solving sudoku using monte carlo marching

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Mathematical Solution to Sudoku Using Monte Carlo

Marching

Jacob H Lashover

September 17, 2015

Baton Rouge, Louisiana

Abstract: As noted in an earlier paper titled Monte Carlo Marching, the author

had previously solved Sudoku puzzles using Monte Carlo Marching (MCM).

(Lashover, 2012) In this paper, details of that MCM solution are reported. The

MCM algorithm is shown to provide exact, integer solutions to 6 X 6 Sudoku

puzzles. A math model is constructed: a convex objective function is developed

from linear constraints, non-linear constraints, and non-smooth constraints. The

linearized objective function is then minimized to zero using MCM which

determines the values of the unknown variables (as many as 19). This MCM

method is unique in that no other analytical solution has been found in the

literature.

The MCM analytical method has now been shown to solve chemical

engineering problems such as calculation of dew points and calculation of fractions

of vapor flashed in chemical streams. These calculations required solution of non-

linear, simultaneous equations. Rufus Isaacs' non-linear model for optimizing

horse racing wagers was also solved in "A Chemical Engineer Goes to the Horse

Races". (Lashover J. H., 2014) These algorithms demonstrate that whether the

equations are linear, non-linear, or non-smooth, MCM, which does not require

derivatives, is a very powerful solution method.

Key Words: Algorithm, analytical solution, constraints, mathematical method,

optimization, mathematical modeling, Sudoku, QB64 computer language,

convergence, non-linear equations, convex objective function, chemical

engineering, and horse racing wagers.

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Table of Contents

Page No. Content

1 Abstract and Key Words

2 - 4 Table of Contents

5 Introduction

6 Discussion

A. Development of Math Model

7 Figure 1. Puzzle No. 1

(Used as demonstration of method.)

Summation equations for rows

Summation equations for columns

8 Summation equations for grids

Product equations for rows

Product equations for columns

9 Product equations for grids

Constraint #3-Equations

Constraint #4-Equation

10 B. Development of Constraint Equations

>Convert row values to strings and concatenate

>Define contents of rows

11 >Count duplicates in rows and convert to convex form

12 Figure 2. Computer output for all rows with details of

Row #1

C. Solution of Model

13 D. Computer Code for Iterative Procedure

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14 >Convex objective function for minimization to zero

16 End of main body of report

17 Appendix A. Development of Constraint Equations

(Complete Section B.)

>Convert grid values to strings and concatenate

>Define grid contents

18 >Count duplicates in grids

>Put into convex form

19 >Convert column values to strings and concatenate

>Define column contents

20 >Count duplicates in columns and put in convex form

>Constraint equations for sums and products of units

>Equations for rows 1 to 6

21 >Constraint equations for columns and grids

>Equations for columns 1 to 6

>Equations for grids 1 to 6

22 >Count total digits of each kind in rows, columns

and grids. Convert to convex form.

23 End of Appendix A.

24 Appendix B. Puzzles No. 2 and No. 3

>Figure 3. Puzzle No. 2

>Figure 4. Puzzle No. 3

25 Appendix C. Puzzle No. 1 Computer Solution

25 32 >Complete Puzzle No. 1 Solution

33 Figure 5. Puzzle No. 1 Objective function values, S,

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33 continued vs. Number of Iterations, J and I


Vs. Number of Iterations, J and I


Vs. Number of Iterations, J and I

36 63 Appendix D. QB64 Computer Code

64 Bibliography

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Introduction:

Sudoku is played on a square matrix, in this case, 6 x 6 or 36 total squares. The

same single integer cannot appear twice in the same row, column or in any of the

six 2 x 3 sub-matrices (called grids in this work). Clues which consist of at least 8

permanent numbers located throughout the total matrix are provided. For a puzzle

to be a true Sudoku puzzle, it can only have one solution. The puzzles are

traditionally solved using logic, and the difficulty of a puzzle is determined by the

depth of logic required.

Sudokus origins lie in a Chinese game called Magic Squares which was first

documented 2000 years ago. All of the rows, columns, and diagonals had to add

to the same number. Numbers could only be used once. In 1514, Albrecht Durer,

a German artist, produced a 4 X 4 game called Melancholia where the sum of all

rows, columns, and diagonals had to total 34. Again, numbers could only be used

once. In 1783, the Swiss mathematician, Leonhard Euler, wrote about a game he

called Latin Squares where he removed the diagonal requirements and made the

game a puzzle of permutations. There are 6! = 720 different ways that 6

numbers can be arranged. Between 1890 and 1920, a game called Magic

Squares appeared in a French newspaper. The game first appeared in 1978 in the

U.S., and in 1984 in Japan where it was given the name Sudoku which is short for a longer expression which means the digits are limited to one occurrence.

There are many variations of Sudoku including matrices of 4 X 4, 6 X 6, 9 X 9,

etc. and sometimes adding additional constraint(s) by including the diagonal(s) of

the matrix (X-Sudoku). This paper presents an analytical solution for 6 X 6

puzzles with six 2 X 3 grids. Three example puzzles with 12, 16, and 13

permanent clues, respectively, are solved. See page 4 for puzzle No. 1. See

Appendix B for puzzles No. 2 and No. 3. While fewer clues generally make a

puzzle more difficult to solve, the ultimate difficulty is presently agreed upon by

most experts to be the number and complexity of the solving techniques required

to find a solution. These techniques include, hidden singles, hidden pairs, hidden

triples, hidden quads, naked singles, naked pairs, naked triples, naked quads, X-

Wing, Swordfish, XY-Wing, etc.

A naked single occurs when only one number can fit the constraints of one

number per row, column, and grid for a square. That naked single is an obvious

solution. A hidden single occurs when only one of many possible solution numbers appears in the list of possible solutions for a row, column, or grid.

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Naked and Hidden Singles are initially searched. Finding these solutions

permits rechecking for newly uncovered singles after each find. This is called

propagation of solutions which may significantly reduce the number of

unknowns and hence computing time. This procedure found 8, 4, and 4 additional

clues for Puzzles 1, 2, and 3, respectively. This left 16, 16, and 19 unidentified

squares in Puzzles 1, 2, and 3, respectively.

The computer used to solve these puzzles was a Dell Inspiron 546 Desktop

with 6 GB of RAM, and an AMD Phenom 9650 Quad-Core Processor operating

at 2.30 GHz and using QB64. Times in seconds required for solution of the three

puzzles were 30, 94, and 108, respectively.

Discussion

A. Development of Math Model

Puzzle No. 1 shown below in Figure 1 was randomly selected from those found in

an internet search. It has 12 clues or known squares out of 36 total squares. These

clues are highlighted in yellow. Solution by MCM makes use of constraints which

insure that the solution will only have one of each of the six possible digits in each

row, column, and grid; and no duplicates. The sum and product of the six digits in

each row, column, and grid must be 21 and 720, respectively. These two

constraints form simultaneous linear and non-linear equations which model the

system being studied. These equations have associative solutions as the sums and

products can equal 21 and 720 regardless of their positions in the square. Third

and fourth constraints limit the total number of each digit to six in each set of rows,

columns, and grids; and insure that there are no duplicates. The MCM objective

function is formed by designing the constraints to be convex and adding them. The

justification for the addition of these constraints is similar to that used in

Maximum Likelihood methods. (Banbura, 2010) The objective function

becomes convex by being built of convex functions and can thus be minimized.

The feasible solutions are bounded from 1 to 6. Further proof of the linearity

(ability to add constraints) of the simultaneous marching schema used in this

work can be found in Analysis of the n-dimensional quadtree decomposition for

arbitrary hyper-rectangles by Cristos Faloutsos et.al.at

http://drum.lib.umd.edu/bitstream/1903/678/2/CS-TR-3381.pdf.

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The model is described by assigning X(k) sequentially to the squares from X(1) to

X(36). The squares are also described in two dimensions by A(Row, Col) where

A(1, 1) = X(1) on down to A(6, 6) = X(36). Having these two nomenclatures

facilitates referring to the grid variables. Short subroutines can be called to convert

k to (Row, Col) and (Row, Col) to k. Grids 1 to 6 are 2 rows and 3 columns

numbered from left to right and top to bottom.

FIGURE 1.

