' Pan5x5.bas hh Mar. 31/01 ' Convert an 'essentially different order-5 pandiagonal magic square to 3 ' other squares by 1-3-5-2-4 row/column permutations, then each of these ' to another square by diagonal to column transformation. ' Then obtain 24 other squares from each of these 4 by cyclical transformation ' ' The resulting solutions are saved in sequential file pan5x5.dat. ' I then open this mixed up text file in MS Word, put it into a table ' sort it (3 columns at a time starting from the right, and save it as ' Pan5x5-3600 B.doc ' ' You can also generate just the 4 variations of each of the 36 magic squares as ' pan5x5s.dat. Open it in Word, put in a table & sort as above. ' I then edit the file back to comma delimited text and save it as Pan5x5sl.txt ' Renaming it to Pan5x5sl.dat makes it readable in Pan5Read.bas CONST false = 0, true = NOT false DEFINT A-Z DECLARE SUB r2left () ' move a column from the right to the left DECLARE SUB b2top () ' move a row from the bottom to the top DECLARE SUB reflect () ' reflect the magic square horizontally DECLARE SUB rotate () ' rotate the square counterclockwise DECLARE SUB checksqr () DECLARE SUB cyclic () ' cyclic through the 25 transformations, ' normalizing (as necessary), and saving DECLARE SUB pan144 () ' normalize (as necessary), and save DIM sqr1(26) ' hold the original DIM sqr2(26) ' hold square after diag to row DIM sqr3(26) ' hold square after 1-3-5-2-4- DIM sqr4(26) ' hold square after diag to row DIM sqn(26) ' holds current cyclical permutated square DIM sqt(26) ' holds temporary square DIM sqt2(26) ' holds temporary square in S. R. COMMON SHARED cntr, sqn(), sqt(), sqt2() ' this array is used also in S.R. cntr = 1 ' temporary only, to test that count actually ' equals 3600, but it spoils formatting CLS PRINT " Pan5x5.bas hh Mar. 31/01" PRINT "Convert an 'essentially different order-5 pandiagonal magic square to 3" PRINT "other squares by 1-3-5-2-4 row/column permutations, then each of these" PRINT "to another square by diagonal to column transformation." PRINT "Then obtain 24 other squares from each of these 4 by cyclical transformation" PRINT PRINT "The resulting solutions are saved in sequential file pan5x5.dat." PRINT "I then open this mixed up text file in MS Word, put it into a table" PRINT "sort it (3 columns at a time starting from the right, and save it as" PRINT "Pan5x5-3600.doc" PRINT PRINT "You can also generate just the 4 variations of each of the 36 magic squares as" PRINT "pan5x5s.dat. Open it in Word, put in a table & sort as above." PRINT "I then edit the file back to comma delimited text and save it as Pan5x5sl.txt" PRINT "Renaming it to Pan5x5sl.dat makes it readable in Pan5Read.bas" PRINT : INPUT "Do you wish to generate only the 144 squares (4 x 36).