ON SOME ENTERTAINING APPLICATIONS OF THE CONCEPT OF SET IN COMPUTER SCIENCE COURSE

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1 Yordzhev K., Kostadnova H. Інформаційні технології в освіті ON SOME ENTERTAINING APPLICATIONS OF THE CONCEPT OF SET IN COMPUTER SCIENCE COURSE Yordzhev K., Kostadnova H. Some aspects of programmng educaton are examned n ths work. It s emphassed, based on the entertanment value, the most approprate examples are chosen to demonstrate the dfferent language constructons data structures. Such an example s the demonstrated algorthm for solvng the wdespread nowadays Sudoku puzzle. Ths s made, because of the connecton wth the term set puttng t nto practce n the programmng. Usng the so bult program there are solved some combnatoral problems, connected to the Sudoku matrces. Keywords: Educaton n programmng, programmng languages, data structures, set, Sudoku matrx, Sudoku puzzle. INTRODUCTION The present work s contemplated to help the lecturer n programmng n hs aspraton for gvng an approprate, nterestng entertanng example of the advantage to use the term set n programmng. For ths purpose the students have to be famlar wth the basc defntons of the set theory to be skllful at the basc operatons wth sets. It follows the well-known fact that to be a good programmer t s necessary (but not effcent) to be a good mathematcan. A classcal example for the use of sets n programmng has become the programmng realzaton of the problem for fndng prme numbers usng the method Seve of Eratosthenes [,5,8]. How to construct a faster algorthm, solvng the problem for recevng all n n bnary matrces, whch contan exactly k ones n each row each column, wth the help of the set theory the operatons over sets, s shown n [10]. We wll examne the entertanng actual problem, whch s very nterestng for the students algorthms for solvng Sudoku. It s wdespread puzzle nowadays, whch presents n the entertanng pages n most of the newspapers magaznes n entertanng web stes. Sudoku, or Su Doku, s a Japanese word (or phrase) meanng somethng lke Number Place. On the other sde, Sudoku matrces fnd an nterestng combnatoral applcaton, for example n the desgn theory [7]. The connecton between the set of all m m permutaton matrces (.e. bnary matrces whch contan ust one 1 n every row every column) the set of all m m Sudoku matrces, s shown n [3]. As far as the authors of ths study know there s not a unversal formula for the number of Sudoku matrces n wth every natural number n. We consder t as an open problem n mathematcs. When n 3 n [4] s shown that there are exactly = = 3 7 = 9! = 0 8 = n number Sudoku matrces

2 On Some Entertanng Applcatons Of The Concept Of Set In Computer Scence Course PROBLEM FORMULATION AND ALGORITHM DESCRIPTION m n. Let Let n s a postve nteger let S s s a square m m matrx, whose elements n ths table are ntegers, belongng to the closed nterval 1,m. The matrx S s dvded to n, n n square submatrces, whch are not ntersected wll be called blocks, wth the help of n 1 horzontal n 1 vertcal lnes (the matrx S, when n 3, s shown on fg. 1). defnton f Let denote by s A, then s 11 s 1 s 13 s 14 s 15 s 16 s 17 s 18 s 19 s 1 s s 3 s 4 s 5 s 6 s 7 s 8 s 9 s 31 s 3 s 33 s 34 s 35 s 36 s 37 s 38 s 39 s 41 s 4 s 43 s 44 s 45 s 46 s 47 s 48 s 49 s 51 s 5 s 53 s 54 s 55 s 56 s 57 s 58 s 59 s 61 s 6 s 63 s 64 s 65 s 66 s 67 s 68 s 69 s 71 s 7 s 73 s 74 s 75 s 76 s 77 s 78 s 79 s 81 s 8 s 83 s 84 s 85 s 86 s 87 s 88 s 89 s 91 s 9 s 93 s 94 s 95 s 96 s 97 s 98 s 99 Fg. 1 A, 1 k, l n the blocks n the above descrbed matrx S. Then by ( k 1) n kn ( l 1) ( l 1). Let s belong to the block A let are known. Then t s easy to guess that k l can be calculated wth the help of the formulas 1 k 1 1 l 1, x the functon: whole part of the real number x. where, as usual, we denote by We say that s set 1,,, m n Z m S, 1, m n s a Sudoku matrx f there s ust one number of the n every row, every column every block. The puzzle named Sudoku s wdespread nowadays. It s gven a Sudoku matrx, n whch some of the elements are erased. The mssng elements wll be equal to 0, f we need ths. The task n the puzzle s to restore the mssng elements of the Sudoku matrx. It s supposed that the authors of the concrete puzzle have chosen the mssng elements, so the problem has only one soluton. Ths condton we wll mss wll not reckon wth t. In ths work we wll buld our programmng product to show the number of every possble soluton. If the task has no soluton ths number has to be zero. The most popular puzzles Sudoku are when n 3,.e. m 9. We are gong to descrbe an algorthm for creatng a computer program, whch fnds all solutons (f there are some) of rom Sudoku. For ths am we wll use the knowledge of the set theory. We examne the sets R, C B, where 1 n s, m, k, l n 1. For every 1,,,m, the set R conssts of all mssng numbers n the -th row of the matrx. Analogously 5