1 2 5 4 6 3 1

2 6 1 3 5 2 4

3 4 2 6 1 5 3

4 1 3 5 2 4 6

5 5 4 1 3 6 2

6 3 6 2 4 1 5

1 2 3 4 5 6

Summation equations for rows:

Row #1. X(1) + X(2) + X(3) + X(4) + X(5) + X(6) = 21 or i=1 to 6 Xi = 21

Row #2. i=7 to 12 Xi = 21

Row #3 i=13 to 18 Xi = 21

Row #4. i=19 to 24 Xi = 21

Row #5. i=25 to 30 Xi = 21

Row #6. i=31 to 36 Xi = 21

Summation equations for columns:

Column #1. X(1) + X(7) + X(13) + X(19) + X(25) + X(31) = 21

Column #2. X(2) + X(8) + X(14) + X(20) + X(26) + X(32) = 21

Column #3. X(3) + X(9) + X(15) + X(21) + X(27) + X(33) = 21

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Column #4. X(4) + X(10) + X(16) + X(22) + X(28) + X(34) = 21

Column #5. X(5) + X(11) + X(17) + X(23) + X(29) + X(35) = 21

Column #6. X(6) + X(12) + X(18) + X(24) + X(30) + X(36) = 21

Summation equations for grids:

Grid #1. X(1) + X(2) + X(3) + X(7) + X(8) + X(9) = 21

Grid #2. X(4) + X(5) + X(6) + X(10) + X(11) + X(12) = 21

Grid #3. X(13) + X(14) + X(15) + X(19) + X(20) + X(21) = 21

Grid #4. X(16) + X(17) + X(18) + X(22) + X(23) + X(24) = 21

Grid #5. X(25) + X(26) + X(27) + X(31) + X(32) + X(33) = 21

Grid #6. X(28) + X(29) + X(30) + X(34) + X(35) + X(36) = 21

Product equations for rows:

Row #1. X(1) x X(2) x X(3) x X(4) x X(5) x X(6) = 720 or i = 1 to 6 Xi =720

Row #2. i = 7 to 12 Xi =720

Row #3. i = 13 to 18 Xi =720

Row #4. i = 19 to 24 Xi =720

Row #5. i = 25 to 30 Xi =720

Row #6. i = 31 to 36 Xi =720

Product equations for columns:

Column #1. X(1) x X(7) x X(13) x X(19) x X(25) x X(31) = 720






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Product equations for grids:

Grid #1. X(1) x X(2) x X(3) x X(7) x X(8) x X(9) = 720






Constraint #3: Total of each digit equal to 6 in each grid, row, and column where TGD(Grids,

Digit), TRD(Rows, Digits), TCD(Columns, Digits) represent the total count of each digit in

grids, rows, and columns, respectively. Details of how these values are obtained follow in

Section A. TOTONES% = total number of ones, TOTTWOS% = total number of twos etc.

TOTONES% = 0: TOTTWOS% = 0: TOTTHREES% = 0: TOTFOURS% = 0:

TOTFIVES% = 0: TOTSIXES% = 0

FOR II = 1 TO 6: JJ = 1: TOTONES% = TOTONES% + TGD(II, JJ) + TRD(II, JJ) +

TCD(II, JJ): NEXT II: T1 = ABS(TOTONES% - 18)

FOR II = 1 TO 6: JJ = 2: TOTTWOS% = TOTTWOS% + TGD(II, JJ) + TRD(II, JJ) +

TCD(II, JJ): NEXT II: T2 = ABS(TOTTWOS% - 18)

FOR II = 1 TO 6: JJ = 3: TOTTHREES% = TOTTHREES% + TGD(II, JJ) + TRD(II, JJ) +

TCD(II, JJ): NEXT II: T3 = ABS(TOTTHREES% - 18)

FOR II = 1 TO 6: JJ = 4: TOTFOURS% = TOTFOURS% + TGD(II, JJ) + TRD(II, JJ) +

TCD(II, JJ): NEXT II: T4 = ABS(TOTFOURS% - 18)

FOR II = 1 TO 6: JJ = 5: TOTFIVES% = TOTFIVES% + TGD(II, JJ) + TRD(II, JJ) +

TCD(II, JJ): NEXT II: T5 = ABS(TOTFIVES% - 18)

FOR II = 1 TO 6: JJ = 6: TOTSIXES% = TOTSIXES% + TGD(II, JJ) + TRD(II, JJ) +

TCD(II, JJ): NEXT II: T6 = ABS(TOTSIXES% - 18)

Constraint #4. Total duplicates in the grids, TOTSUMGD, rows, TOTSUMRD, and columns,

TOTSUMCD, are added to determine the total duplicates. Details are in Section B below, and

Appendix A.

Total Duplicates = TOTSUMGD + TOTSUMRD + TOTSUMCD

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B. Development of Constraint Equations

Details of how the third and fourth constraints are developed are presented here. When the

proposed solution values for the squares are converted to strings using the QB64 STR$

function they can be concatenated into a new string containing all of the proposed values in a

row, column or grid unit. The INSTR function can then be used to determine the quantity and

location of each of the six possible digits in the unit. This code is, no doubt, similar to that used

by Microsoft EXCEL in its COUNTIF function. The code below demonstrates this operation:

REM CONVERT ROW VALUES TO STRINGS AND CONCATENATE

FOR ROW% = 1 TO 6: XX$(ROW%) = "": NEXT ROW%

FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6

GOSUB 40 'CONVERT ROW% AND COL% TO K

XL$ = STR$(X(K)): XL$ = LTRIM$(XL$)

XX$(ROW%) = XX$(ROW%) + XL$

NEXT COL%

PRINT " ROW%= "; ROW%; " ROW STRING= "; XX$(ROW%)

NEXT ROW%

REM DEFINE CONTENTS OF ROWS

FOR ROW% = 1 TO 6: FOR II = 1 TO 6: CTR(ROW%, II) = 0: NEXT II: NEXT ROW%

FOR ROW% = 1 TO 6

FOR II = 1 TO 6

DIG$ = STR$(II): DIG$ = LTRIM$(DIG$) LTRIM$ insures no blank spaces to the left or

beginning of the string

POSITION% = INSTR(XX$(ROW%), DIG$)

DO WHILE POSITION% 0

PRINT " ROW= "; ROW%; " DIGIT= "; II; " POSITION= "; POSITION%

CTR(ROW%, II) = CTR(ROW%, II) + 1

POSITION% = INSTR(POSITION% + 1, XX$(ROW%), DIG$)

LOOP

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IF CTR(ROW%, II) >= 1 THEN

PRINT " COUNT= "; CTR(ROW%, II); " FOR DIGIT= "; II

ELSE

PRINT " ROW= "; ROW%; " CONTAINS NO "; II

END IF

NEXT II

NEXT ROW%

REM COUNT DUPLICATES IN ROWS

FOR II = 1 TO 6: SUMRD(II) = 0: NEXT II

FOR II = 1 TO 6

FOR DIGITS = 1 TO 6

SUMRD(II) = SUMRD(II) + ABS(TRD(II, JJ) - 1) Convert to convex form

NEXT JJ

NEXT II

TOTSUMRD = 0

FOR II = 1 TO 6

TOTSUMRD = TOTSUMRD + SUMRD(II)

NEXT II

PRINT: PRINT " TOTSUMRD= "; TOTSUMRD

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Figure 2. Computer output for all rows with details for Row #1

The count of the number of digits, 1 to 6, plus the number of duplicates can be

determined from the above results. The code for columns and grids is similar and

can be found in Appendix A.

C. Solution of the model

Lower (LL) and Upper (UL) limits or bounds are set for the puzzle variables at 1

and 6. The differences between these limits are called the ranges of the variables

for each square and can be different for each square. A Crosshatch subroutine

(eliminates digits which are already in use) is used to check for the set of feasible

values for each unknown square. In the inner loop, I, values of X(k) for each

iteration are chosen randomly from these sets of feasible variables for each square.

The convex constraint equations are evaluated for the set of X(k)s selected and

inserted into the objective function. If the values chosen provide the minimum

value of the objective function so far, they are retained as the best solution to date.

If not, another similar iteration is performed.

A predetermined number of iterations is executed before the second loop, J, is

entered. This loop raises the lower bounds, LL, of the hyper-rectangles for each

X(k) toward the upper bounds, UL, based on the best solution of each so far. The

range, R, of each hyper-rectangle is individually reduced to center its limits around

the best solution. A focus factor, H, is used in the calculations to reduce the

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rectangle widths by a power. When the power is two (2) the range is bisected each

time. The equation describing the rectangle area is f(x) = R/(HJ) where R is the

range of solutions and H=1.3 with J = 1, 2, 3,.n, the number of the outer loop

iteration. In the case of bisection, where H =2, the first iteration when J = 1 splits

the range in half. In the second iteration where J = 2, the remaining range is

reduced by 2^2 or one fourth. The limit of this function as J approaches infinity is

0 where the entire range of solutions has been examined.

The three puzzles chosen for this report are Easy, Medium Hard and Hard.

The focus factors for each puzzle were chosen from experience to be 1.3. The

number of iterations in the inner loop, I, was increased as the difficulty of the

puzzle increased. Values of 5000, 30,000, and 40,000 iterations of I were used,

respectively, as the puzzle difficulty increased. Completion times for the three

puzzles were 30, 94, and 108 seconds, respectively, with no pauses. Removing the

comment code () on the Input Press ENTER to continue, Yes will permit the user to watch the algorithm move the objective function values to zero.

