By Alexander Soifer

ISBN-10: 0387754695

ISBN-13: 9780387754697

The etudes awarded listed below are now not easily these of Czerny, yet are greater in comparison to the etudes of Chopin, not just technically challenging and addressed to a number of particular talents, yet even as owning an outstanding good looks that characterizes the simplest of art... hold this booklet to hand as you intend your subsequent challenge fixing seminar.

—Don Chakerian


Alexander Soifer’s Geometrical Etudes in Combinatorial Mathematics is worried with attractive arithmetic, and it'll most likely occupy a different and everlasting position within the mathematical literature, hard and encouraging either amateur and specialist readers with incredible and delightful difficulties and theorems… He conveys the enjoyment of discovery in addition to a person, and he has selected an issue that may stand the try out of time.

—Cecil Rousseau


Each time I checked out Geometrical Etudes in Combinatorial Mathematics i discovered anything that used to be new and incredible to me, even after greater than fifty years operating in combinatorial geometry. the recent version has been accelerated (and up-to-date the place needed), through numerous new pleasant chapters. The cautious and sluggish advent of subject matters and effects is both inviting for rookies and for jaded experts. i'm hoping that the charm of the e-book will allure many younger mathematicians to the visually appealing difficulties that retain you guessing how the questions may be replied within the end.

—Branko Grünbaum


All of Alexander Soifer’s books may be seen as first-class and crafty entrees to arithmetic within the MAPS mode... diversified humans may have varied personal tastes between them, yet this is anything that Geometric Etudes does greater than the others: after bringing the reader right into a subject through posing attention-grabbing difficulties, ranging from a totally common point, it then is going deep. The intensity accomplished is so much astounding in bankruptcy four, on Combinatorial Geometry, that can be used as half or all of a graduate direction at the topic, however it is additionally lovely notable in bankruptcy three, on graph thought, and in bankruptcy 2, the place the countless pigeon gap precept (infinitely many pigeons, finitely many holes) is used to turn out theorems in a major subset of the set of basic theorems of analysis.

—Peter D. Johnson, Jr.


This attention-grabbing and pleasant booklet … is written either for mature mathematicians attracted to a little bit unconventional geometric difficulties and particularly for gifted younger scholars who're attracted to engaged on unsolved difficulties that are simply understood by means of rookies and whose recommendations might be won't require loads of wisdom yet could require loads of ingenuity ... i like to recommend this publication very warmly.

—Paul Erdos

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Extra resources for Geometric Etudes in Combinatorial Mathematics

Example text

4. 3), then the difference between the number of black squares and the number of white squares is equal to either 0 or ˙m. Fig. 5. Prove that if an a b rectangle is tiled with Ltetrominoes, then the number of tiles is even. This exercise shows that the area of a rectangle tiled by Ltetrominoes must be divisible by 8. 4). The following problem arises naturally: What other rectangles can be tiled by tetrominoes of this type? Can you solve this problem on your own? If not, try to solve the following exercises to pave the way for a successful train of thought.

We cut the given rectangle into two rectangles that are kq n and r1 n. Since the kq n rectangle can be tiled using linear k-ominoes, it contains an equal number of squares of each color. The given rectangle contains an equal number of squares of every color as well; therefore, the r1 n rectangle has the same property. Case 2: m D kq r1 . In this case we attach the r1 n rectangle to the given rectangle to obtain a kq n rectangle and extend the coloring of the given rectangle to a diagonal cyclic k-coloring of the kq n board.

1. Let a; b be integers such that 2 Ä a Ä 3 and a Ä b. The a b rectangle can be tiled by L-trominoes if and only if ab is divisible by 6. Equivalently: the a b rectangle with 2 Ä a Ä 3 and a Ä b can be tiled by L-trominoes if and only if one of the numbers a; b is divisible by 2 and the other is divisible by 3. Now a new direction of research is clear: we will consider a b rectangles in cases when 4 Ä a Ä b. But first we would like to share a couple of “philosophical” observations. Suppose we are trying to solve a combinatorial problem.

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Geometric Etudes in Combinatorial Mathematics by Alexander Soifer

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