What do you think?
Rate this book
Problems for Mathematicians, Young and Old
Book by Paul R. Halmos
- GenresMathematics
318 pages, Paperback
First published December 1, 1991
2 people are currently reading
112 people want to read
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 2 of 2 reviews
December 8, 2017
Paul Richard Halmos [1916-2006] was a Hungarian-Jewish-born American mathematician who made fundamental contributions to mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great expositor of mathematics and an advocate for education and effective teaching.
In this 318-page gem of a book, Halmos has collected a wide array of problems (126 pp.), complete with hints (18 pp.) and solutions (174 pp.). The best way to review this book is to present a sample problem from each of its 14 chapters. I will exclude chapter 13, “Mappings,” because it contains no problem that can be solved without resorting to advance math; the 13 problems that follow can be solved by anyone trained in basic math and mathematical reasoning.
Chapter 1, “Combinatorics” (Tennis tournament): Consider an elimination tennis tournament with 1025 participants. In each round, the remaining players are paired, and if the number of players is odd, one of them gets a bye. For example, in the first round, 512 matches are played and one player gets a bye. How many matches must be played to determine a champion?
Chapter 2, “Calculus” (Railroad track): We have a perfectly straight and flat railroad track of length 2640 m. Suppose we add 1 m of rail to the middle of the track, while keeping its endpoints fixed. The additional length causes the track to bulge upward in the shape of a circular arc. How far will the middle of the track rise above the original level?
Chapter 3, “Puzzles” (Factorials ending in 0s): The first factorial that ends in 0 is 5! = 120. To have the factorial end in two 0s, we need to go to 10! = 3,628,800. How many 0s does 1,000,000! have at its end?
Chapter 4, “Numbers” (Irrational punch): We have a weird paper punch such that when its center is placed at a point on paper, all points whose distance to that center are irrational numbers are removed. How many punchings are needed to remove every point?
Chapter 5, “Geometry” (Shortest road to connect 4 houses): Four houses are located on flat land at the corners of a square of side length 1 km. What is the shortest road system that would enable each inhabitant to visit the other three?
Chapter 6, “Tilings” (Tiling a checkerboard with dominoes): You have a standard 8-by-8 checkerboard from which 2 squares at opposite corners have been removed. Given 31 dominoes, each of which covers exactly 2 adjacent squares on the board, can you cover all 62 squares of the board with the 31 dominoes?
Chapter 7, “Probability” (Fair and loaded dice): A fair or honest die is one for which the probability of each of the outcomes 1 through 6 is 1/6. Assume we can load a die as we please, for example, to have the probabilities 3/8, 1/4, 1/8. 1/9, 1/12, 1/18 for outcomes 1 through 6 (these probabilities add up to 1). Is there a way to load two different dice so that when they are rolled together the sum outcomes 2 through 12 each occurs with the same probability 1/11?
Chapter 8, “Analysis” (Infinite mathematical expressions): What is the value of the following expression in the limit when there are an infinite number of sqrt(2) terms?
sqrt(2) ^ (sqrt(2) ^ (sqrt(2) ^ (sqrt(2) ^ … )))
Chapter 9, “Matrices” (Square-root of a matrix): The square-root of a matrix A is a matrix B such that A = B^2. Not every matrix has a square-root. Does a 3-by-3 matrix with all entries 0, except the top right element which is 1, have a square-root? How about a matrix of all-0s, with its only 1 entry in the middle of its top row?
Chapter 10, “Algebra” (Polynomial arithmetic): If a polynomial in x vanishes when x = 2, then it must be divisible by x – 2. Is it also true that if a polynomial in x and y vanishes when x = y, then it is divisible by x – y?
Chapter 11, “Sets” (Lines on a plane): Obviously, an infinite set of lines are needed to cover a plane. Can the set of such lines whose union is the plane be countably infinite?
Chapter 12, “Spaces” (Map coloring): It has been known for sure since 1976 (and suspected long before then) that any planar map of countries can be colored using no more than 4 colors, so that no two countries sharing a border are of the same color. Show that if the countries do not have arbitrary borders but, instead, are formed by a set of intersecting circles, then 2 colors would suffice.
Chapter 14, “Measures” (Fair sharing of a cake among 3 people): When two people want to divide a cake so that they are both convinced of the division’s fairness, the “you cut, I choose” scheme is used. One person cuts the cake into two pieces that s/he thinks is a fair division and the other person picks one of the pieces. Can we extend this scheme to 3 people? This is a much harder problem than fair division into 2 pieces. There exists a scheme that appears fair, but does not work upon further reflection.
In this 318-page gem of a book, Halmos has collected a wide array of problems (126 pp.), complete with hints (18 pp.) and solutions (174 pp.). The best way to review this book is to present a sample problem from each of its 14 chapters. I will exclude chapter 13, “Mappings,” because it contains no problem that can be solved without resorting to advance math; the 13 problems that follow can be solved by anyone trained in basic math and mathematical reasoning.
Chapter 1, “Combinatorics” (Tennis tournament): Consider an elimination tennis tournament with 1025 participants. In each round, the remaining players are paired, and if the number of players is odd, one of them gets a bye. For example, in the first round, 512 matches are played and one player gets a bye. How many matches must be played to determine a champion?
Chapter 2, “Calculus” (Railroad track): We have a perfectly straight and flat railroad track of length 2640 m. Suppose we add 1 m of rail to the middle of the track, while keeping its endpoints fixed. The additional length causes the track to bulge upward in the shape of a circular arc. How far will the middle of the track rise above the original level?