D. Computer code for iterative procedure

MM = 1E+30: H = 1.3

RANDOMIZE (-1111)

COUNTI& = 0: COUNT& = 0

TS# = TIMER

FOR J = 1 TO 20

FOR I& = 1 TO ZZZ

FOR K = 1 TO 36

COUNT& = COUNT& + 1: COUNT1% = 0

GOSUB 35 convert k to row and column

IF Z(K) >= 1 THEN X(K) = Z(K): GOTO 292

IF AA(K) - UL(K) / H ^ J < LL(K) THEN GOTO 250

GOTO 260

250 L(K) = LL(K)

255 GOTO 265

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260 L(K) = AA(K) - UL(K) / H ^ J

265 IF (AA(K) + UL(K) / H ^ J > UL(K)) THEN GOTO 280

270 GOTO 290

280 R = UL(K) - L(K)

285 GOTO 295

290 R = AA(K) + UL(K) / H ^ J - L(K)

295 X(K) = INT(L(K) + (RND * R) + 0.5): COUNT1% = COUNT1% + 1

IF COUNT1% > 50 THEN INPUT " PRESS ENTER TO CONTINUE", YES: GOTO

292 ' OUT OF DIGITS AT K= "; K

FLAG5 = 1: GOSUB 700 'CHECK FOR FEASIBILITY

IF FLAG5 = 0 THEN GOTO 295

292 PRINT " K= "; K; " Z= "; Z(K); " X= "; X(K); " COUNT= "; COUNT1%

300 NEXT K

REM CONVEX OBJECTIVE FUNCTION FOR MINIMIZATION TO ZERO

See Appendix A for the calculation of the following values:

PT = Total sum of the rows, CT = the total sum of the columns, and GT = the total

sum of the grids. PMT = the product of all of the rows, CMT = the product of all of

the columns, and GMT = the product of all of the grids. T1 through T6 = the sum

of the six digits for all rows, columns, and grids. TOTSUMs are the total duplicates for the rows, columns, and grids. All of these values are in convex form.

S = PT + CT + GT + PMT + CMT + GMT + T1 + T2 + T3 + T4 + T5 + T6 + TOTSUMRD

+ TOTSUMCD + TOTSUMGD

PRINT: PRINT " S= "; S; "MM= "; MM; "J= "; J; " I= "; I&

IF S = 0 THEN GOTO 556 Found solution!

IF S < MM THEN GOTO 660

GOTO 670

660 MM = S new minimum

FOR K = 1 TO 36

AA(K) = X(K)

GOSUB 35

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A(ROW%, COL%) = X(K)

333 'PRINT " K= "; K; " X= "; X(K); " ROW= "; ROW%; " COL= "; COL%; " AA= ";

AA(K)

NEXT K

670 COUNTI& = COUNTI& + 1 ': PRINT: PRINT " COUNTI= "; COUNTI&

'INPUT "PRESS ENTER TO CONTINUE", YES

444 NEXT I&

CLS

'PRINT: PRINT " PTT= "; PTT; " CTT= "; CTT; " GT= "; GT

'PRINT: PRINT " PMTT= "; PMTT; " CMTT= "; CMTT; " GMT= "; GMT

'PRINT: PRINT " TOTSUMGD= "; TOTSUMGD; " TOTSUMRD= "; TOTSUMRD; "

TOTSUMCD= "; TOTSUMCD

'PRINT: PRINT " T1= "; T1; " T2= "; T2; " T3= "; T3; " T4= "; T4; " T5= "; T5; " T6= "; T6

'PRINT: PRINT " S= "; S; "MM= "; MM

'PRINT: PRINT " J= "; J; " I= "; I&; " S= "; S; "MM= "; MM

COUNTI& = 0

COUNTJ& = COUNTJ& + 1: PRINT: PRINT " COUNTJ= "; COUNTJ&

'INPUT " PRESS ENTER TO CONTINUE", YES

555 NEXT J

556 CLS: PRINT: PRINT " FINAL RESULTS!"

PRINT: PRINT " S= "; S; " MM= "; MM

PRINT: PRINT " J= "; J; " I= "; I&; " S= "; S; " MM= "; MM

PRINT: PRINT " COUNTJ = "; COUNTJ&; " COUNTI= "; COUNTI&

BEEP: BEEP

TF# = TIMER

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PRINT: PRINT " START TIME= "; TS#; " FINISH TIME= "; TF#

ELAPSE# = (TF# - TS#): PRINT " ELAPSED TIME= "; ELAPSE#; " SECONDS"

INPUT "PRESS ENTER TO CONTINUE", YES

GOSUB 190 Print solution to computer screen

GOSUB 200 Print solution to hard drive

INPUT "PRESS ENTER TO CONTINUE", YES

GOTO 999 END

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Appendix A: Development of Constraint Equations

REM CONVERT GRID VALUES TO STRINGS AND CONCATENATE

FOR GRID%% = 1 TO 6: XY$(GRID%%) = "": NEXT GRID%%

XY$(1) = LTRIM$(STR$(X(1))) + LTRIM$(STR$(X(2))) + LTRIM$(STR$(X(3))) +

LTRIM$(STR$(X(7))) + LTRIM$(STR$(X(8))) + LTRIM$(STR$(X(9)))











REM DEFINE GRID CONTENTS

FOR GRID% = 1 TO 6: FOR II = 1 TO 6: CTG(GRID%, II) = 0: NEXT II: NEXT GRID%

FOR GRID% = 1 TO 6

FOR II = 1 TO 6

DIG$ = STR$(II): DIG$ = LTRIM$(DIG$)

POSITION% = INSTR(XY$(GRID%), DIG$)


PRINT " GRID= "; GRID%; " DIGIT= "; II; " POSITION= "; POSITION%

CTG(GRID%, II) = CTG(GRID%, II) + 1

POSITION% = INSTR(POSITION% + 1, XY$(GRID%), DIG$)

LOOP

IF CTG(GRID%, II) >= 1 THEN

PRINT " COUNT= "; CTG(GRID%, II); " FOR DIGIT= "; II

ELSE

PRINT " GRID= "; GRID%; " CONTAINS NO "; II

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END IF

NEXT II

NEXT GRID%

REM COUNT DUPLICATES INGRIDS

FOR GRID% = 1 TO 6: FOR II = 1 TO 6: TGD(GRID%, II) = 0: NEXT II: NEXT GRID%

FOR GRID% = 1 TO 6

FOR II = 1 TO 6

TGD(GRID%, II) = TGD(GRID%, II) + CTG(GRID%, II)

PRINT: PRINT " GRID= "; GRID%; " DIGIT= "; II; " CTG= "; CTG(GRID%, II)

NEXT II

PRINT " GRID= "; GRID%; " TGD(GRID%, II)= "; TGD(GRID%, 1); TGD(GRID%,

2); TGD(GRID%, 3); TGD(GRID%, 4); TGD(GRID%, 5); TGD(GRID%, 6)

NEXT GRID%

REM PUT INTO CONVEX FORM

FOR II = 1 TO 6: SUMGD(II) = 0: NEXT II

FOR II = 1 TO 6

FOR JJ = 1 TO 6

SUMGD(II) = SUMGD(II) + ABS(TGD(II, JJ) - 1)convert to convex form

NEXT JJ

NEXT II

TOTSUMGD = 0

FOR II = 1 TO 6

TOTSUMGD = TOTSUMGD + SUMGD(II)

NEXT II

PRINT: PRINT " TOTSUMGD= "; TOTSUMGD

19

REM CONVERT COLUMN VALUES TO STRINGS AND CONCATENATE

FOR COL% = 1 TO 6: YY$(COL%) = "": NEXT COL%

FOR COL% = 1 TO 6

FOR ROW% = 1 TO 6

GOSUB 40


YY$(COL%) = YY$(COL%) + XL$

NEXT ROW%

PRINT " COL%= "; COL%; " COLUMN STRING= "; YY$(COL%)

NEXT COL%

REM DEFINE CONTENTS OF COLUMNS

FOR COL% = 1 TO 6: FOR II = 1 TO 6: CTC(COL%, II) = 0: NEXT II: NEXT COL%

FOR COL% = 1 TO 6

FOR II = 1 TO 6


POSITION% = INSTR(YY$(COL%), DIG$)


PRINT " COL= "; COL%; " DIGIT= "; II; " POSITION= "; POSITION%

CTC(COL%, II) = CTC(COL%, II) + 1

POSITION% = INSTR(POSITION% + 1, YY$(COL%), DIG$)