(y/Y. anything else is no)"; flag\$ IF flag\$ = "y" OR flag\$ = "Y" THEN flg = 1 'IF flg THEN ' PRINT "144" 'ELSE ' PRINT "3600" 'END IF IF flg THEN ' open the correct data file OPEN "pan5x5s.dat" FOR OUTPUT AS #1 ' the output data file for the short list ELSE OPEN "pan5x5.dat" FOR OUTPUT AS #1 ' the output data file for the complete list END IF FOR essential = 1 TO 36 ' *** start of main loop *** FOR i = 1 TO 25 ' put an essentially different square from READ sqr1(i) ' data statement into Sqr1 NEXT IF flg THEN FOR k = 1 TO 25 ' put Sqn into Sqt for normalizing & print sqt(k) = sqr1(k) NEXT pan144 ' call S.R. to normalize (if necessary) and save END IF ' ---- convert (original) sqr1 diagonal to column and put in Sqr2 ---- sqr2(1) = sqr1(1): sqr2(2) = sqr1(7): sqr2(3) = sqr1(13): sqr2(4) = sqr1(19) sqr2(5) = sqr1(25): sqr2(6) = sqr1(10): sqr2(7) = sqr1(11): sqr2(8) = sqr1(17) sqr2(9) = sqr1(23): sqr2(10) = sqr1(4): sqr2(11) = sqr1(14): sqr2(12) = sqr1(20) sqr2(13) = sqr1(21): sqr2(14) = sqr1(2): sqr2(15) = sqr1(8): sqr2(16) = sqr1(18) sqr2(17) = sqr1(24): sqr2(18) = sqr1(5): sqr2(19) = sqr1(6): sqr2(20) = sqr1(12) sqr2(21) = sqr1(22): sqr2(22) = sqr1(3): sqr2(23) = sqr1(9): sqr2(24) = sqr1(15) sqr2(25) = sqr1(16) IF flg THEN FOR k = 1 TO 25 ' put Sqr2 into Sqt for normalizing & print sqt(k) = sqr2(k) NEXT pan144 ' call S.R. to normalize (if necessary) and save END IF ' ---- convert Sqr1 row/column 1-3-5-2-4 and put in Sqr3 ---- sqr3(1) = sqr1(1): sqr3(2) = sqr1(3): sqr3(3) = sqr1(5): sqr3(4) = sqr1(2) sqr3(5) = sqr1(4): sqr3(6) = sqr1(11): sqr3(7) = sqr1(13): sqr3(8) = sqr1(15) sqr3(9) = sqr1(12): sqr3(10) = sqr1(14): sqr3(11) = sqr1(21): sqr3(12) = sqr1(23) sqr3(13) = sqr1(25): sqr3(14) = sqr1(22): sqr3(15) = sqr1(24): sqr3(16) = sqr1(6) sqr3(17) = sqr1(8): sqr3(18) = sqr1(10): sqr3(19) = sqr1(7): sqr3(20) = sqr1(9) sqr3(21) = sqr1(16): sqr3(22) = sqr1(18): sqr3(23) = sqr1(20): sqr3(24) = sqr1(17) sqr3(25) = sqr1(19) IF flg THEN FOR k = 1 TO 25 ' put Sqr3 into Sqt for normalizing & print sqt(k) = sqr3(k) NEXT pan144 ' call S.R. to normalize (if necessary) and save END IF ' ---- convert sqr3 diagonal to column and put in Sqr4 ---- sqr4(1) = sqr3(1): sqr4(2) = sqr3(7): sqr4(3) = sqr3(13): sqr4(4) = sqr3(19) sqr4(5) = sqr3(25): sqr4(6) = sqr3(10): sqr4(7) = sqr3(11): sqr4(8) = sqr3(17) sqr4(9) = sqr3(23): sqr4(10) = sqr3(4): sqr4(11) = sqr3(14): sqr4(12) = sqr3(20) sqr4(13) = sqr3(21): sqr4(14) = sqr3(2): sqr4(15) = sqr3(8): sqr4(16) = sqr3(18) sqr4(17) = sqr3(24): sqr4(18) = sqr3(5): sqr4(19) = sqr3(6): sqr4(20) = sqr3(12) sqr4(21) = sqr3(22): sqr4(22) = sqr3(3): sqr4(23) = sqr3(9): sqr4(24) = sqr3(15) sqr4(25) = sqr3(16) IF flg THEN FOR k = 1 TO 25 ' put Sqr4 into Sqt for normalizing & print sqt(k) = sqr4(k) NEXT pan144 ' call S.R. to normalize (if necessary) and save