STEP1 : Arrange 10 distinct digits and 2 blank spaces into 3x4 matrix.
Example:
STEP2 : Now sum up the numbers row by row (in our example SumRows=12+8+4567+3+9=4599).
STEP3 : Next sum up the numbers column by column (in our example SumCols=143+25+60+879=1107).
STEP4 : Evaluate the ratio R= (bigger sum)/ (smaller sum).
In our example 4599/1107=4.14543
You are requested to arrange the 12 symbols so that the value of R will be as close as possible to:
a) 1.000
b) 2.000
c) Phi (the golden ratio =1.618034)
QB64 enabled this to run in under 1/2 hour:
DECLARE FUNCTION total# (x$)
DEFDBL A-Z
DECLARE SUB permute (a$)
OPEN "closer3.txt" FOR OUTPUT AS #2
CLS
a$ = "0123456789 ": h$ = a$
best1diff = .01: best2diff = .01: bestphidiff = .01
phi = (1 + SQR(5)) / 2
DO
b$ = MID$(a$, 1, 1) + MID$(a$, 5, 1) + MID$(a$, 9, 1) + " "
b$ = b$ + MID$(a$, 2, 1) + MID$(a$, 6, 1) + MID$(a$, 10, 1) + " "
b$ = b$ + MID$(a$, 3, 1) + MID$(a$, 7, 1) + MID$(a$, 11, 1) + " "
b$ = b$ + MID$(a$, 4, 1) + MID$(a$, 8, 1) + MID$(a$, 12, 1) + " "
a2$ = MID$(a$, 1, 4) + " " + MID$(a$, 5, 4) + " " + MID$(a$, 9, 4) + " "
tot1 = total(a2$)
tot2 = total(b$)
rat = tot1 / tot2: IF rat < 1 THEN rat = 1 / rat
diff1 = ABS(1 - rat): diff2 = ABS(2 - rat): diffphi = ABS(phi - rat)
IF diff1 <= best1diff OR diff2 <= best2diff OR diffphi <= bestphidiff THEN
IF diff1 < best1diff THEN best1diff = diff1: PRINT rat, diff1
IF diff2 < best2diff THEN best2diff = diff2: PRINT rat, diff2
IF diffphi < bestphidiff THEN bestphidiff = diffphi: PRINT rat, diffphi
PRINT #2, a$; "; "; rat
END IF
prv$ = MID$(a$, 1, 1)
permute a$
IF MID$(a$, 1, 1) <> prv$ THEN PRINT a$
LOOP UNTIL a$ = h$
CLOSE 2
DEFDBL A-Z
FUNCTION total (x$)
x2$ = x$
t = 0
DO
ix = INSTR(x2$, " ")
t = t + VAL(MID$(x2$, 1, ix - 1))
x2$ = LTRIM$(MID$(x2$, ix))
LOOP UNTIL x2$ = ""
total = t
END FUNCTION
Out of the 12!/2 (= 239,500,800) permutations of the digits and spaces, there are many, many ways of getting exactly 1 or 2 (149,516 and 208,600 ways, respectively), such as
For 1:
134
9 87
0526
278
4 69
1350
70 9
3 42
1586
For 2:
1 69
350
2784
278
4 69
3051
859
024
3671
The 116 ways of getting 1.618025751072961, the closest possible to phi (~= 1.618033988749895) are:
1857
602
934
2397
586
410
2397
856
140
2479
053
861
2497
035
861
2586
397
410
2586
793
014
2793
586
014
2856
397
140
2965
341
087
3279
5 68
0 41
3279
5 68
1 40
3279
8 65
0 41
3279
8 65
1 40
3297
5 86
1 04
3297
5 86
4 01
3297
6 85
1 04
3297
6 85
4 01
397
586
2410
397
856
2140
4 97
3205
1 86
4 97
3205
6 81
479
053
2861
479
2061
853
479
2063
851
497
035
2861
497
2061
835
497
2065
831
497
2510
386
497
2516
380
5 68
3279
0 41
5 68
3279
1 40
5 69
0 71
3248
5 69
1 70
3248
5 69
3241
0 78
5 69
3241
8 70
5 78
3214
0 96
5 78
3214
6 90
5 78
3241
0 69
5 78
3241
9 60
5 86
3297
1 04
5 86
3297
4 01
5 96
1 07
3284
5 96
3214
0 78
5 96
3214
8 70
5 96
7 01
3284
586
397
2410
586
793
2014
6 59
3241
0 87
6 59
3241
7 80
6 85
3297
1 04
6 85
3297
4 01
6 95
1 07
3284
6 95
3214
0 78
6 95
3214
8 70
6 95
7 01
3284
65 9
70 1
3248
65 9
71 0
3248
69 5
70 1
3248
69 5
71 0
3248
7 49
3250
1 86
7 49
3250
6 81
7 94
3205
1 86
7 94
3205
6 81
749
2061
583
749
2063
581
793
586
2014
794
2061
538
794
2068
531
794
2513
086
794
2516
083
8 65
3279
0 41
8 65
3279
1 40
8 75
3214
0 96
8 75
3214
6 90
8 75
3241
0 69
8 75
3241
9 60
856
397
2140
857
602
1934
9 47
3250
1 86
9 47
3250
6 81
9 56
3241
0 87
9 56
3241
7 80
9 65
0 71
3248
9 65
1 70
3248
9 65
3241
0 78
9 65
3241
8 70
947
2061
385
947
2065
381
95 6
01 7
3284
95 6
07 1
3284
96 5
01 7
3284
96 5
07 1
3284
965
341
2087
974
2061
358
974
2068
351
397
586
2410
397
856
2140
397
2586
410
397
2856
140
479
053
2861
479
2053
861
497
035
2861
497
2035
861
586
397
2410
586
793
2014
586
2397
410
586
2793
014
793
586
2014
793
2586
014
856
397
2140
856
2397
140
857
602
1934
857
1602
934
965
341
2087
965
2341
087
|
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Posted by Charlie
on 2012-07-01 18:36:12 |
computer solution
