[...]... PROBLEMS The problems in this volume are related by the fact that in nearly all of them we are required to answer a question of "how many?" or "in how many ways?" Such problems are called combinatorial, as they are exercises in calculating the number of different combinations of various objects The branch of mathematics which deals with such problems is called combinatorial analysis In the solutions. .. greatest possible number of knights is used Some other combinatorial problems connected with arrangements of chess pieces can be found in L Y Okunev's booklet, Combinatorial Problems on the Chessboard (ONTJ, Moscow and Leningrad, 1935) IV GEOMETRIC PROBLEMS INVOLVING COMBINATORIAL ANALYSIS Some of the problems in this group are concerned with convex sets A set in the plane or in three-dimensional space... relations involving the numbers Bnk VI PROBLEMS ON COMPUTING PROBABILITIES A very important class of combinatorial problems is concerned with the computation of probabilities This section is devoted to some of these problems, and the following general remarks are intended to provide the background necessary for their solution In science and engineering we often deal with experiments (or observations or... + 1, 2 + 2 + I + I The partitions in which no part is divisible by 3 are 1 Some of Euler's results are contained in problems 53b, 145, 164 PROBLEMS 10 5 + 1,4 + 2,4+ I + 1,2 + 2 + 2, 2 + 2 + I + 1,2+ I + 1 + I + 1, 1 + 1 + 1 + 1 + 1 + l III COMBINATORIAL PROBLEMS ON THE CHESSBOARD The problems of this section involve various configurations of chess pieces on a chessboard We will consider not only the... positive integral solutions of the equation x + y + z = n satisfy the inequalities x ~ y + z,y ~ x + z, z ~ x + y? Here solutions differing only in the order of the terms are to be considered as different 30." How many incongruent triangles are there with perimeter n if the lengths of the sides are integers? II The representation of integers as sums and products 9 3la.· How many different solutions in positive... Infinitorum Euler proves his theorems by the use of an interesting general method (the "method of generating functions"); these proofs are different from the more elementary ones presented in this book as solutions to problems 32 and 33 In problems 32 and 33 representations of a number n as a sum which differ only in the order of the terms are considered to be the same Such representations are called... positive integers does the equation Xl + X z + Xa + + Xm = n have? b How many solutions in nonnegative integers does the equation Xl + Xi + Xa + + xm = n have? Remark Problem 27 is a special case of problem 31a (that corresponding to m = 3) To conclude this set of problems we will present four general theorems dealing with the representation of numbers as sums of positive integers The first three... do not intersect within the circle? Problem 54 will reoccur later in another connection (see problem 84a) At that point some related problems (84b and 84c) will be given; for more general results, see the remark at the end of the solution of problem 84c 55a A circle is divided into p equal sectors, where p is a prime number In how many different ways can these p sectors be colored with n given colors... the square on which the king lies is marked with a circle and the squares controlled by the king are marked with crosses.) A knight controls those squares which can be reached by moving one square horizontally or vertically and one square diagonally away from the square occupied by the knight (See fig 2b; the square occupied by the o b Fig 2 2 In accordance with what has been said, we count the square... of money in the bank? Here k is an integer in the range 0 ;£ k ;£ m + n - 1 VII EXPERIMENTS WITH INFINITELY MANY POSSIBLE OUTCOMES In the preceding section we dealt with experiments having a finite number of equally likely possible outcomes In that case the probability • It is not hard to see that a polygon with an odd number of sides can never be decomposed into quadrilaterals as required in the problem . alt="" CHALLENGING MATHEMATICAL PROBLEMS WITH ELEMENTARY SOLurIONS Volume I Combinatorial Analysis and Probability Theory