In the process of casting out nines, both sides of the latter equation are computed, and if they are not equal the original addition must have been faulty. Repunits and repdigits[ edit ] Main article: Repunit Repunits are integers that are represented with only the digit 1. For example, one thousand, one hundred and eleven is a repunit.
Mathematical recreations comprise puzzles and games that vary from naive amusements to sophisticated problems, some of which have never been solved. They may involve arithmetic, algebra, geometry, theory of numbers, graph theory, topology, matrices, group theorycombinatorics dealing with problems of arrangements or designsset theorysymbolic logicor probability theory.
Any attempt to classify this colourful assortment of material is at best arbitrary. Included in this article are the history and the main types of number games and mathematical recreations and the principles on which they are based.
Details, including descriptions of puzzles, games, and recreations mentioned in the article, will be found in the references listed in the bibliography.
At times it becomes difficult to tell where pastime ends and serious mathematics begins. An innocent puzzle requiring the traverse of a path may lead to technicalities of graph theory; a simple problem of counting parts of a geometric figure may involve combinatorial theory; dissecting a polygon may involve transformation geometry and group theory; logical inference problems may involve matrices.
A problem regarded in medieval times—or before electronic computers became commonplace—as very difficult may prove to be quite simple when attacked by the mathematical methods of today.
Mathematical recreations have a universal appeal. The urge to solve a puzzle is manifested alike by young and old, by the unsophisticated as well as the sophisticated. An outstanding English mathematician, G. Hardyobserved that professional puzzle makers, aware of this propensityexploit it diligently, knowing full well that the general public gets an intellectual kick out of such activities.
The relevant literature has become extensive, particularly since the beginning of the 20th century. Some of it is repetitious, but surprisingly enough, successive generations have found the older chestnuts to be quite delightful, whether dressed in new clothes or not.
Much newly created material is continually being added. A few survived from the ancient Greeks and Romans: Such activities were most prominent on the Continent, particularly in Italy and Germany.
Kinds of problems The problems in general were of two kinds: The first required little or no mathematical skill, merely general intelligence and ingenuity, as for example, so-called decanting and difficult crossings problems.
A typical example of the former is how to measure out one quart of a liquid if only an eight- a five- and a three-quart measure are available.
Difficult crossings problems are exemplified by the dilemma of three couples trying to cross a stream in a boat that will hold only two persons, with each husband too jealous to leave his wife in the company of either of the other men. Many variants of both types of problems have appeared over the years.
Some examples Problems involving computation also took on a variety of forms; some were as follows: Finding a number Think of a number, triple it, and take half the product; triple this and take half the result; then divide by 9. The quotient will be one-fourth the original number. The chessboard problem How many grains of wheat are required in order to place one grain on the first square, 2 on the second, 4 on the third, and so on for the 64 squares?
The lion in the well This is typical of many problems dealing with the time required to cover a certain distance at a constant rate while at the same time progress is hindered by a constant retrograde motion.
There is a lion in a well whose depth is 50 palms. In how many days will he get out of the well? Courier problems These are typified by the movements of bodies at given rates in which some position of these bodies is given and the time required for them to arrive at some other specified position is demanded.
Pioneers and imitators The 17th century produced books devoted solely to recreational problems not only in mathematics but frequently in mechanics and natural philosophy as well.
The latter passed through five editions, the last as late as ; it was the forerunner of similar collections of recreations to follow. The emphasis was placed on arithmetic rather than geometric puzzles. Among the outstanding problems given by Bachet were questions involving number bases other than 10; card tricks; watch-dial puzzles depending on numbering schemes; the determination of the smallest set of weights that would enable one to weigh any integral number of pounds from one pound to 40, inclusive; and difficult crossings or ferry problems.The Editor's Blog is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by .
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In mathematics and computing, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A–F (or alternatively a–f) to represent values ten to fifteen.. Hexadecimal numerals are widely used by computer system designers and programmers, as they provide a.
Self-describing numbers the digit in each position is equal to the number of times that that digit appears in the number. For example, is a four-digit self describing number: # write the first 4 self-describing numbers every write (self_describing_numbers () \ 4) end.