Magic Multiplication for the HP-41

Overview
 

-This program takes 2 magic squares and calculates a "product" of these matrices that is also a magic square.
 

  Formula:     A is an mxm matrix   B is an nxn matrix  then  C = A*B = [ C_i,j ]  is defined as

   C_i,j = n² A + b_i,j 1_mxm          where       1_mxm  is the mxm matrix whose all elements equal 1

-So,  C_i,j  is also an  mxm matrix  and  C is an mnxmn matrix built with n² submatrices C_i,j
 
 

Program Listing
 
 

Data Registers:              R00 = n²               ( Registers Rbb thru Ree & Rbb' thru Ree' are to be initialized before executing "MG*MG" )

                                      •  Rbb .................... •  Ree = A
                                      •  Rbb' ................... •  Ree' = B

>>> When the program stops,   RBB .................... •  REE = C         IF   CF 02

Flag:  F02                        CF 02  <->  the coefficients of C are stored in RBB thru REE
                                         SF 02   <->  the coefficients of C are stored in a DATA file which must be the current file
Subroutines: /
 
 
 

01  LBL "MG*MG"  02  FS? 02  03  SEEKPT  04  STO O              05  RDN  06  STO N  07  FRC  08  ISG X  09  INT  10  STO 00  11  ST* 00  12  STO Q  13  X<>Y  14  STO M  15  FRC  16  ISG X  17  INT

18  STO P              19  ENTER  20  LBL 01  21  RCL IND M  22  SIGN  23  CLX  24  RCL 00  25  ST* L  26  CLX  27  RCL IND N  28  ST+ L  29  X<> L  30  FC? 02  31  STO IND O  32  FS? 02  33  SAVEX  34  CLX

35  SIGN  36  ST+ M  37  ST+ O  38  RDN  39  DSE P  40  GTO 01  41  R^  42  ST+ N  43  RDN  44  STO P              45  ST- M  46  DSE Q  47  GTO 01  48  ST+ M  49  RCL 00  50  SQRT  51  STO Q

52  ST- N  53  RDN  54  DSE Y  55  GTO 01  56  R^  57  ST+ N  58  RDN  59  STO P              60  STO Y  61  X^2  62  ST- M  63  X<> L  64  DSE Z  65  GTO 01  66  RCL 00  67  SQRT  68  *

69  ENTER  70  X^2  71  CHS  72  RCL O              73  ST+ Y  74  DSE X  75   E3  76  /  77  +  78  X<>Y  79   E5  80  /  81  +  82  CLA  83  END

 
     ( 138 bytes / SIZE ??? )
 
 

STACK	INPUTS	OUTPUTS
Z	bbb.eee,mm	/
Y	bbb'.eee',nn	/
X	BBB	BBB.EEE,NN

    with  N = m x n

Example1:    With the following 4x4 magic square

   01   14   04   15     store these 16 numbers      R01   R05   R09   R13
   08   11   05   10                                              R02   R06   R10   R14
   13   02   16   03              in registers                R03   R07   R11   R15            respectively:           control number = 1.01604
   12   07   09   06                                              R04   R08   R12   R16

-Let's calculate  A*A,  we can choose  BBB = 17

  CF 02  1.01604   ENTER^   ENTER^   17   XEQ "MG*MG"  >>>>   17.27216                ---Eecution time = 2m29s---

-And we get in R17 thru R272 the following magic square

 

17	225	65	241	30	238	78	254	20	228	68	244	31	239	79	255
129	177	81	161	142	190	94	174	132	180	84	164	143	191	95	175
209	33	257	49	222	46	270	62	212	36	260	52	223	47	271	63
193	113	145	97	206	126	158	110	196	116	148	100	207	127	159	111
24	232	72	248	27	235	75	251	21	229	69	245	26	234	74	250
136	184	88	168	139	187	91	171	133	181	85	165	138	186	90	170
216	40	264	56	219	43	267	59	213	37	261	53	218	42	266	58
200	120	152	104	203	123	155	107	197	117	149	101	202	122	154	106
29	237	77	253	18	226	66	242	32	240	80	256	19	227	67	243
141	189	93	173	130	178	82	162	144	192	96	176	131	179	83	163
221	45	269	61	210	34	258	50	224	48	272	64	211	35	259	51
205	125	157	109	194	114	146	98	208	128	160	112	195	115	147	99
28	236	76	252	23	231	71	247	25	233	73	249	22	230	70	246
140	188	92	172	135	183	87	167	137	185	89	169	134	182	86	166
220	44	268	60	215	39	263	55	217	41	265	57	214	38	262	54
204	124	156	108	199	119	151	103	201	121	153	105	198	118	150	102

 
-Here, the magic constant = 2312
-But if you subtract 16 to all these elements, you get a normal magic square, with all the integers between 1 and 256, magic constant = 2056

-Since A is a panmagic square,  C is a panmagic square too !
 