3 Yordzhev K., Kostadnova H. we defne the sets C, 1,,, m correspondngly for the mssng numbers n the -th column B, k, l 1,,, n correspondngly for the mssng numbers n the blocks A of S. When the algorthm starts workng t traverses many tmes all of the elements s S, such that s 0,.e. these are the elements, whch real values we have to fnd. Let s 0 let s Ak l. We assume P R C B. Then the followng three cases are possble: ) P (empty set). The task has no soluton n ths case; ) P d, d Z m 1,,, m,.e. P : the number of the elements of P s equal to 1 ( P s a set contanng one element). Then the only one possblty for s s s d,.e. we have found the unknown value of s n ths case. After ths we remove the common element d from the sets R, an element); C B, then we contnue to the next zero element of the matrx S (f there s such ) P. Then we can not say anythng about the unknown value of s we move on the next mssng (zero) element of the matrx S. We traverse all zero elements of the matrx S untl one the followng events occur: е1) For some, {1,,, m} s true s 0, but P R C B ; е) All elements n S become postve; е3) All zero elements of S are traversed, but t does not occur nether event е1, nor event е. In other words for all the remanng zero elements n S, the above descrbed case s always true. In case that t occurs one of the events е1 or е, then the procedure stops ts work vsualzes the obtaned result. In case that event e3 occurs, then the algorthm has to contnue workng by usng other methods, for example t has to apply the tral error method. In the concrete case ths method conssts of the followng thngs: 1 1 We choose rom s S, such that s 0 let k 1, l 1. Let P R C Bk l { d1, d,, dt}. Then for every d r P, r 1,,, t we assume s d r. Such an assumng we call a rom tral. We count the number of all rom trals, untl the soluton s found, n the programmng realzaton of the algorthm. After ths we solve the problem for fndng the unknown elements of the Sudoku matrx, whch contan one element less than the prevous matrx. It s comfortable to use a recurson here. The base of the recurson,.e. we go out of the procedure f there occurs event e1 or e. It s absolutely sure that t wll happen (.e. there wll not be an nfnte cycle ), because when we do the rom trals we reduce the number of the zero elements by 1. The above descrbed algorthm s realzed n the Pascal programmng language, because of the two man reasons: 1) Pascal s chosen, of the most educatonal nsttutons n the system of the secondary educaton, as a frst programmng language. Ths means that the basc am of ths work s realzed to help the teacher n hs preparaton n the process of fndng nterestng applcaton of the teachng materal. That s the way the presented materal becomes more nterestng for the pupls. ) There are bult n nstruments used to work wth sets n the Pascal programmng language. The above descrbed deas can be realzed n other rom programmng language, for example C++. But n ths case we have to look for addtonal nstruments to work wth sets for 6