Chapter 3, “Puzzles” (Factorials ending in 0s): The first factorial that ends in 0 is 5! = 120. To have the factorial end in two 0s, we need to go to 10! = 3,628,800. How many 0s does 1,000,000! have at its end?
Chapter 4, “Numbers” (Irrational punch): We have a weird paper punch such that when its center is placed at a point on paper, all points whose distance to that center are irrational numbers are removed. How many punchings are needed to remove every point?
Chapter 5, “Geometry” (Shortest road to connect 4 houses): Four houses are located on flat land at the corners of a square of side length 1 km. What is the shortest road system that would enable each inhabitant to visit the other three?
Chapter 6, “Tilings” (Tiling a checkerboard with dominoes): You have a standard 8-by-8 checkerboard from which 2 squares at opposite corners have been removed. Given 31 dominoes, each of which covers exactly 2 adjacent squares on the board, can you cover all 62 squares of the board with the 31 dominoes?
Chapter 7, “Probability” (Fair and loaded dice): A fair or honest die is one for which the probability of each of the outcomes 1 through 6 is 1/6. Assume we can load a die as we please, for example, to have the probabilities 3/8, 1/4, 1/8. 1/9, 1/12, 1/18 for outcomes 1 through 6 (these probabilities add up to 1). Is there a way to load two different dice so that when they are rolled together the sum outcomes 2 through 12 each occurs with the same probability 1/11?
Chapter 8, “Analysis” (Infinite mathematical expressions): What is the value of the following expression in the limit when there are an infinite number of sqrt(2) terms?
sqrt(2) ^ (sqrt(2) ^ (sqrt(2) ^ (sqrt(2) ^ … )))
Chapter 9, “Matrices” (Square-root of a matrix): The square-root of a matrix A is a matrix B such that A = B^2. Not every matrix has a square-root. Does a 3-by-3 matrix with all entries 0, except the top right element which is 1, have a square-root? How about a matrix of all-0s, with its only 1 entry in the middle of its top row?
Chapter 10, “Algebra” (Polynomial arithmetic): If a polynomial in x vanishes when x = 2, then it must be divisible by x – 2. Is it also true that if a polynomial in x and y vanishes when x = y, then it is divisible by x – y?
Chapter 11, “Sets” (Lines on a plane): Obviously, an infinite set of lines are needed to cover a plane. Can the set of such lines whose union is the plane be countably infinite?
Chapter 12, “Spaces” (Map coloring): It has been known for sure since 1976 (and suspected long before then) that any planar map of countries can be colored using no more than 4 colors, so that no two countries sharing a border are of the same color. Show that if the countries do not have arbitrary borders but, instead, are formed by a set of intersecting circles, then 2 colors would suffice.
Chapter 14, “Measures” (Fair sharing of a cake among 3 people): When two people want to divide a cake so that they are both convinced of the division’s fairness, the “you cut, I choose” scheme is used. One person cuts the cake into two pieces that s/he thinks is a fair division and the other person picks one of the pieces. Can we extend this scheme to 3 people? This is a much harder problem than fair division into 2 pieces. There exists a scheme that appears fair, but does not work upon further reflection.
August 13, 2015
Cet ouvrage montre bien la puissance d'Halmos en tant qu'écrivain. Avec des exemples bien choisis dans les plus divers domaines des mathématiques, le pupil d'Erdos fait un excellent exposé pour les mathématiciens de tous les âges. Le livre est composé de 14 chapitres, chacun traitant d'un sujet à travers des problèmes, en même temps que des brefs indications et des très claires solutions. Egalement interessantes sont les motivations avant chaque problème du livre.
Mais fait attention! Le livre est vraimenet pour tous les âges. Parfois il faut sortir sont livre d'Algèbre ou Topologie pour se rappeler de quelques définitions ou théorèmes.
1. Combinatoire
Des problèmes classiques, plusieurs problèmes d'entretien (au moins en finance), theorème de ramsey sont dest stars du chapitre. Belle demonstration du theorème de Erdos Szekeres.
2. Suites et fonctions
Avec très peu de résultats en Analyse réel l'auteur arrive a des resultats intéressants, comme la suite de cos iterés ou la méthode de Newton Raphson. Le problème 2F est trés intéressant.
3. Jeux Mathématiques
Malgré le fait qu'il n'y ait pas de jeux, le chapitre reste intéresant. Ils s'agit des problèmes classiques.
4. Nombres
Problèmes relativement faciles de la Théorie des nombres ( application des Restes Chinois, Petit Theorème de Fermat, etc.)
5. Géometrie
Problèmes intéressants, même pour les non-aimant de la matière ( comme moi.)
6. Pavages
Techniques de coloriage et des jeux (enfin ceux là). Il mentionne la technique du "ou bien, ou bien", bien utile: une sorte de preuve par contradiction, où on sépare en cas.
7. Probabilités
Ceux qui aiment vont se régaler. Les problèmes ont des résultats les plus inattendus de tout le livre (e.g. 7E).
8. Analyse
9. Matrices
Excellente révision de Algèbre linéaire (théorème spectral, etc)
10. Algèbre
Sors bien ton livre, surtout si tu es rouillé comme moi.
11. Ensembles
12. Espaces
Des problèmes intérssants, maintenant pour les "petits".
13. Applications
Actions, morphismes
14. Mesures
A ne pas manquer, comme le chapitre 7.
Displaying 1 - 2 of 2 reviews