Example2:    With the same 4x4 matrix as above and the 5x5 matrix below

   01   14   22   10   18
   07   20   03   11   24
   13   21   09   17   05    that we can store, in column order, in  R21 thru R45, control number = 21.04505
   19   02   15   23   06
   25   08   16   04   12

-The "product" C = A*B is a 20x20 matrix, so there is not enough room in main memory:
-Create a DATA file, size = 401

                              SF 02
              1.01604   ENTER^
            21.04505   ENTER^
                   1         XEQ "MG*MG"  >>>>   1.40020                             ---Execution time = 4m25s---

-And the data file contains the 20x20 magic square:
 
 

26	351	101	376	39	364	114	389	47	372	122	397	35	360	110	385	43	368	118	393
201	276	126	251	214	289	139	264	222	297	147	272	210	285	135	260	218	293	143	268
326	51	401	76	339	64	414	89	347	72	422	97	335	60	410	85	343	68	418	93
301	176	226	151	314	189	239	164	322	197	247	172	310	185	235	160	318	193	243	168
32	357	107	382	45	370	120	395	28	353	103	378	36	361	111	386	49	374	124	399
207	282	132	257	220	295	145	270	203	278	128	253	211	286	136	261	224	299	149	274
332	57	407	82	345	70	420	95	328	53	403	78	336	61	411	86	349	74	424	99
307	182	232	157	320	195	245	170	303	178	228	153	311	186	236	161	324	199	249	174
38	363	113	388	46	371	121	396	34	359	109	384	42	367	117	392	30	355	105	380
213	288	138	263	221	296	146	271	209	284	134	259	217	292	142	267	205	280	130	255
338	63	413	88	346	71	421	96	334	59	409	84	342	67	417	92	330	55	405	80
313	188	238	163	321	196	246	171	309	184	234	159	317	192	242	167	305	180	230	155
44	369	119	394	27	352	102	377	40	365	115	390	48	373	123	398	31	356	106	381
219	294	144	269	202	277	127	252	215	290	140	265	223	298	148	273	206	281	131	256
344	69	419	94	327	52	402	77	340	65	415	90	348	73	423	98	331	56	406	81
319	194	244	169	302	177	227	152	315	190	240	165	323	198	248	173	306	181	231	156
50	375	125	400	33	358	108	383	41	366	116	391	29	354	104	379	37	362	112	387
225	300	150	275	208	283	133	258	216	291	141	266	204	279	129	254	212	287	137	262
350	75	425	100	333	58	408	83	341	66	416	91	329	54	404	79	337	62	412	87
325	200	250	175	308	183	233	158	316	191	241	166	304	179	229	154	312	187	237	162

 
-Here, the magic constant = 4510
-But if you subtract 25 to all these elements, you get a normal magic square, with all the integers between 1 and 400, magic constant = 4010

-Since A is a panmagic square,  C is a panmagic square too !

-Here is a short routine to visualize the content of a data file:
 
 
 

01  LBL "VWD"  02  SEEKPT  03   E3  04  *  05  INT  06   E3  07  /  08  SIGN  09  LBL 01  10  GETX  11  VIEW X  12  FC? 21  13  PSE  14  ISG L  15  GTO 01  16  END

 
-In the example above, key in

    1.40020   XEQ "VWD"

  or simply   1.400   XEQ "VWD"
 

Notes:

-"MG*MG"  uses the synthetic registers P & Q
-So, don't interrupt this routine: it would change the content of these tegisters.

-P & Q may also be replaced by R01 & R02 ( in this case, all  bbb > 002 )
-Thus, you could replace line 31 by VIEW X  if  the matrix C has more than 600 elements.
 ( VIEW X  also changes registers P & Q )
 
 

Reference:

[1] Yangkok Kim, Jaechil Yoo  - "An algorithm for constructing magic squares" - Discrete Applied Mathematics 156 (2008) 2804–2809