LOOP

IF CTC(COL%, II) >= 1 THEN

PRINT " COUNT= "; CTC(COL%, II); " FOR DIGIT= "; II

ELSE

PRINT " COL= "; COL%; " CONTAINS NO "; II

END IF

NEXT II

NEXT COL%

20

REM COUNT DUPLICATES IN COLUMNS

FOR II = 1 TO 6: SUMCD(II) = 0: NEXT II

FOR II = 1 TO 6

FOR JJ = 1 TO 6

SUMCD(II) = SUMCD(II) + ABS(TCD(II, JJ) - 1) convert to convex

NEXT JJ

NEXT II

TOTSUMCD = 0

FOR II = 1 TO 6

TOTSUMCD = TOTSUMCD + SUMCD(II)

NEXT II

PRINT: PRINT " TOTSUMCD= "; TOTSUMCD

REM CONSTRAINT EQUATIONS FOR SUMS AND PRODUCTS OF UNITS

REM EQUATIONS FOR ROWS 1.6

REM ROW #1

P1 = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) sums of variables

PM1 = X(1) * X(2) * X(3) * X(4) * X(5) * X(6) products of variables

PRINT " PM1= "; PM1

PM1 = ABS(PM1 - 720) convert to convex form

P1 = ABS(P1 - 21) convert to convex form

PRINT: PRINT " PM1= "; PM1; " P1= "; P1

REM ROW #6

P6 = X(31) + X(32) + X(33) + X(34) + X(35) + X(36)

PM6 = X(31) * X(32) * X(33) * X(34) * X(35) * X(36)

PRINT " PM6= "; PM6

PM6 = ABS(PM6 - 720)

21

P6 = ABS(P6 - 21)

PRINT: PRINT " PM6= "; PM6; " P6= "; P6

PT = P1 + P2 + P3 + P4 + P5 + P6

PMT = PM1 + PM2 + PM3 + PM4 + PM5 + PM6

PRINT: PRINT " PT= "; PT; " PMT= "; PMT

REM EQUATIONS FOR COLUMNS 1.6

REM COLUMN #1

C1 = X(1) + X(7) + X(13) + X(19) + X(25) + X(31)

CM1 = X(1) * X(7) * X(13) * X(19) * X(25) * X(31)

PRINT " CM1= "; CM1

CM1 = ABS(CM1 - 720)

C1 = ABS(C1 - 21)

PRINT: PRINT " CM1= "; CM1; " C1= "; C1

REM COLUMN #6

C6 = X(6) + X(12) + X(18) + X(24) + X(30) + X(36)

CM6 = X(6) * X(12) * X(18) * X(24) * X(30) * X(36)

PRINT " CM6= "; CM6

CM6 = ABS(CM6 - 720)

C6 = ABS(C6 - 21)

PRINT: PRINT " CM6= "; CM6; " C6= "; C6

CT = C1 + C2 + C3 + C4 + C5 + C6

CMT = CM1 + CM2 + CM3 + CM4 + CM5 + CM6

PRINT: PRINT " CT= "; CT; " CMT= "; CMT

REM EQUATIONS FOR GRIDS 16

REM GRID% #1

G1 = X(1) + X(2) + X(3) + X(7) + X(8) + X(9)

22

GM1 = X(1) * X(2) * X(3) * X(7) * X(8) * X(9)

PRINT " GM1= "; GM1

GM1 = ABS(GM1 - 720)

G1 = ABS(G1 - 21)

PRINT: PRINT " GM1= "; GM1; " G1= "; G1

REM GRID% #6

G6 = X(28) + X(29) + X(30) + X(34) + X(35) + X(36)

GM6 = X(28) * X(29) * X(30) * X(34) * X(35) * X(36)

PRINT " GM6= "; GM6

GM6 = ABS(GM6 - 720)

G6 = ABS(G6 - 21)

PRINT: PRINT " GM6= "; GM6; " G6= "; G6

GT = G1 + G2 + G3 + G4 + G5 + G6

GMT = GM1 + GM2 + GM3 + GM4 + GM5 + GM6

PRINT: PRINT " GT= "; GT; " GMT= "; GMT

REM COUNT TOTAL DIGITS OF EACH KIND IN GRIDS, ROWS AND COLUMNS AND

CONVERT TO CONVEX FORM




TCD(II, JJ): NEXT II: T1 = ABS(TOTONES% - 18) The same six digits appear in each unit









23



PRINT: PRINT " PT= "; PT; " CT= "; CT; " GT= "; GT

PRINT: PRINT " PMT= "; PMT; " CMT= "; CMT; " GMT= "; GMT

PRINT: PRINT " TOTSUMGD= "; TOTSUMGD; " TOTSUMRD= "; TOTSUMRD; "


PRINT: PRINT " T1= "; T1; " T2= "; T2; " T3= "; T3; " T4= "; T4; " T5= "; T5; " T6= "; T6

24

Appendix B: Puzzles No. 2 and No. 3

FIGURE 3

1 6 2 3 4 5 1

2 1 5 4 3 6 2

3 3 6 2 5 1 4

4 4 1 5 6 2 3

5 2 3 6 1 4 5

6 5 4 1 2 3 6

1 2 3 4 5 6

FIGURE 4.