END IF IF flg THEN GOTO again ' end the program ' --step through 25 cyclical transformations - first Sqr1, then sqr2, sqr3 and sqr4 -- ' Sqn() has to be preserved for all cyclical transformations. ' So change to Sqt() for normalizing and printing the 25 individual squares FOR i = 1 TO 25 ' first put Sqr1 into Sqn sqn(i) = sqr1(i) NEXT cyclic ' call cyclic transformations S. R. FOR i = 1 TO 25 ' put Sqr2 into Sqn sqn(i) = sqr2(i) NEXT cyclic ' cyclic also handles normalization FOR i = 1 TO 25 ' put Sqr3 into Sqn sqn(i) = sqr3(i) NEXT cyclic ' cyclic also handles printing & file output FOR i = 1 TO 25 ' put Sqr4 into Sqn sqn(i) = sqr4(i) NEXT cyclic ' call cyclic transformations S. R. again: NEXT essential ' go for rest of the 36 squares CLOSE #1 END DATA 1,7,13,19,25,14,20,21,2,8,22,3,9,15,16,10,11,17,23,4,18,24,5,6,12 DATA 1,7,13,20,24,15,19,21,2,8,22,3,10,14,16,9,11,17,23,5,18,25,4,6,12 DATA 1,7,13,24,20,14,25,16,2,8,17,3,9,15,21,10,11,22,18,4,23,19,5,6,12 DATA 1,7,13,25,19,15,24,16,2,8,17,3,10,14,21,9,11,22,18,5,23,20,4,6,12 DATA 1,7,14,18,25,13,20,21,2,9,22,4,8,15,16,10,11,17,24,3,19,23,5,6,12 DATA 1,7,14,20,23,15,18,21,2,9,22,4,10,13,16,8,11,17,24,5,19,25,3,6,12 DATA 1,7,14,23,20,13,25,16,2,9,17,4,8,15,21,10,11,22,19,3,24,18,5,6,12 DATA 1,7,14,25,18,15,23,16,2,9,17,4,10,13,21,8,11,22,19,5,24,20,3,6,12 DATA 1,7,15,18,24,13,19,21,2,10,22,5,8,14,16,9,11,17,25,3,20,23,4,6,12 DATA 1,7,15,19,23,14,18,21,2,10,22,5,9,13,16,8,11,17,25,4,20,24,3,6,12 DATA 1,7,15,23,19,13,24,16,2,10,17,5,8,14,21,9,11,22,20,3,25,18,4,6,12 DATA 1,7,15,24,18,14,23,16,2,10,17,5,9,13,21,8,11,22,20,4,25,19,3,6,12 DATA 1,7,18,14,25,13,24,5,6,17,10,16,12,23,4,22,3,9,20,11,19,15,21,2,8 DATA 1,7,18,15,24,13,25,4,6,17,9,16,12,23,5,22,3,10,19,11,20,14,21,2,8 DATA 1,7,18,24,15,19,25,11,2,8,12,3,9,20,21,10,16,22,13,4,23,14,5,6,17 DATA 1,7,18,25,14,20,24,11,2,8,12,3,10,19,21,9,16,22,13,5,23,15,4,6,17 DATA 1,7,19,13,25,14,23,5,6,17,10,16,12,24,3,22,4,8,20,11,18,15,21,2,9 DATA 1,7,19,15,23,14,25,3,6,17,8,16,12,24,5,22,4,10,18,11,20,13,21,2,9 DATA 1,7,19,23,15,18,25,11,2,9,12,4,8,20,21,10,16,22,14,3,24,13,5,6,17 DATA 1,7,19,25,13,20,23,11,2,9,12,4,10,18,21,8,16,22,14,5,24,15,3,6,17 DATA 1,7,20,13,24,15,23,4,6,17,9,16,12,25,3,22,5,8,19,11,18,14,21,2,10 DATA 1,7,20,14,23,15,24,3,6,17,8,16,12,25,4,22,5,9,18,11,19,13,21,2,10 DATA 1,7,20,23,14,18,24,11,2,10,12,5,8,19,21,9,16,22,15,3,25,13,4,6,17 DATA 1,7,20,24,13,19,23,11,2,10,12,5,9,18,21,8,16,22,15,4,25,14,3,6,17 DATA 1,7,23,14,20,13,19,5,6,22,10,21,12,18,4,17,3,9,25,11,24,15,16,2,8 DATA 1,7,23,15,19,13,20,4,6,22,9,21,12,18,5,17,3,10,24,11,25,14,16,2,8 DATA 1,7,23,19,15,18,14,5,6,22,10,21,17,13,4,12,3,9,25,16,24,20,11,2,8 DATA 1,7,23,20,14,18,15,4,6,22,9,21,17,13,5,12,3,10,24,16,25,19,11,2,8 DATA 1,7,24,13,20,14,18,5,6,22,10,21,12,19,3,17,4,8,25,11,23,15,16,2,9 