4 On Some Entertanng Applcatons Of The Concept Of Set In Computer Scence Course example the assocatve contaners set multset realzed n Stard Template Lbrary (STL) [1,5]. It can be used the template class set of the system of computer algebra Symbolc C++, programmng code s gven n detals n [9]. Of course can be bult another class set, specfc methods of ths class can be descrbed, as a tranng. Ths s a good exercse, havng n mnd the fact that the cardnal number of the basc ( unversal ) set s not very bg. For example the stard puzzle Sudoku has basc set the set of the ntegers from 1 to 9 plus the empty set. SOME COMBINATORIAL APPLICATION Obvously, f for solvng gven Sudoku, there s no need to use the tral error method,.e. the problem can be solved by usng set theoretcal operatons (0 rom trals), then ths Sudoku has a unque soluton. Such a Sudoku s shown on fgure. The opposte proposton s not true, as t s n the next example, shown on fgure 3, where there s a unque soluton program has made 18 rom trals only to receve ths soluton, as the number of the rom trals (after the fnal soluton s receved, the program contnues workng f t s necessary) s Fg. Fg. 3 Sometmes the Sudoku authors do not examne f the soluton s unque. For example the Sudoku, shown on fgure 4 (s publshed n a newspaper s rubrc Easy Sudoku, the name of the newspaper we wll not menton) has 4 dfferent solutons, the program does 16 rom trals untl the fnal soluton s obtaned, the common number of the rom trals s 18. The Sudoku puzzle shown on fgure 5 has no soluton, although on a frst sght there s not contradcton n the condton. The program has to make 1 rom trals to receve the result that ths puzzle has no soluton Fg. 4 Fg. 5 7

5 Yordzhev K., Kostadnova H. There are even more combnatoral results wth dfferent Sudoku matrces, receved by the help of the above descrbed algorthm some experments. There are Sudoku matrces of the knd shown on fgure 6; There are 6 80 Sudoku matrces of the knd shown on fgure 7; There are 680 Sudoku matrces of the knd shown on fgure 8; There are 1 78 Sudoku matrces of the knd shown on fgure 9; There are Sudoku matrces of the knd shown on fgure 10; There are 8 Sudoku matrces of the knd shown on fgure 11; 1 3 * * * * * * * * * * * * * * * * * * * * * 1 3 * * * * * * * * * * * * * * * * * * * * * 1 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 3 1 * * * * * * * * * * * * * * * * * * Fg. 6 Fg * * * * * * * * * 5 6 * * * * * * * 1 8 * * * * * * * * * * * * * * * * Fg. 8 Fg * * * * * * * * * * * * * * * * * * * * * 9 1 * * * * * * 1 * * 7 * * 4 * * * * * 8 * * 5 * * * 3 * * 9 * * 6 7 * * 4 * * 1 * * * 8 * * 5 * * * * * 9 * * 6 * * 3 4 * * 1 * * 7 * * * 5 * * * * 8 * * * 6 * * 3 * * 9 Fg. 10 Fg. 11 8

6 On Some Entertanng Applcatons Of The Concept Of Set In Computer Scence Course REFERENCES 1. Collns W. J., Data structures the stard template lbrary. McGraw-Hll, 00.. Dahl D. J., Dkstra E. W., Hoare C. A. R., Structured Programmng. Academc Press Inc., Dahl G., Permutaton matrces related to Sudoku. Lnear Algebra ts Applcatons, 009, 430, Felgenhauer B., Jarvs F. Enumeratng Possble Sudoku Grds. Preprnt avalable at 5. Jensen, K., Wrth N., Pascal User Manual Report. 3 rd ed., Sprnger-Verlag, Lschener R., STL Pocket Reference. O Relly Meda, Mo Hu-Dong, Xu Ru-Gen, Sudoku Square a New Desgn n Feld Experment. Acta Agronomca Snca, 008, 34(9), Prce D., UCSD Pascal A Consderate Approach. Prentce-Hall, Sh T. K., Steeb W.-H., Hardy Y., Symbolc C++: An Introducton to Computer Algebra usng Obect-Orented Programmng, Sprnger, Yordzhev, K., Stefanov S., On Some Applcatons of the Consept of Set n Computer Scence Course. Mathematcs Educatons n Mathematcs, 003, v.3,