1 3 4 2 5 1 6

2 1 5 6 4 2 3

3 6 2 1 3 4 5

4 5 3 4 2 6 1

5 4 6 3 1 5 2

6 2 1 5 6 3 4

1 2 3 4 5 6

25

Appendix C: Puzzle No. 1 Computer Solution

QMCMSU28.BAS

PUZZLE#1

09-17-2015 20:47:22

2 3 4 4 4 1

5 1 4 4 2 4

4 2 6 1 3 3

1 3 5 2 3 3

5 4 1 3 6 2

3 2 2 4 1 5

PTT= 14 CTT= 16 GT= 14

PMTT= 1634 CMTT= 1920 GMT= 1486

TOTSUMGD= 16 TOTSUMRD= 16 TOTSUMCD= 16

T1= 0 T2= 3 T3= 6 T4= 9 T5= 6 T6= 12

S= 5168 MM= 5168

J= 1 I= 1 S= 5168 MM= 5168

2 5 3 5 5 1

6 1 3 4 2 6

4 2 6 1 4 4

1 3 5 2 4 4

5 4 2 3 6 2

3 6 1 4 1 5

PTT= 5 CTT= 5 GT= 5

PMTT= 1542 CMTT= 900 GMT= 868


T1= 0 T2= 0 T3= 3 T4= 6 T5= 0 T6= 3

26

S= 3371 MM= 3371

J= 1 I= 2 S= 3371 MM= 3371

2 5 4 5 3 1

6 1 3 5 2 4

4 2 6 1 5 4

1 3 5 2 3 4

5 4 1 3 6 2

3 2 2 4 1 5

PTT= 9 CTT= 7 GT= 7

PMTT= 1200 CMTT= 860 GMT= 840


T1= 0 T2= 3 T3= 0 T4= 3 T5= 3 T6= 9

S= 2965 MM= 2965

J= 1 I= 10 S= 2965 MM= 2965

2 6 4 6 4 1

6 1 3 5 2 4

4 2 6 1 4 4

1 3 5 2 4 4

5 4 2 3 6 2

3 6 1 4 1 5

PTT= 6 CTT= 2 GT= 4

PMTT= 1800 CMTT= 272 GMT= 592


T1= 0 T2= 0 T3= 6 T4= 12 T5= 6 T6= 0

S= 2734 MM= 2734

J= 1 I= 47 S= 2734 MM= 2734

27

2 6 3 5 5 1

5 1 4 4 2 4

4 2 6 1 4 4

1 3 5 2 4 3

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 5 CTT= 7 GT= 3

PMTT= 668 CMTT= 984 GMT= 416


T1= 0 T2= 0 T3= 3 T4= 9 T5= 0 T6= 6

S= 2131 MM= 2131

J= 1 I= 93 S= 2131 MM= 2131

2 3 4 5 5 1

6 1 3 4 2 3

4 2 6 1 4 4

1 3 5 2 3 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 4 CTT= 4 GT= 4

PMTT= 636 CMTT= 528 GMT= 552


T1= 0 T2= 0 T3= 3 T4= 3 T5= 3 T6= 3

S= 1760 MM= 1760

J= 1 I= 221 S= 1760 MM= 1760

2 5 4 5 4 1

6 1 4 4 2 3

28

4 2 6 1 4 4

1 3 5 2 3 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 2 CTT= 4 GT= 4

PMTT= 452 CMTT= 624 GMT= 624


T1= 0 T2= 0 T3= 3 T4= 9 T5= 3 T6= 3

S= 1752 MM= 1752

J= 1 I= 474 S= 1752 MM= 1752

2 5 4 4 5 1

5 1 3 4 2 4

4 2 6 1 5 3

1 3 5 2 3 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 3 CTT= 5 GT= 3

PMTT= 500 CMTT= 636 GMT= 380


T1= 0 T2= 0 T3= 0 T4= 3 T5= 3 T6= 6

S= 1563 MM= 1563

J= 1 I= 2548 S= 1563 MM= 1563

2 3 4 6 5 1

5 1 3 5 2 3

4 2 6 1 3 4

1 3 5 2 4 6

29

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 3 CTT= 3 GT= 5

PMTT= 414 CMTT= 408 GMT= 684


T1= 0 T2= 0 T3= 3 T4= 0 T5= 0 T6= 3

S= 1539 MM= 1539

J= 1 I= 3647 S= 1539 MM= 1539

2 6 3 4 4 1

4 1 4 5 2 4

4 2 6 1 4 4

1 3 5 2 4 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 2 CTT= 6 GT= 2

PMTT= 272 CMTT= 912 GMT= 272


T1= 0 T2= 0 T3= 6 T4= 15 T5= 6 T6= 3

S= 1528 MM= 1528

J= 2 I= 194 S= 1528 MM= 1528

2 5 4 6 3 1

4 1 3 5 2 4

4 2 6 1 5 3

1 3 5 2 4 4

5 4 1 3 6 2

3 6 2 4 1 5

30

PTT= 4 CTT= 4 GT= 4

PMTT= 480 CMTT= 480 GMT= 480


T1= 0 T2= 0 T3= 0 T4= 6 T5= 0 T6= 6

S= 1476 MM= 1476

J= 2 I= 281 S= 1476 MM= 1476

2 5 3 6 5 1

6 1 4 4 2 3

4 2 6 1 5 4

1 3 5 2 4 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 3 CTT= 3 GT= 1

PMTT= 564 CMTT= 624 GMT= 240


T1= 0 T2= 0 T3= 3 T4= 3 T5= 0 T6= 0

S= 1453 MM= 1453

J= 2 I= 680 S= 1453 MM= 1453

2 5 3 6 4 1

5 1 4 5 2 4

4 2 6 1 4 4

1 3 5 2 4 4

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 2 CTT= 2 GT= 4

PMTT= 368 CMTT= 248 GMT= 568

31


T1= 0 T2= 0 T3= 6 T4= 12 T5= 0 T6= 6

S= 1246 MM= 1246

J= 2 I= 1068 S= 1246 MM= 1246

2 5 4 5 3 1

6 1 3 6 2 4

4 2 6 1 4 3

1 3 5 2 3 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 4 CTT= 2 GT= 2

PMTT= 588 CMTT= 288 GMT= 288


T1= 0 T2= 0 T3= 3 T4= 0 T5= 3 T6= 0

S= 1190 MM= 1190

J= 2 I= 1672 S= 1190 MM= 1190

2 5 4 5 4 1

6 1 3 5 2 3

4 2 6 1 4 4

1 3 5 2 4 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 1 CTT= 1 GT= 1

PMTT= 308 CMTT= 168 GMT= 168


T1= 0 T2= 0 T3= 3 T4= 6 T5= 0 T6= 3

32

S= 681 MM= 681

J= 3 I= 1827 S= 681 MM= 681

2 5 4 5 3 1

6 1 3 6 2 4

4 2 6 1 5 3

1 3 5 2 4 6

5 4 1 3 6 2

3 6 2 4 1 5

PTT= 2 CTT= 0 GT= 0

PMTT= 264 CMTT= 0 GMT= 0


T1= 0 T2= 0 T3= 0 T4= 0 T5= 0 T6= 0

S= 270 MM= 270

J= 3 I= 2801 S= 270 MM= 270

SOLUTION ACHIEVED! I= 124 J= 5

QMCMSU27.BAS

PUZZLE#1

09-17-2015 20:48:23

2 5 4 6 3 1

6 1 3 5 2 4

4 2 6 1 5 3

1 3 5 2 4 6

5 4 1 3 6 2

3 6 2 4 1 5

ELAPSED TIME= 14.5 SECONDS

33

S J I

5168 1 1

3371 1 2

2965 1 10

2734 1 47

2131 1 93

1760 1 221

1752 1 474 S-Blue

1563 1 2548

1539 1 3647

1528 2 194

1476 2 281

1453 2 680

1246 2 1068

1190 2 1672

681 3 1827

J-

Orange I-Gray

270 3 2801

0 5 124

Figure 5. Puzzle #1 Objective Function Value, S vs. No. of

Iterations, J and I

34

S J I

14191 1 1

13127 1 3

8225 1 4

6920 1 10

4570 1 14

4452 1 20

4172 1 21 S-Blue

3263 1 75

3084 1 103

2775 1 271

1534 1 1090

1154 1 2765

1101 1 16581

920 2 7242

658 2 17211

J-

Orange I-Gray

555 3 2621

540 3 18244

375 4 1939

270 4 22302

0 4 24458


Iterations, J and I

0

5000

10000

15000

20000

25000

30000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

S vs J & I

Series1 Series2 Series3

35

S J I

44122 1 1

8819 1 2

5111 1 7

4829 1 338

3507 1 653

2749 1 2200 S-Blue

2626 1 3140

2196 1 6211

2081 2 2597

1902 2 26269

1854 3 588

1818 3 2062

J-

Orange I-Gray

1587 3 3422

1526 3 5514

1519 3 6102

1458 3 17536

1378 3 18486

1309 3 19265

1110 3 21610

1094 3 25745

790 3 37194

447 4 1995

0 4 2587


Iterations, J and I

0

10000

20000

30000

40000

50000

1 3 5 7 9 11 13 15 17 19 21 23

S vs J & I

Series1 Series2 Series3

36

Appendix D: QB64 Computer Code

REM SOLVE SUDOKU USING MCM

REM 01/17/13 J H LASHOVER REV. 10/08/2013 & 08/09/2015 & 09/17/2015

REM FOR 6 X 6 PUZZLES

CLS

PRINT: PRINT " WELCOME TO QMCMSU28.BAS"

DEFDBL A, Z

DEFINT I, J, K

DIM X(36), L(36), Y(36), A(6, 6), Z(36), AA(36), LL(36), UL(36), CTG(6, 6), CTC(6, 6),

CTR(6, 6) AS LONG

DIM ZX(6, 6), SUMGD(6), SUMRD(6), SUMCD(6), TGD(6, 6), TRD(6, 6), TCD(6, 6), P(6),

PM(6), C(6), CM(6) AS LONG

DIM YY$(6), XY$(6), XX$(6), YYY$(6, 6)

DIM XXY(6, 6), YYX(6, 6) AS LONG

DIM CT(6, 6), NF(6, 6) AS LONG

REM SELECT A SAMPLE PUZZLE OR ENTER YOUR OWN

10 OP1 = 0: INPUT " PRESS ENTER TO SELECT A SAMPLE PUZZLE, OR ENTER '1' TO

INPUT YOUR OWN. OP1= ", OP1

IF OP1 = 0 THEN GOTO 11

IF OP1 = 1 THEN GOTO 30

PRINT " PLEASE ENTER A ZERO OR '1': GOTO 10"

REM INPUT PERMANENT PUZZLE NUMBERS FOR FOUR SAMPLE PUZZLES

11 OP2 = 0: INPUT " NUMBER OF PUZZLE TO SOLVE: 1, 2, OR 3: NO.= ", OP2

IF OP2 5 THEN PRINT " PLEASE ENTER NUMBER BETWEEN 1 AND 3.": GOTO 11

37

ON OP2 GOTO 1, 2, 3

1 REM SAMPLE PUZZLE NO 1

ZZZ = 5000

ZX(1, 1) = 2: ZX(1, 6) = 1

ZX(2, 2) = 1: ZX(2, 5) = 2

ZX(3, 3) = 6: ZX(3, 4) = 1

ZX(4, 3) = 5: ZX(4, 4) = 2

ZX(5, 2) = 4: ZX(5, 5) = 6

ZX(6, 1) = 3: ZX(6, 6) = 5

GOTO 36


ZZZ = 30000

ZX(1, 1) = 6: ZX(1, 4) = 4: ZX(1, 6) = 1

ZX(2, 2) = 5: ZX(2, 3) = 4: ZX(2, 4) = 3: ZX(2, 5) = 6: ZX(2, 6) = 2

ZX(3, 3) = 2: ZX(3, 6) = 4

ZX(4, 5) = 2

ZX(5, 1) = 2: ZX(5, 4) = 1: ZX(5, 5) = 4

ZX(6, 3) = 1: ZX(6, 6) = 6

GOTO 36


ZZZ = 40000

ZX(1, 2) = 4

ZX(2, 1) = 1: ZX(2, 3) = 6: ZX(2, 4) = 4: ZX(2, 6) = 3

38

ZX(3, 1) = 6: ZX(3, 3) = 1: ZX(3, 4) = 3

ZX(4, 5) = 6

ZX(5, 4) = 1: ZX(5, 6) = 2

ZX(6, 1) = 2: ZX(6, 3) = 5

GOTO 36

30 REM INPUT PUZZLE DATA

132 INPUT " ROW#= ", I

IF I = 7 THEN GOTO 36

INPUT " COLUMN#= ", J

INPUT " NUMBER IN CELL= ", ZX(I, J)