DATA 1,7,24,15,18,14,20,3,6,22,8,21,12,19,5,17,4,10,23,11,25,13,16,2,9 DATA 1,7,24,18,15,19,13,5,6,22,10,21,17,14,3,12,4,8,25,16,23,20,11,2,9 DATA 1,7,24,20,13,19,15,3,6,22,8,21,17,14,5,12,4,10,23,16,25,18,11,2,9 DATA 1,7,25,13,19,15,18,4,6,22,9,21,12,20,3,17,5,8,24,11,23,14,16,2,10 DATA 1,7,25,14,18,15,19,3,6,22,8,21,12,20,4,17,5,9,23,11,24,13,16,2,10 DATA 1,7,25,18,14,20,13,4,6,22,9,21,17,15,3,12,5,8,24,16,23,19,11,2,10 DATA 1,7,25,19,13,20,14,3,6,22,8,21,17,15,4,12,5,9,23,16,24,18,11,2,10 SUB b2top ' ----- move a row from bottom to top ----- sqt(1) = sqn(21): sqt(2) = sqn(22): sqt(3) = sqn(23): sqt(4) = sqn(24) sqt(5) = sqn(25): sqt(6) = sqn(1): sqt(7) = sqn(2): sqt(8) = sqn(3) sqt(9) = sqn(4): sqt(10) = sqn(5): sqt(11) = sqn(6): sqt(12) = sqn(7) sqt(13) = sqn(8): sqt(14) = sqn(9): sqt(15) = sqn(10): sqt(16) = sqn(11) sqt(17) = sqn(12): sqt(18) = sqn(13): sqt(19) = sqn(14): sqt(20) = sqn(15) sqt(21) = sqn(16): sqt(22) = sqn(17): sqt(23) = sqn(18): sqt(24) = sqn(19) sqt(25) = sqn(20) FOR k = 1 TO 25 ' now put sqt back into sqn sqn(k) = sqt(k) NEXT END SUB SUB checksqr 'This sub routine checks 8 lines of the square for magic ' It is actually to test for typos in data statements. L1 = sqt(1) + sqt(2) + sqt(3) + sqt(4) + sqt(5) L2 = sqt(6) + sqt(7) + sqt(8) + sqt(9) + sqt(10) L3 = sqt(11) + sqt(12) + sqt(13) + sqt(14) + sqt(15) L4 = sqt(16) + sqt(17) + sqt(18) + sqt(19) + sqt(20) L5 = sqt(21) + sqt(22) + sqt(23) + sqt(24) + sqt(25) l6 = sqt(1) + sqt(7) + sqt(13) + sqt(19) + sqt(25) ' leading diagonal L7 = sqt(1) + sqt(6) + sqt(11) + sqt(16) + sqt(21) ' 1st column L8 = sqt(3) + sqt(8) + sqt(13) + sqt(18) + sqt(23) ' 3rd column IF L1 + L2 + L3 + L4 + L5 + l6 + L7 + L8 <> 520 THEN STOP ' I didn't bother with other 4 lines END SUB SUB cyclic ' Cycle through row and column transformations by moving columns from right hand ' side to left side. Then move a row from bottom to top, and repeat column moves, etc. ' This S. R. also looks after normalization and printing of the 25 panmagic squares. ' Magic squares enter cyclic S.R. in sqn() ' Within cyclic, they enter b2top & r2left S.R. in sqn(), but for normalizing and ' printing they are put into sqt(). FOR i = 1 TO 5 ' move a row from the bottom to the top FOR j = 1 TO 5 ' move a column from the right to the left r2left ' call column exchange S. R. FOR k = 1 TO 25 ' put Sqn into Sqt for normalizing & print sqt(k) = sqn(k) NEXT DO IF sqt(1) < sqt(5) AND sqt(1) < sqt(21) AND sqt(1) < sqt(25) THEN EXIT DO rotate LOOP IF sqt(2) > sqt(6) THEN reflect ' reflect if necessary PRINT cntr; ' solution counter (screen only) FOR cell = 1 TO 25 ' write to screen and file PRINT USING "###"; sqt(cell); PRINT #1, sqt(cell); NEXT cntr = cntr + 1: PRINT : PRINT #1, ' increment counter and do a line feed