented the same way.

Lucía Guerrero
Lucía Guerrero
Numerade Educator
01:14

Problem 9

Show that the conclusion of Exercise 8 need not hold when $a$ is not a factor of $b$

Edward Downes
Edward Downes
Numerade Educator
03:27

Problem 10

Verify that there is no magic square of order 2 .

James Chok
James Chok
Numerade Educator
00:49

Problem 11

Use de la Loubère's method to construct a magic square of order $7 .$

AG
Ankit Gupta
Numerade Educator
01:00

Problem 12

Use de la Loubère's method to construct a magic square of order $9 .$

Katelyn Vandeaver
Katelyn Vandeaver
Numerade Educator
00:59

Problem 13

Construct a magic square of order $6 .$

Ali Soave
Ali Soave
Numerade Educator
01:22

Problem 14

Show that a magic square of order 3 must have a 5 in the middle position. Deduce that there are exactly 8 magic squares of order $3 .$

Liuxi Sun
Liuxi Sun
Numerade Educator
00:35

Problem 15

Can the following partial square be completed to obtain a magic square of order $4 ?$
$$
\left[\begin{array}{lll}
2 & 3 & \\
4 & &
\end{array}\right]
$$

Julie Silva
Julie Silva
Numerade Educator
01:32

Problem 16

Show that the result of replacing every integer $a$ in a magic square of order $n$ with $n^{2}+1-a$ is a magic square of order $n$.

Julian Wong
Julian Wong
Numerade Educator
03:21

Problem 17

Let $n$ be a positive integer divisible by 4 , say $n=4 m$. Consider the following construction of an $n$ -by-n array:
(1) Proceeding from left to right and from first row to nth row, fill in the places of the array with the integers $1,2, \ldots, n^{2}$ in order.
(2) Partition the resulting square array into $m^{2} 4$ -by-4 smaller arrays. Replace each number $a$ on the two diagonals of each of the 4 -by-4 arrays with its "complement" $n^{2}+1-a$.
Verify that this construction produces a magic square of order $n$ when $n=4$ and $n=8$. (Actually it produces a magic square for each $n$ divisible by 4.)

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:06

Problem 18

Show that there is no magic cube of order $2 .$

Carson Merrill
Carson Merrill
Numerade Educator
00:50

Problem 19

Show that there is no magic cube of order 4 .

Nick Johnson
Nick Johnson
Numerade Educator
01:15

Problem 20

Show that the following map of 10 countries $\{1,2, \ldots, 10\}$ can be colored with three but no fewer colors. If the colors used are red, white, and blue, determine the number of different colorings.

Nick Johnson
Nick Johnson
Numerade Educator
01:23

Problem 21

(a) Does there exist a magic hexagon of order 2 ? That is, is it possible to arrange the numbers $1,2, \ldots, 7$ in the following hexagonal array so that all of the nine "line" sums (the sum of the numbers in the hexagonal boxes penetrated by a line through midpoints of opposite sides) are the same?
(b) " Construct a magic hexagon of order 3 ; that is, arrange the integers $1,2, \ldots, 19$ in a hexagonal array (three integers on a side) in such a way that all of the fifteen "line" sums are the same (namely, 38 ).

Selena Armstrong
Selena Armstrong
Numerade Educator
01:34

Problem 22

Construct a pair of orthogonal Latin squares of order $4 .$

Linh Vu
Linh Vu
Numerade Educator
01:08

Problem 23

Construct Latin squares of orders 5 and $6 .$

Melissa Munoz
Melissa Munoz
Numerade Educator
00:36

Problem 24

Find a general method for constructing a Latin square of order $n$.

Amy Jiang
Amy Jiang
Numerade Educator
01:33

Problem 25

A 6-by-6 chessboard is perfectly covered with 18 dominoes. Prove that it is possible to cut it either horizontally or vertically into two nonempty pieces without cutting through a domino; that is, prove that there must be a fault line.

James Chok
James Chok
Numerade Educator
00:30

Problem 26

Construct a perfect cover of an 8 -by-8 chessboard with dominoes having no fault-line.

Prathan Jarupoonphol
Prathan Jarupoonphol
Numerade Educator
01:01

Problem 27

Determine all shortest routes from $A$ to $B$ in the system of intersections and streets (graph) in the following diagram. The numbers on the streets represent the lengths of the streets measured in terms of some unit.

WZ
Wen Zheng
Numerade Educator
08:18

Problem 28

Consider 3 -heap Nim with heaps of sizes 1,2, and 4 . Show that this game is unbalanced and determine a first move for player 1 .

Chris Trentman
Chris Trentman
Numerade Educator
08:18

Problem 29

Is 4-heap Nim with heaps of sizes $22,19,14$, and 11 balanced or unbalanced? Player I's first move is to remove 6 coins from the heap of size 19. What should player II's first move be?

Chris Trentman
Chris Trentman
Numerade Educator
08:18

Problem 30

Consider 5 -heap Nim with heaps of sizes $10,20,30,40$, and $50 .$ Is this game balanced? Determine a first move for player $1 .$

Chris Trentman
Chris Trentman
Numerade Educator
08:18

Problem 31

Show that player I can always win a Nim game in which the number of heaps with an odd number of coins is odd.

Chris Trentman
Chris Trentman
Numerade Educator
08:18

Problem 32

Show that in an unbalanced game of Nim in which the largest unbalanced bit is the $j$ th bit, player I can always balance the game by removing coins from any heap the base 2 numeral of whose number has a 1 in the $j$ th bit.

