Both a challenge to mathematically inclined readers and a useful supplementary text for high school and college courses, One Hundred Problems in Elementary Mathematics presents an instructive, stimulating collection of problems. Many problems address such matters as numbers, equations, inequalities, points, polygons, circles, ellipses, space, polyhedra, and spheres. An equal number deal with more amusing or more practical subjects, such as a picnic ham, blood groups, rooks on a chessboard, and the doings of the ingenious Dr. Abracadabrus. Are the problems in this book really elementary? Perhaps not in the lay reader’s sense, for anyone who desires to solve these problems must know a fair amount of mathematics, up to calculus. Nevertheless, Professor Steinhaus has given complete, detailed solutions to every one of his 100 problems, and anyone who works through the solutions will painlessly learn an astonishing amount of mathematics. A final chapter provides a true test for the most proficient readers: 13 additional unsolved problems, including some for which the author himself does not know the solutions.
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Yes, you can access One Hundred Problems in Elementary Mathematics by Hugo Steinhaus in PDF and/or ePUB format, as well as other popular books in Mathematik & Spiele in der Mathematik. We have over one million books available in our catalogue for you to explore.
We construct a sequence of numbers as follows : The first number is 2, the next is 3,
2 . 3 = 6,
the third number of the sequence is 6,
3 . 6 = 18,
the fourth number is 1, and the fifth is 8,
6 . 1 = 6,1 . 8 = 8,
the sixth number is 6, then follows 8, etc.
This is the sequence which we obtain:
The little arcs under the numbers denote the multiplication carried out, the result of which follows the last digit of the sequence. For example, we ought to multiply now 6 by 8 and write down the numbers of the result, namely 4, 8. There will never be a shortage of numbers for multiplication, since the number of arcs is increased by one with each multiplication and the result will yield at least one and often two digits, so that there always appears at least one new digit.
Prove that numbers 5, 7 and 9 never appear in this sequence.
2. An interesting property of numbers
Let us first write an arbitrary natural number (for example, 2583), and then add the squares of its digits (22+52+82+32 = 102). Next, we do the same with the number obtained (12+02+22 = 5), and proceed in the same way (52 = 25, 22+52 = 29, 22+92 = 85, ...).
Prove that unless this procedure leads to the number 1 (in which case the number 1 will of course recur indefinitely, it must lead to the number 145, and the following cycle will occur again and again:
145, 42, 20, 4, 16, 37, 58, 89.
3. Division by 11
Prove that the number
55k+1+45k+2+35k
is divisible by 11 for every natural k.
4. The divisibility of numbers
The number
3105+4105
is divisible by 13, 49, 181 and 379, and is not divisible either by 5 or by 11.
How can this result be confirmed?
5. A simplified form of Fermat’s theorem
If x, y, z and n are natural numbers, and n ≧ z, then the relation xn+yn = zn does not hold.
6. Distribution of numbers
Find ten numbers x1, x2, ..., x10 such that
(i) the number x1 is contained in the closed interval <0, 1>,
(ii) the numbers x1 and x2 lie in different halves of the interval <0, 1>,
(iii) the numbers x1, x2 and x3 lie each in different thirds of the interval <0, 1>,
(iv) the numbers x1, x2, x3 and x4 lie each in a different quarter of the interval, etc. and, finally
(v) the numbers x1, x2, ..., x10 lie each in a different tenth of the closed interval <0, 1>.
7. Generalization
Is the above problem solvable if instead of 10 numbers and 10 conditions there are n numbers and n conditions (where n is an arbitrary natural number)?
8. Proportions
Consider numbers A, B, C, p, q, r whose mutual dependence is expressed as follows:
A : B = p,B : C = q,C : A = r.
Write the proportion
A : B : C =
:
:
is such a way that in the empt...
Table of contents
Cover
Title Page
Copyright Page
Contents
Foreword
Preface
Chapter I: Problems on Numbers, Equations and Inequalities
Chapter II: Problems on Points, Polygons, Circles and Ellipses
Chapter III: Problems on Space, Polyhedra and Spheres
Chapter IV Practical and Non-Practical Problems
Chapter V Problems on Chess, Volleyball and Pursuit
Chapter VI Mathematical Adventures of Dr. Abracadabrus