GOTO 132

36 REM START MAIN PROGRAM

INPUT " NAME OF FILE TO STORE OUTPUT= ", FI$

INPUT " COMMENTS= "; FII$

OPEN "C:\QB64\" + FI$ + ".TXT" FOR OUTPUT AS #1

PRINT #1,: PRINT #1, "QMCMSU28.BAS"

PRINT #1, FII$

PRINT #1, DATE$, TIME$

REM INITIALIZE VARIABLES

46 GOSUB 5000

47 GOSUB 10000

IF FLAG = 1 THEN GOTO 47

FOR ROW% = 1 TO 6

39

FOR COL% = 1 TO 6


UL(K) = 6: LL(K) = 1: AA(K) = 3

IF ZX(ROW%, COL%) 0 THEN X(K) = ZX(ROW%, COL%): GOTO 71

GOTO 68

71 AA(K) = X(K): Z(K) = X(K): A(ROW%, COL%) = X(K)

68 NEXT COL%

70 NEXT ROW%

MM = 1E+30: H = 1.3

RANDOMIZE (-1111)

COUNTI& = 0: COUNT& = 0

TS# = TIMER

FOR J = 1 TO 20

FOR I& = 1 TO ZZZ

FOR K = 1 TO 36

COUNT& = COUNT& + 1: COUNT1% = 0

GOSUB 35

IF Z(K) >= 1 THEN X(K) = Z(K): GOTO 292

IF AA(K) - UL(K) / H ^ J < LL(K) THEN GOTO 250

GOTO 260

250 L(K) = LL(K)

255 GOTO 265

260 L(K) = AA(K) - UL(K) / H ^ J

40

265 IF (AA(K) + UL(K) / H ^ J > UL(K)) THEN GOTO 280

270 GOTO 290

280 R = UL(K) - L(K)

285 GOTO 295

290 R = AA(K) + UL(K) / H ^ J - L(K)

295 X(K) = INT(L(K) + (RND * R) + 0.5): COUNT1% = COUNT1% + 1

IF COUNT1% > 50 THEN INPUT " PRESS ENTER TO CONTINUE", YES: GOTO

292 'PRINT: PRINT " OUT OF DIGITS AT K= "; K

FLAG5 = 1: GOSUB 700 'CHECK FOR FEASIBILITY

IF FLAG5 = 0 THEN GOTO 295

292 'PRINT " K= "; K; " Z= "; Z(K); " X= "; X(K); " COUNT= "; COUNT1%


300 NEXT K

REM EQUATIONS FOR ROWS

CLS: PTT = 0: PMTT = 0

FOR JJ = 1 TO 6: P(JJ) = 0: PM(JJ) = 1: NEXT JJ

KK = 0: LLL = 1

FOR II = 1 TO 6

KK = KK + 6

FOR JJ = LLL TO KK

LLL = LLL + 1

P(II) = P(II) + X(JJ)

PM(II) = PM(II) * X(JJ)

NEXT JJ

P(II) = ABS(P(II) - 21)

PM(II) = ABS(PM(II) - 720)

'PRINT: PRINT II, P(II), PM(II)

PTT = PTT + P(II)

41

PMTT = PMTT + PM(II)

NEXT II

'PRINT: PRINT " PTT= "; PTT; " PMTT= "; PMTT


REM EQUATIONS FOR COLUMNS

CLS: CTT = 0: CMTT = 0

FOR JJ = 1 TO 6: C(JJ) = 0: CM(JJ) = 1: NEXT JJ

KK = 30: LLL = 1

FOR II = 1 TO 6

KK = KK + 1

FOR JJ = LLL TO KK STEP 6

C(II) = C(II) + X(JJ)

CM(II) = CM(II) * X(JJ)

NEXT JJ

LLL = LLL + 1

C(II) = ABS(C(II) - 21)

CM(II) = ABS(CM(II) - 720)

'PRINT: PRINT II, C(II), CM(II)

CTT = CTT + C(II)

CMTT = CMTT + CM(II)

NEXT II

'PRINT: PRINT " CTT= "; CTT; " CMTT= "; CMTT


42

REM EQUATIONS FOR SUB GRIDS

CLS

103 REM GRID% #1

G1 = X(1) + X(2) + X(3) + X(7) + X(8) + X(9)

GM1 = X(1) * X(2) * X(3) * X(7) * X(8) * X(9)

'PRINT " GM1= "; GM1

GM1 = ABS(GM1 - 720)

G1 = ABS(G1 - 21)

'PRINT: PRINT " GM1= "; GM1; " G1= "; G1

104 REM GRID% #2

G2 = X(4) + X(5) + X(6) + X(10) + X(11) + X(12)

GM2 = X(4) * X(5) * X(6) * X(10) * X(11) * X(12)


GM2 = ABS(GM2 - 720)

G2 = ABS(G2 - 21)


105 REM GRID% #3

G3 = X(13) + X(14) + X(15) + X(19) + X(20) + X(21)

GM3 = X(13) * X(14) * X(15) * X(19) * X(20) * X(21)


GM3 = ABS(GM3 - 720)

G3 = ABS(G3 - 21)


43

106 REM GRID% #4

G4 = X(16) + X(17) + X(18) + X(22) + X(23) + X(24)

GM4 = X(16) * X(17) * X(18) * X(22) * X(23) * X(24)


GM4 = ABS(GM4 - 720)

G4 = ABS(G4 - 21)


107 REM GRID% #5

G5 = X(25) + X(26) + X(27) + X(31) + X(32) + X(33)

GM5 = X(25) * X(26) * X(27) * X(31) * X(32) * X(33)


GM5 = ABS(GM5 - 720)

G5 = ABS(G5 - 21)


108 REM GRID% #6

G6 = X(28) + X(29) + X(30) + X(34) + X(35) + X(36)

GM6 = X(28) * X(29) * X(30) * X(34) * X(35) * X(36)


GM6 = ABS(GM6 - 720)

G6 = ABS(G6 - 21)


GT = G1 + G2 + G3 + G4 + G5 + G6

GMT = GM1 + GM2 + GM3 + GM4 + GM5 + GM6

'PRINT: PRINT " GT= "; GT; " GMT= "; GMT


44

REM ALGORITHM FOR DETERMINING NUMBER OF DUPLICATES IN A UNIT

REM GRIDS

CLS

FOR GRID%% = 1 TO 6: XY$(GRID%%) = "": NEXT GRID%%













FOR GRID% = 1 TO 6

'PRINT " GRID#= "; GRID%; " GRID STRING= "; XY$(GRID%)


NEXT GRID%

GOSUB 100 'COUNT DUPLICATES IN GRIDS

REM ROWS

CLS


FOR ROW% = 1 TO 6

45

FOR COL% = 1 TO 6




NEXT COL%

'PRINT " ROW%= "; ROW%; " ROW STRING= "; XX$(ROW%)


NEXT ROW%

GOSUB 500 'COUNT DUPLICATES IN ROWS

REM COLUMNS

CLS


FOR COL% = 1 TO 6

FOR ROW% = 1 TO 6

GOSUB 40



NEXT ROW%

'PRINT " COL%= "; COL%; " COLUMN STRING= "; YY$(COL%)


NEXT COL%

GOSUB 600 'COUNT DUPLICATES IN COLUMNS

REM SET UP CONVEX EQUATIONS FOR UNITS

REM GRIDS

46

CLS: CCG = 0: FOR GRID% = 1 TO 6: FOR II = 1 TO 6: TGD(GRID%, II) = 0: NEXT II:

NEXT GRID%

FOR GRID% = 1 TO 6

FOR II = 1 TO 6

TGD(GRID%, II) = TGD(GRID%, II) + CTG(GRID%, II)

'PRINT: PRINT " GRID= "; GRID%; " DIGIT= "; II; " CTG= "; CTG(GRID%, II)

NEXT II

'PRINT " GRID= "; GRID%; " TGD(GRID%, II)= "; TGD(GRID%, 1); TGD(GRID%,

2); TGD(GRID%, 3); TGD(GRID%, 4); TGD(GRID%, 5); TGD(GRID%, 6)


NEXT GRID%

REM ROWS

CLS: CCR = 0: FOR ROW% = 1 TO 6: FOR II = 1 TO 6: TRD(ROW%, II) = 0: NEXT II:

NEXT ROW%

FOR ROW% = 1 TO 6

FOR II = 1 TO 6

TRD(ROW%, II) = TRD(ROW%, II) + CTR(ROW%, II)

'PRINT: PRINT " ROW= "; ROW%; " DIGIT= "; II; " CTR= "; CTR(ROW%, II)

NEXT II

'PRINT " ROW= "; ROW%; " TRD(ROW%, II)= "; TRD(ROW%, 1); TRD(ROW%, 2);

TRD(ROW%, 3); TRD(ROW%, 4); TRD(ROW%, 5); TRD(ROW%, 6)


NEXT ROW%

REM COLUMNS

CLS: CCC = 0: FOR COL% = 1 TO 6: FOR II = 1 TO 6: TCD(COL%, II) = 0: NEXT II:

NEXT COL%

FOR COL% = 1 TO 6

FOR II = 1 TO 6

47

TCD(COL%, II) = TCD(COL%, II) + CTC(COL%, II)

'PRINT: PRINT " COL= "; COL%; " DIGIT= "; II; " CTC= "; CTC(COL%, II)

NEXT II

'PRINT " COL= "; COL%; " TCD(COL%, II)= "; TCD(COL%, 1); TCD(COL%, 2);

TCD(COL%, 3); TCD(COL%, 4); TCD(COL%, 5); TCD(COL%, 6)


NEXT COL%

REM PUT INTO CONVEX FORM

FOR II = 1 TO 6: SUMGD(II) = 0: NEXT II

FOR II = 1 TO 6

FOR JJ = 1 TO 6

SUMGD(II) = SUMGD(II) + ABS(TGD(II, JJ) - 1)

NEXT JJ

NEXT II

TOTSUMGD = 0

FOR II = 1 TO 6

TOTSUMGD = TOTSUMGD + SUMGD(II)

NEXT II

'PRINT: PRINT " TOTSUMGD= "; TOTSUMGD


FOR II = 1 TO 6: SUMRD(II) = 0: NEXT II

FOR II = 1 TO 6

FOR JJ = 1 TO 6

SUMRD(II) = SUMRD(II) + ABS(TRD(II, JJ) - 1)

NEXT JJ

48

NEXT II

TOTSUMRD = 0

FOR II = 1 TO 6

TOTSUMRD = TOTSUMRD + SUMRD(II)

NEXT II

'PRINT: PRINT " TOTSUMRD= "; TOTSUMRD


FOR II = 1 TO 6: SUMCD(II) = 0: NEXT II

FOR II = 1 TO 6

FOR JJ = 1 TO 6

SUMCD(II) = SUMCD(II) + ABS(TCD(II, JJ) - 1)

NEXT JJ

NEXT II

TOTSUMCD = 0

FOR II = 1 TO 6

TOTSUMCD = TOTSUMCD + SUMCD(II)

NEXT II

'PRINT: PRINT " TOTSUMCD= "; TOTSUMCD





TCD(II, JJ): NEXT II: T1 = ABS(TOTONES% - 18)



49









CLS





'PRINT: PRINT " T1= "; T1; " T2= "; T2; " T3= "; T3; " T4= "; T4; " T5= "; T5; " T6= ";

T6

REM CALCULATE OBJECTIVE FUNCTION

732 S = PTT + CTT + GT + PMTT + CMTT + GMT + T1 + T2 + T3 + T4 + T5 + T6 +

TOTSUMGD + TOTSUMRD + TOTSUMCD:

PRINT: PRINT " S= "; S; " MM= "; MM; " J= "; J; " I= "; I&

734 'INPUT " PRESS ENTER TO CONTINUE", YES

IF S = 0 THEN PRINT #1,: PRINT #1, " SOLUTION ACHIEVED!"; " I= "; I&; " J= "; J:

GOTO 556

IF S < MM THEN GOTO 660

GOTO 670

660 MM = S

FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6

50

GOSUB 40

IF COL% 6 THEN GOTO 1905

PRINT #1, X(K): GOTO 1911

1905 PRINT #1, X(K);

1911 NEXT COL%

PRINT #1,

NEXT ROW%

PRINT #1,: PRINT #1, " PTT= "; PTT; " CTT= "; CTT; " GT= "; GT

PRINT #1,: PRINT #1, " PMTT= "; PMTT; " CMTT= "; CMTT; " GMT= "; GMT

PRINT #1,: PRINT #1, " TOTSUMGD= "; TOTSUMGD; " TOTSUMRD= ";

TOTSUMRD; " TOTSUMCD= "; TOTSUMCD

PRINT #1,: PRINT #1, " T1= "; T1; " T2= "; T2; " T3= "; T3; " T4= "; T4; " T5= "; T5; "

T6= "; T6

PRINT #1,: PRINT #1, " S= "; S; " MM= "; MM

PRINT #1,: PRINT #1, " J= "; J; " I= "; I&; " S= "; S; " MM= "; MM

FOR K = 1 TO 36

AA(K) = X(K)

GOSUB 35

A(ROW%, COL%) = X(K)

333 'PRINT #1, " K= "; K; " X= "; X(K)

NEXT K

670 COUNTI& = COUNTI& + 1 ': PRINT: PRINT " COUNTI= "; COUNTI&


51

444 NEXT I&

CLS





'PRINT: PRINT " T1= "; T1; " T2= "; T2; " T3= "; T3; " T4= "; T4; " T5= "; T5; " T6= "; T6

'PRINT: PRINT " S= "; S; " MM= "; MM

'PRINT: PRINT " J= "; J; " I= "; I&; " S= "; S; " MM= "; MM

COUNTI& = 0

COUNTJ& = COUNTJ& + 1: PRINT: PRINT " COUNTJ= "; COUNTJ&


'OP1 = 0: INPUT " ENTER '1' TO QUIT AND DUMP PRESENT SOLUTION. OP1= ", OP1

'IF OP1 = 1 THEN GOSUB 1000: GOTO 999

555 NEXT J

556 CLS: PRINT: PRINT " FINAL RESULTS!"

PRINT: PRINT " S= "; S; " MM= "; MM

PRINT: PRINT " J= "; J; " I= "; I&; " S= "; S; " MM= "; MM

PRINT: PRINT " COUNTJ = "; COUNTJ&; " COUNTI= "; COUNTI&

BEEP: BEEP

52

TF# = TIMER

PRINT: PRINT " START TIME= "; TS#; " FINISH TIME= "; TF#

ELAPSE# = (TF# - TS#): PRINT " ELAPSED TIME= "; ELAPSE#; " SECONDS"

PRINT " ELAPSED TIME= "; ELAPSE#; " SECONDS"

INPUT " PRESS ENTER TO CONTINUE", YES

GOSUB 190

GOSUB 200


GOTO 999

190 REM SUB TO PRINT OUTPUT

CLS

PRINT: PRINT " QMCMSU28.AS"

FOR K = 1 TO 36

GOSUB 35

777 IF K = 18 THEN PRINT " PRESS ENTER TO CONTINUE", YES

PRINT " K= "; K; " X= "; X(K); " ROW= "; ROW%; " COL= "; COL%; " AA= "; AA(K)

NEXT K


FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6

GOSUB 40


PRINT USING "###"; X(K): GOTO 910

905 PRINT USING "###"; X(K);

910 NEXT COL%

53

PRINT

NEXT ROW%


RETURN

200 REM SUB TO PRINT OUTPUT

'INPUT " NAME OF FILE TO STORE OUTPUT= ", FI$

'INPUT " COMMENTS= "; FII$

'OPEN "C:\QB64\" + FI$ + ".TXT" FOR OUTPUT AS #1

PRINT #1,: PRINT #1, "QMCMSU28.BAS"

PRINT #1, FII$

PRINT #1, DATE$, TIME$

FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6

GOSUB 40


PRINT #1, X(K): GOTO 210

205 PRINT #1, X(K);

210 NEXT COL%

PRINT #1,

NEXT ROW%

PRINT #1,: PRINT #1, "ELAPSED TIME= "; ELAPSE#; " SECONDS"

CLOSE #1


RETURN

35 REM CONVERT K TO ROW AND COLUMN--ALTERNATIVE CONVERSION OF K TO

ROW AND COLUMN

IF K > 6 THEN GOTO 69

54

ROW% = 1: COL% = K

GOTO 63

69 ROW% = K / 6 + 0.49999: ROW% = CINT(ROW%)

COL% = K - ((ROW% - 1) * 6)

63 'PRINT " K= "; K; " ROW= "; ROW%; " COL= "; COL%


RETURN

40 REM CONVERT ROW AND COLUMN TO K

K = ((ROW% - 1) * 6) + COL% 'CONVERT I,J TO SINGLE NUMBER

'PRINT " K= "; K; " ROW= "; ROW%; "COL= "; COL%


RETURN

1000 REM SUB TO DUMP MEMORY

CLS

PRINT: PRINT " DUMP X'S"

FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6

GOSUB 40


PRINT USING "##"; X(K): GOTO 1005

1010 PRINT USING "##"; X(K);

1005 NEXT COL%

PRINT

NEXT ROW%


55

PRINT: PRINT " DUMP UL'S"

FOR II = 1 TO 6

FOR JJ = 1 TO 6

KK = ((II - 1) * 6) + JJ 'CONVERT II,JJ TO SINGLE NUMBER

IF JJ 6 THEN GOTO 5010

PRINT UL(KK): GOTO 5005

5010 PRINT UL(KK);

5005 NEXT JJ

PRINT

NEXT II


PRINT: PRINT " DUMP LL'S"

FOR II = 1 TO 6

FOR JJ = 1 TO 6

KK = ((II - 1) * 6) + JJ 'CONVERT II,JJ TO SINGLE NUMBER

IF JJ 6 THEN GOTO 6010

PRINT L(KK): GOTO 6005

6010 PRINT L(KK);

6005 NEXT JJ

PRINT

NEXT II


RETURN

56

100 REM SUB TO COUNT DUPLICATES IN A GRID

FOR GRID% = 1 TO 6: FOR II = 1 TO 6: CTG(GRID%, II) = 0: NEXT II: NEXT GRID%

FOR GRID% = 1 TO 6

FOR II = 1 TO 6


POSITION% = INSTR(XY$(GRID%), DIG$)


'PRINT " GRID= "; GRID%; " DIGIT= "; II; " POSITION= "; POSITION%

CTG(GRID%, II) = CTG(GRID%, II) + 1

POSITION% = INSTR(POSITION% + 1, XY$(GRID%), DIG$)

LOOP

IF CTG(GRID%, II) >= 1 THEN

'PRINT " COUNT= "; CTG(GRID%, II); " FOR DIGIT= "; II

ELSE

'PRINT " GRID= "; GRID%; " CONTAINS NO "; II

END IF

NEXT II


NEXT GRID%

RETURN

500 REM SUB TO COUNT DUPLICATES IN A ROW

FOR ROW% = 1 TO 6: FOR II = 1 TO 6: CTR(ROW%, II) = 0: NEXT II: NEXT ROW%

FOR ROW% = 1 TO 6

FOR II = 1 TO 6


POSITION% = INSTR(XX$(ROW%), DIG$)


57

'PRINT " ROW= "; ROW%; " DIGIT= "; II; " POSITION= "; POSITION%

CTR(ROW%, II) = CTR(ROW%, II) + 1

POSITION% = INSTR(POSITION% + 1, XX$(ROW%), DIG$)

LOOP

IF CTR(ROW%, II) >= 1 THEN

'PRINT " COUNT= "; CTR(ROW%, II); " FOR DIGIT= "; II

ELSE

'PRINT " ROW= "; ROW%; " CONTAINS NO "; II

END IF

NEXT II


NEXT ROW%

RETURN

600 REM SUB TO COUNT DUPLICATES IN A COLUMN

FOR COL% = 1 TO 6: FOR II = 1 TO 6: CTC(COL%, II) = 0: NEXT II: NEXT COL%

FOR COL% = 1 TO 6

FOR II = 1 TO 6


POSITION% = INSTR(YY$(COL%), DIG$)


'PRINT " COL= "; COL%; " DIGIT= "; II; " POSITION= "; POSITION%

CTC(COL%, II) = CTC(COL%, II) + 1

POSITION% = INSTR(POSITION% + 1, YY$(COL%), DIG$)

LOOP

IF CTC(COL%, II) >= 1 THEN

'PRINT " COUNT= "; CTC(COL%, II); " FOR DIGIT= "; II

ELSE

58

'PRINT " COL= "; COL%; " CONTAINS NO "; II

END IF

NEXT II


NEXT COL%

RETURN

5000 REM SUB TO ASSIGN SEQUENTIAL VALUES TO ARRAY XXY() AND YYX()

FROM 1 TO 6 AND ZERO YYY$ ARRAY

FOR II = 1 TO 6

KK = 0

FOR JJ = 1 TO 6

KK = KK + 1

XXY(II, JJ) = KK: YYX(JJ, II) = KK: YYY$(II, JJ) = ""

'PRINT " XXY= "; XXY(II, JJ); " YYX= "; YYX(II, JJ); " II= "; II; " JJ= "; JJ


NEXT JJ

NEXT II

REM CHECK FOR GIVEN VALUES AND ASSIGN THEM TO 'A' ARRAY WHILE

SETTING THEIR XXY()'S AND YYX()S TO ZERO

REM THE ZERO VALUES OF XXY() AND YYX()SHOW THAT THE DIGIT HAS

ALREADY BEEN USED

FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6

GOSUB 40 'CONVERT ROW, COL TO K

X(K) = 0

IF ZX(ROW%, COL%) = 0 THEN GOTO 67

59

X(K) = ZX(ROW%, COL%): XXY(ROW%, ZX(ROW%, COL%)) = 0: YYX(COL%,

ZX(ROW%, COL%)) = 0

67 NEXT COL%

NEXT ROW%

RETURN

700 REM SUB TO CHECK FEASIBILITY OF X(K)

GOSUB 35 'CONVERT K TO ROW%, COL%

'PRINT: PRINT " K= "; K; " YYY$= "; YYY$(ROW%, COL%); " X(K)= "; X(K): INPUT "

PRESS ENTER TO CONTINUE", YES

UNIT$ = STR$(X(K)): UNIT$ = LTRIM$(UNIT$)

IF INSTR(YYY$(ROW%, COL%), UNIT$) 0 THEN GOTO 799

FLAG5 = 0

799 RETURN

10000 REM X(K) FEASIBILITY CHECK

CLS: FLAG = 0

FOR GRID% = 1 TO 6: XY$(GRID%) = "": YY$(GRID%) = "": XX$(GRID%) = "": NEXT

GRID%











60



'FOR I = 1 TO 6

'PRINT: 'PRINT " GRID= "; I; " XY$(I)= "; XY$(I)

'NEXT I



FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6




NEXT COL%

'PRINT " ROW%= "; ROW%; " ROW STRING= "; XX$(ROW%)

NEXT ROW%



FOR COL% = 1 TO 6

FOR ROW% = 1 TO 6

GOSUB 40



NEXT ROW%

'PRINT " COL%= "; COL%; " COLUMN STRING= "; YY$(COL%)

NEXT COL%


61

REM X(K) FEASIBILITY CHECK

FOR ROW% = 1 TO 6: FOR COL% = 1 TO 6: CT(ROW%, COL%) = 0: YYY$(ROW%,

COL%) = "": NEXT COL%: NEXT ROW%

FOR ROW% = 1 TO 6

FOR COL% = 1 TO 6

GOSUB 40 'CONVERT ROW, COL TO K

IF ZX(ROW%, COL%) 0 THEN GOTO 3401

FOR XL = 1 TO 6

IF XXY(ROW%, XL) = 0 THEN GOTO 3400

IF YYX(COL%, XL) = 0 THEN GOTO 3400

302 IF ROW% = 1 OR ROW% = 2 THEN GOTO 303

IF ROW% = 3 OR ROW% = 4 THEN GOTO 304

IF ROW% = 5 OR ROW% = 6 THEN GOTO 305

303 IF COL% >= 1 AND COL% = 4 AND COL% = 1 AND COL% = 4 AND COL%

62

305 IF COL% >= 1 AND COL% = 4 AND COL%

63

38 d = VAL(YYY$(ROW%, COL%)): ZX(ROW%, COL%) = d: XY(ROW%, d) = 0

YX(COL%, d) = 0: FLAG = 1

'PRINT: PRINT " NEW VAL FOR ROW= "; ROW%; " COL= "; COL%; " VAL= "; D


37 NEXT COL%

NEXT ROW%

3500 RETURN

999 END

64

Bibliography

Banbura, M. A. (2010, May). Maximum Likelihood Estimation of Factor Models on Data Sets

with Arbitrary Pattern of Missing Data. Social Science Research Network , Retrieved

Sept. 1, 2012 from SSRN Electronic Library (Working Paper Series No. 1189/May

2010:http://ssrn.com/abstract_id=1598302. Retrieved from

http://ssrn.com/abstract_id=1598302.

Isaacs, R. (1953). Optimal Horse Race Bets. American Math Monthly, 60, No. 5, 310-315.

Lashover, J. H. (2012, November 12). Monte Carlo Marching. Academia.edu and

ResearchGate.net.

Lashover, J. H. (2014, August 28). A Chemical Engineer Goes to the Horse Races. Retrieved

from www.academia.edu and www.ResearchGate.com.

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