The Museum of HP Calculators
The Museum of HP Calculators
This program is Copyright © 2004 by Jean-Marc Baillard and is used here by permission.
This program is supplied without representation or warranty of any kind. Jean-Marc Baillard and The Museum of HP Calculators therefore assume no responsibility and shall have no liability, consequential or otherwise, of any kind arising from the use of this program material or any part thereof.
-An hypermatrix [ ai,j,k ] ( 1<= i <= n ; 1 <=
j <= m ; 1 <= k <= p ) is actually equivalent to a p-list
of n x m matrices.
-Two programs are listed herafter:
1°) "HYPMAT" creates an n x m x p hypermatrix
provided its elements verify ai,j,k = f(i,j,k) where
f is a known function.
2°) "T" allows to compute the elements of a p x p x p magic
cube, where p is a prime > 2 ( and p < 999 )
-If p = 3 or 5 the
cube is an Andrews magic cube ( the sums equal the magic constant p(p3+1)
in the 3 directions and the 4 space diagonals )
-If p = 7
the cube is perfect pandiagonal ( all the orthogonal sections are panmagic
squares )
-If p = 11 , 13 , ... ,
we have a panmagic cube ( all the orthogonal or diagonal sections - broken
or unbroken - are magic squares )
1°) Creating an Hypermatrix
-The elements of the hypermatrix [ ai,j,k ] are stored
as shown below.
-If n.m.p > 318 , replace line 42 by VIEW X and the coefficients will
be displayed ( from the last to the first )
Data Registers: • R00 = n.mmm,ppp = n + m/1,000 + p/1,000,000 ( this register is to be initialized before executing "HYPMAT" )
and when the program stops:
R01 = a1,1,1 Rn+1
= a1,2,1 .......................................
Rnm-n+1 = a1,m,1
R02 = a2,1,1 Rn+2
= a2,2,1 .......................................
Rnm-n+2 = a2,m,1
..................................................................................................................
Rn = an,1,1
R2n = an,2,1 ........................................
Rnm = an,m,1
Rnm+1 = a1,1,2 Rnm+n+1
= a1,2,2 .......................................
R2nm-n+1 = a1,m,2
Rnm+2 = a2,1,2 Rnm+n+2
= a2,2,2 .......................................
R2nm-n+2 = a2,m,2
..................................................................................................................................
Rnm+n = an,1,2 Rnm+2n
= an,2,2 ........................................
R2nm = an,m,2
..........................................................................................................
..........................................................................................................
..........................................................................................................
Rnm(p-1)+1 = a1,1,p
Rnm(p-1)+n+1 = a1,2,p .......................................
Rnmp-n+1 = a1,m,p
Rnm(p-1)+2 = a2,1,p
Rnm(p-1)+n+2 = a2,2,p .......................................
Rnmp-n+2 = a2,m,p
....................................................................................................................................................
Rnm(p-1)+n = an,1,p
Rnm(p-1)+2n = an,2,p ........................................
Rnmp = an,m,p
Flag: /
Z = k
Subroutine: a program called "T" ( global
label ) changing the stack from Y = j
to X' = ai,j,k without disturbing synthetic
register M
X = i
-In other words, it must calculate ai,j,k
= f(i,j,k) assuming i , j , k are in registers X , Y , Z (
respectively ) upon entry.
01 LBL "HYPMAT"
02 RCL 00
03 INT
04 LASTX
05 FRC
06 E3
07 *
08 INT
09 *
10 STO 01
11 RCL 00
12 E6
13 ST* Y
14 SQRT
15 MOD
16 *
17 STO M
18 LBL 01
19 RCL M
20 1
21 -
22 STO Y
23 RCL 01
24 /
25 INT
26 RCL Y
27 RCL 01
28 MOD
29 RCL 00
30 INT
31 /
32 INT
33 R^
34 RCL 00
35 INT
36 MOD
37 1
38 ST+ T
39 ST+ Z
40 +
41 XEQ "T"
42 STO IND M
43 DSE M
44 GTO 01
45 CLA
this line is superfluous if the subroutine doesn't disturb registers N
& O.
46 END
( 71 bytes / SIZE nmp+1 )
Example: n = 4 ; m = 5 ; p = 3 and ai,j,k = ( i2 + j3 + k4 ) modulo (pi)
-Load this short routine in program memory:
LBL "T"
X^2
X<>Y
ENTER^
X^2
*
+
X<>Y
X^2
X^2
+
PI
MOD
RTN
-Then 4.005003 STO 00 XEQ "HYPMAT"
yields ( in 75 seconds )
R01 = 3 ; R02 = 2.8584 ; R03 = 1.5752 ; ..... ; R21 = 2.2920
; .....; R41 = 1.3186 ; ..... ; R60 = 2.0885
2°) Magic Cubes ( of order p , assuming p is a
prime and p > 2 )
-The following program calculates the elements of a magic cube
ai,j,k assuming i , j , k are in registers X , Y
, Z ( respectively ) upon entry.
Data Registers:
• R00 = p.ppp,ppp ( R00 is to be initialized
)
R01 to Rp3 = the coefficents of the magic cube
Flags: /
Subroutines: /
01 LBL "T"
02 1
03 ST- T
04 ST- Z
05 -
06 STO N
07 RDN
08 STO O
09 RCL 00
10 INT
11 7
12 -
13 X<0?
14 GTO 01
15 X=0?
16 GTO 02
17 X<> T
18 ST+ X
19 +
20 ST+ X
21 RCL N
22 +
23 RCL X
24 RCL Z
25 +
26 RCL X
27 RCL O
28 +
29 GTO 03
30 LBL 01
31 X<> T
32 +
33 RCL N
34 ST- Y
35 ST+ X
36 RCL Y
37 +
38 RCL O
39 ST+ X
40 CHS
41 X<>Y
42 ST+ Y
43 R^
44 ST+ X
45 -
46 GTO 03
47 LBL 02
48 CLX
49 RCL N
50 +
51 ST+ X
52 +
53 CHS
54 RCL N
55 CHS
56 X<>Y
57 ST+ Y
58 RCL O
59 CHS
60 X<>Y
61 ST+ Y
62 R^
63 ST+ X
64 -
65 RCL N
66 +
67 LBL 03
68 RCL 00
69 INT
70 MOD
71 X<>Y
72 LASTX
73 ST* Z
74 MOD
75 +
76 X<>Y
77 RCL 00
78 INT
79 ST* Z
80 MOD
81 +
82 1
83 +
84 END
( 125 bytes )
Example: 3.003003
STO 00 XEQ "HYPMAT" stores the following Andrews magic cube
1
23
18
R01
R10
R19
17
3
22
R04
R13
R22
15
24
7 16
20 2
R02 R07
R11 R16
R20 R25
19
14
9
in registers
R05
R14
R23
respectively.
26
8
12 21
4 13
R03 R08
R12 R17
R21 R26
6
25
11
R06
R15
R24
10
5
27
R09
R18
R27
-As you can check, the sums equal 42 in the x- , y- , z- directions
and the space diagonals 1 + 14 + 27 = 26 + 14 + 2 = 10 + 14 +
18 = 24 + 14 + 4 = 42
Notes:
-If p = 7 , 11 , 13 , 17 , .... , replace line 42 by VIEW X in the "HYPMAT"
listing to see the successive coefficients of the magic cube.
-A perfect magic cube of order 5 has been discovered last year, but
this program cannot calculate its elements.
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