NEXT j ' move next column b2top ' call row exchange S. R. NEXT i END SUB SUB pan144 'This S.R. generates only the 3 squares derived from each of the 36 "essentially" ' different" squares, normalizes as necessary, and prints to data file P144.dat ' FOR k = 1 TO 25 ' put Sqn into Sqt for normalizing & print ' sqt(k) = sqn(k) ' NEXT DO IF sqt(1) < sqt(5) AND sqt(1) < sqt(21) AND sqt(1) < sqt(25) THEN EXIT DO rotate LOOP IF sqt(2) > sqt(6) THEN reflect ' reflect if necessary PRINT cntr; : PRINT #1, cntr; ' solution counter FOR cell = 1 TO 25 ' write to screen and file PRINT USING "###"; sqt(cell); PRINT #1, sqt(cell); NEXT cntr = cntr + 1: PRINT : PRINT #1, ' increment counter and do a line feed END SUB SUB r2left ' ----- move a column from right to left ----- sqt(1) = sqn(5): sqt(2) = sqn(1): sqt(3) = sqn(2): sqt(4) = sqn(3) sqt(5) = sqn(4): sqt(6) = sqn(10): sqt(7) = sqn(6): sqt(8) = sqn(7) sqt(9) = sqn(8): sqt(10) = sqn(9): sqt(11) = sqn(15): sqt(12) = sqn(11) sqt(13) = sqn(12): sqt(14) = sqn(13): sqt(15) = sqn(14): sqt(16) = sqn(20) sqt(17) = sqn(16): sqt(18) = sqn(17): sqt(19) = sqn(18): sqt(20) = sqn(19) sqt(21) = sqn(25): sqt(22) = sqn(21): sqt(23) = sqn(22): sqt(24) = sqn(23) sqt(25) = sqn(24) FOR k = 1 TO 25 ' now put sqt back into sqn sqn(k) = sqt(k) NEXT END SUB SUB reflect ' this subroutine is called if it is necessary to reflect the magic square ' it reflects around the leading diagonal sqt2(1) = sqt(1): sqt2(2) = sqt(6): sqt2(3) = sqt(11): sqt2(4) = sqt(16) sqt2(5) = sqt(21): sqt2(6) = sqt(2): sqt2(7) = sqt(7): sqt2(8) = sqt(12) sqt2(9) = sqt(17): sqt2(10) = sqt(22): sqt2(11) = sqt(3): sqt2(12) = sqt(8) sqt2(13) = sqt(13): sqt2(14) = sqt(18): sqt2(15) = sqt(23): sqt2(16) = sqt(4) sqt2(17) = sqt(9): sqt2(18) = sqt(14): sqt2(19) = sqt(19): sqt2(20) = sqt(24) sqt2(21) = sqt(5): sqt2(22) = sqt(10): sqt2(23) = sqt(15): sqt2(24) = sqt(20) sqt2(25) = sqt(25) FOR k = 1 TO 25 ' now put sqt2 back into sqt sqt(k) = sqt2(k) NEXT checksqr ' check that this is magic ' I put it in to check for typos END SUB SUB rotate ' this subroutine is called if it is necessary to rotate the magic square ' it rotates counter-clockwise. It may need to be called up to 3 times. sqt2(1) = sqt(5): sqt2(2) = sqt(10): sqt2(3) = sqt(15): sqt2(4) = sqt(20) sqt2(5) = sqt(25): sqt2(6) = sqt(4): sqt2(7) = sqt(9): sqt2(8) = sqt(14) sqt2(9) = sqt(19): sqt2(10) = sqt(24): sqt2(11) = sqt(3): sqt2(12) = sqt(8) sqt2(13) = sqt(13): sqt2(14) = sqt(18): sqt2(15) = sqt(23): sqt2(16) = sqt(2) sqt2(17) = sqt(7): sqt2(18) = sqt(12): sqt2(19) = sqt(17): sqt2(20) = sqt(22) sqt2(21) = sqt(1): sqt2(22) = sqt(6): sqt2(23) = sqt(11): sqt2(24) = sqt(16) sqt2(25) = sqt(21) FOR k = 1 TO 25 ' now put sqt2 back into sqt sqt(k) = sqt2(k) NEXT checksqr ' put in to check for data statement typos END SUB