International Journal of Pure and Applied Mathematics Volume 113 No. 13 2017, 122 – 131 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu

A NEW APPROACH IN ASSIGNMENT PROBLEM

P. Sumathi1, V. Preethy2 1Department of Mathematics, Bharath University Chennai - 600 073, Tamilnadu, India psumathi16@rediffmail.com 2Department of Mathematics, Bharath University Chennai - 600 073, Tamilnadu, India [email protected]

Abstract This paper discusses the Kaprekar constant and Kaprekar numbers in number theory. A brief account of kaprekar procedure and its number of convergence of different digits of number is also being explained. An interesting innovation of connecting Kaprekar procedure and Kaprekar constant with assignment problem to find the optimum solution and the corresponding number of iterations to the respective numbers will be converged is also being portrayed. AMS Subject Classification: 11E76, 11E81, 90C08. Key Words and Phrases: Kaprekar Number, Kaprekar Constant, Kaprekar Procedure, Assignment problem.

1 Introduction

Number theory is an ancient subject but we still cannot answer the integers in Number theory. It has extensive applications in recent 30 years. There are different types of numbers which add beauty to number theory. One among such fascinating numbers is Kaprekar numbers whose brief account has been portrayed.

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This paper deals about the Kaprekar number and Kaprekar constant .and a new innovative method is proposed using Kaprekar procedure in assignment problem. In this paper Section 2 deals with the concept of Kaprekar Numbers with examples. Section 3 and Section 4 portrays the Algebraic expression of Kaprekar Numbers and the Procedure for splitting the Kaprekar Numbers in two cases. A brief account on Kaprekar Constant and Kaprekar Procedure is dealt in Section 5 and Section 6. In Section 7, an innovative approach of combining Kaprekar Procedure with Assignment problem is clearly discussed.

2 Kaprekar Numbers

A Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split up into two parts such that add up to the original number again. In other words, a Kaprekar number n is such that n2 can be split into two so that the two parts sum to n.

Example 1.

452 =2025 20+25=45 7032 =494209 494+209=703

3 Algebraic Expression of Kaprekar Number

Let n be a k-digit integer in base b. Then n is said to be a Kaprekar number in base b, if n2 has the property such that when we add the number formed by its right hand digits to that formed by its left hand digits, we get n. In algebraic form, an integer n such that in a given base b has

k 1 − 2 i n = dib i=0 X where, K - number of digits in square of the given number.

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di - digits with d0 the least significant digit and dk 1 the most − significant digit.

Example 2. Consider the number n = 297, n2 = 88209. Apply the formula, k 1 − 2 i n = dib i=0 X Here,

k =5,d0 =9,d1 =0,d2 =2,d3 =8,d4 =8,b = 10 Hence,

k 1 − 2 i 297 = dib i=0 X0 1 2 0 1 = d0b + d1b + d2b + d3b + d4b =9 100 +0 102 +2 102 +8 103 +8 104 × × × × × = 9+0+200+8000+80000 2972 = 88209

4 Procedure for Splitting of Square of a Kaprekar Number

Consider a Kaprekar number. Let it be denoted as n. Find the square of n. Let it be n2. Let the number of digits in n2 be k. Now, here arises two cases, Case (1): If k is odd, then

[k/2] k 1 − i i [k/2] 1 n = dib + dib − − i=0 X i=[Xk/2]+1 Case (ii): If k is even, then

k 2 1 k 1 − − i i k n = dib + dib − 2 i=0 i= k X X2

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5 Kaprekar Constant

A number that remain unchanged on applying the Kaprekar process on it is known as Kaprekar constant.

5.1 Kaprekar constant for various digit numbers The Kaprekar constant for Three digit numbers is 495 Four digit numbers is 6174 Six digit numbers is 549945, 631764 Eight digit number is 63317664, 97508421 Nine digit number is 554999445, 864197532 Ten digit number is 6333176664, 9753086421

6 Kaprekar Process

Procedure for three digit numbers to reach the Kaprekar constant Choose any three digit number consisting of three distinct digits. Rearrange the digits into descending order and then into ascending order including leading zeroes. Find the difference of these digits. It may be 495. If not, repeat the same procedure to the answer obtained from the differences of those digits. The end result will terminate to a number 495 which is the Kaprekar Constant for three digit numbers. The Procedure is similar for different digits of numbers.

7 Coupling Kaprekar Procedure with Assignment Problem

This concept provokes an interesting gateway of getting solution of number of iterations of four digit numbers to reach the Kaprekar constant in assignment problem procedure.

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Algorithm: Step 1: Consider any four digit numbers at least two of its number is distinct. Step 2: The number of iterations for those numbers to reach the Kaprekar constant 6174 should be calculated initially using simple method. Step 3: The numbers are arranged in 4 4 matrix by placing × any two of the numbers in ascending order and the remaining two numbers in descending order. Step 4: The actual numbers are taken along the column wise and the corresponding iterations are taken along row wise outside the matrix. Step 5: Now by applying the procedure of the assignment problem, the optimum solution is obtained and the corresponding number of iterations to the respective numbers will be converged.

Example 3. Consider the following numbers with their corresponding number of iterations 1467-1, 3087-2, 1432-3, 5238-4 Step 1:

Step 2: Row Minima

Step 3: Column Minima

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Number of lines = Order of matrix, Hence the optimality is not obtained. 6 Add 1 to the numbers in the lines of intersection and sub 1 to the numbers not on the line.

Step 4:

Number of lines = Order of matrix Hence the optimality is obtained. Step 5:

Finally the numbers are converged to their respective number of iterations. 1467 1 3087 → 2 → 1432 3 5238 → 4 →

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8 Conclusion

In this paper Kaprekar Constant and Kaprekar Numbers were discussed in detailed with some examples. An interesting new innovative method of connecting Kaprekar procedure and Kaprekar constant with assignment problem was proposed. Using assignment problem the optimum solution is obtained and the corresponding number of iterations to the respective numbers will be converged.

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[9] P. Sumathi, S. Paulraj, Identification of redundant constraints in large s cale linear programming problems with minimal computational effort, Applied Mathematical Sciences, 7, No. 80 (2013), 3963-3974.

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[15] P. Sumathi, A New Tactic for Finding Irrelevant Constraints in Linear Programming Problems, International Journal of Scientific and Engineering Research, 6, No. 3 (2015), 66-69.

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[17] K. Yuvarekha, V. Nandakumar, Dr. P. Sumathi, Characteri- zation of planar graph with edge series, International journal of Research in Science, Engineering and Techonology, 3, No. 12 (2015), 18101-18104.

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[18] K. Yuvarekha, V. Nandhakumar, Dr. P. Sumathi, Dual of planar graph with crossing number, International journal of Research in Science, Engineering and Techonology, 4, No. 9 (2015), 8299-8301.

[19] K. Karunagaran, P. Sumathi, V. Nandakumar, Theory and Applications of Butterfly Network in Graph Theory, International journal of Research in Science, Engineering and Techonology, 4, No. 6 (2015), 4067-4070.

[20] K. Karunagaran, P. Sumathi, V. Nandakumar, Theory and Applications of Benes Network in Graph Theory, International journal of Research in Science, Engineering and Techonology, 4, No. 5 (2015), 3350-3352.

[21] Kaialash M. Patil, Nikhil P. Shah, Kaprekar Numbers and its analo g euqations, 7, No, 6 (2016).

[22] P. Sumathi, A new approach to solve linear programming problems with intercept values, Journal of Information and Optimization Sciences, 37, No. 4 (2016), 95-510.

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