Chris Trentman
Chris Trentman
Numerade Educator
08:18

Problem 33

Suppose we change the object of Nim so that the player who takes the last coin loses (the misère version). Show that the following is a winning strategy: Play as in ordinary Nim until all but exactly one heap contains a single coin. Then remove either all or all but one of the coins of the exceptional heap so as to leave an odd number of heaps of size 1 .

Chris Trentman
Chris Trentman
Numerade Educator
04:57

Problem 34

A game is played between two players, alternating turns as follows: The game starts with an empty pile. When it is his turn, a player may add either 1,2, 3. or 4 coins to the pile. The person who adds the 100 th coin to the pile is the winner. Determine whether it is the first or second player who can guarantee a win in this game. What is the winning strategy?

Steven Clarke
Steven Clarke
Numerade Educator
01:17

Problem 35

Suppose that in Exercise 34 , the player who adds the 100 th coin loses. Now who wins, and how?

Hoan Nguyen
Hoan Nguyen
Numerade Educator
01:47

Problem 36

Eight people are at a party and pair off to form four teams of two. In how many ways can this be done? (This is sort of an "unstructured" domino-covering problem.)

Vysakh M
Vysakh M
Numerade Educator
01:37

Problem 37

A Latin square of order $n$ is idempotent provided the integers $\{1,2, \ldots, n\}$ occur in the diagonal positions $(1,1),(2,2), \ldots,(n, n)$ in the order $1,2, \ldots, n$, and is symmetric provided the integer in position $(i, j)$ equals the integer in position $(j, i)$ whenever $i \neq j .$ There is no symmetric, idempotent Latin square of order 2. Construct a symmetric, idempotent Latin square of order 3. Show that there is no symmetric, idempotent Latin square of order $4 .$ What about order $n$ in general, where $n$ is even?

Srilakshmi E K
Srilakshmi E K
Numerade Educator
02:19

Problem 38

Take any set of $2 n$ points in the plane with no three collinear, and then arbitrarily color each point red or blue. Prove that it is always possible to pair up the red points with the blue points by drawing line segments connecting them so that no two of the line segments intersect.

Charlotte Ihme
Charlotte Ihme
Numerade Educator
03:06

Problem 39

Consider an $n$ -by- $n$ board and $L$ -tetrominoes ( 4 squares joined in the shape of an L). Show that if there is a perfect cover of the $n$ -by- $n$ board with $L$ -tetrominoes, then $n$ is divisible by 4 . What about $m$ -by- $n$ -boards?

Aman Gupta
Aman Gupta
Numerade Educator
00:45

Problem 40

Solve the following Sudoku puzzle,
$$
\begin{array}{||l|l|l||l|l|l|l|l|l|}
& & & 5 & & & & & 6 \\
\hline & & 8 & & & & & & 7 \\
\hline 7 & 5 & & & 6 & 4 & & & \\
\hline & 3 & 6 & & 8 & & 2 & 4 & 5 \\
\hline & 2 & & 3 & & 9 & & 6 & \\
\hline 5 & 1 & 7 & & 2 & & 8 & 3 & \\
\hline & & & 2 & 4 & & & 7 & 8 \\
\hline 4 & & & & & & 3 & & \\
\hline 1 & & & & & 3 & & &
\end{array}
$$

Linda Hand
Linda Hand
Numerade Educator
02:07

Problem 41

Solve the following Sudoku puzzle,
$$
\begin{array}{|c|c|c||c|c|c||c|c|c||}
& & & & & & & & \\
\hline 7 & & & 1 & 5 & 4 & & & 8 \\
\hline 2 & & 5 & 9 & & 8 & 1 & & 6 \\
\hline & & 6 & 7 & & 3 & 4 & & \\
\hline & 3 & & & & & & 2 & \\
\hline & & 7 & 2 & & 9 & 6 & & \\
\hline & & 7 & 2 & & 9 & 6 & & \\
\hline 8 & & 3 & 4 & & 2 & 9 & & 5 \\
\hline 5 & & & 8 & 7 & 6 & & & 2
\end{array}
$$

Sneha Ravi
Sneha Ravi
Numerade Educator
01:33

Problem 42

Let $S_{n}$ denote the staircase board with $1+2+\cdots+n=n(n+1) / 2$ squares. For example, $S_{4}$ is
$$
\begin{array}{|l|l|l|l|}
\hline & \times & \times & \times \\
\hline & & \times & \times \\
\hline & & & \times \\
\hline & & & \\
\hline
\end{array}
$$
Prove that $S_{n}$ does not have a perfect cover with dominoes for any $n \geq 1$.

James Chok
James Chok
Numerade Educator
00:42

Problem 43

Consider a block of wood in the shape of a cube, 3 feet on an edge. It is desired to cut the cube into 27 smaller cubes, 1 foot on an edge. One way to do this is to make 6 cuts, 2 in each direction, while keeping the cube in one block. Is it possible to use fewer cuts if the pieces can be rearranged between cuts?

Maranda Parks
Maranda Parks
Numerade Educator
03:56

Problem 44

Show how to cut a cube, 3 feet on an edge, into 27 cubes, 1 foot on an edge, using exactly 6 cuts but making a nontrivial rearrangement of the pieces between two of the cuts.

Charles Carter
Charles Carter
Numerade Educator