The 9 numbers and 7 Xs of the set (1,2,3,4,5,6,7,8,9,X,X,X,X,X,X,X ) were placed in a 4x4 grid to create a matrix as follows:
X 4 8 9
5 6 X 7
1 X X X
3 2 X X
Consider the Xs as black squares in a crossword and evaluate the sum of the sums taken per row:
Sr=489+(56+7)+1+32=585.
Same operation per column:
Sc= 513+(46+2)+8+97=666
Evaluate the ratio
r= Sr/ Sc=585/666= 0.878378...
Your task :
Distribute the 9 non-zero digits and 7 black squares in a 4x4 grid so
that
the ratio r, calculated as in the example above will be as close
to the value of pi (=3.14159265…) as possible.
HaPPy Pi day, every Person.
355/113 is a good approximation to pi, so if you could get the totals to be these two, you've got a good approximation, 3.141592920353983.
333/106, the previous level in a continued fraction approximation, is not as good, 3.141509433962264.
Of course multiples of the numerator and denominator of each of these, such as 710/226, would be the same good approximations.
Combining the two numerators and two denominators can give good approximations also. For example, the worst approximation shown below, 3.14160263 which is 2862/911, is (9*355+1*333)/(9*113+1*106). All are shown reduced to simplest form.
discrepancy value numerator/denominator
0.00000127 3.14159138 18083/5756
0.00000130 3.14159135 17728/5643
0.00000133 3.14159132 17373/5530
0.00000137 3.14159129 17018/5417
0.00000140 3.14159125 16663/5304
0.00000144 3.14159122 16308/5191
0.00000148 3.14159118 15953/5078
0.00000152 3.14159114 15598/4965
0.00000156 3.14159110 15243/4852
0.00000160 3.14159105 14888/4739
0.00000165 3.14159101 14533/4626
0.00000169 3.14159096 14178/4513
0.00000174 3.14159091 13823/4400
0.00000180 3.14159086 13468/4287
0.00000185 3.14159080 13113/4174
0.00000186 3.14159452 17417/5544
0.00000190 3.14159455 17062/5431
0.00000191 3.14159074 12758/4061
0.00000193 3.14159458 16707/5318
0.00000197 3.14159068 12403/3948
0.00000197 3.14159462 16352/5205
0.00000200 3.14159466 15997/5092
0.00000204 3.14159061 12048/3835
0.00000204 3.14159470 15642/4979
0.00000209 3.14159474 15287/4866
0.00000211 3.14159054 11693/3722
0.00000213 3.14159478 14932/4753
0.00000217 3.14159483 14577/4640
0.00000219 3.14159047 11338/3609
0.00000222 3.14159488 14222/4527
0.00000226 3.14159039 10983/3496
0.00000227 3.14159493 13867/4414
0.00000232 3.14159498 13512/4301
0.00000235 3.14159030 10628/3383
0.00000238 3.14159503 13157/4188
0.00000244 3.14159021 10273/3270
0.00000244 3.14159509 12802/4075
0.00000250 3.14159515 12447/3962
0.00000254 3.14159012 9918/3157
0.00000257 3.14159522 12092/3849
0.00000264 3.14159001 9563/3044
0.00000264 3.14159529 11737/3736
0.00000271 3.14159536 11382/3623
0.00000275 3.14158990 9208/2931
0.00000279 3.14159544 11027/3510
0.00000281 3.14158984 18061/5749
0.00000287 3.14158978 8853/2818
0.00000287 3.14159553 10672/3397
0.00000294 3.14158972 17351/5523
0.00000296 3.14159562 10317/3284
0.00000300 3.14158965 8498/2705
0.00000306 3.14159571 9962/3171
0.00000307 3.14158958 16641/5297
0.00000315 3.14158951 8143/2592
0.00000316 3.14159581 9607/3058
0.00000322 3.14158943 15931/5071
0.00000327 3.14159593 9252/2945
0.00000330 3.14158935 7788/2479
0.00000339 3.14158927 15221/4845
0.00000339 3.14159605 8897/2832
0.00000347 3.14158918 7433/2366
0.00000352 3.14159618 8542/2719
0.00000357 3.14158909 14511/4619
0.00000359 3.14159624 16729/5325
0.00000366 3.14158899 7078/2253
0.00000366 3.14159632 8187/2606
0.00000374 3.14159639 16019/5099
0.00000376 3.14158889 13801/4393
0.00000382 3.14159647 7832/2493
0.00000387 3.14158879 6723/2140
0.00000390 3.14159655 15309/4873
0.00000398 3.14158867 13091/4167
0.00000399 3.14159664 7477/2380
0.00000408 3.14159673 14599/4647
0.00000410 3.14158855 6368/2027
0.00000417 3.14159682 7122/2267
0.00000418 3.14158847 18749/5968
0.00000422 3.14158843 12381/3941
0.00000427 3.14158839 18394/5855
0.00000427 3.14159692 13889/4421
0.00000436 3.14158830 6013/1914
0.00000438 3.14159703 6767/2154
0.00000445 3.14158820 17684/5629
0.00000449 3.14159714 13179/4195
0.00000450 3.14158816 11671/3715
0.00000455 3.14158811 17329/5516
0.00000460 3.14159726 6412/2041
0.00000465 3.14158801 5658/1801
0.00000473 3.14159738 12469/3969
0.00000475 3.14158790 16619/5290
0.00000481 3.14158785 10961/3489
0.00000486 3.14158779 16264/5177
0.00000486 3.14159751 6057/1928
0.00000498 3.14158768 5303/1688
0.00000500 3.14159765 11759/3743
0.00000510 3.14158756 15554/4951
0.00000514 3.14159780 5702/1815
0.00000516 3.14158750 10251/3263
0.00000522 3.14158743 15199/4838
0.00000525 3.14159790 16751/5332
0.00000530 3.14159795 11049/3517
0.00000535 3.14158730 4948/1575
0.00000535 3.14159801 16396/5219
0.00000547 3.14159812 5347/1702
0.00000549 3.14158716 14489/4612
0.00000556 3.14158709 9541/3037
0.00000558 3.14159824 15686/4993
0.00000563 3.14158702 14134/4499
0.00000564 3.14159830 10339/3291
0.00000567 3.14158698 18727/5961
0.00000571 3.14159836 15331/4880
0.00000579 3.14158687 4593/1462
0.00000584 3.14159849 4992/1589
0.00000591 3.14158675 18017/5735
0.00000595 3.14158671 13424/4273
0.00000597 3.14159862 14621/4654
0.00000603 3.14158662 8831/2811
0.00000604 3.14159869 9629/3065
0.00000611 3.14159877 14266/4541
0.00000612 3.14158654 13069/4160
0.00000616 3.14158649 17307/5509
0.00000626 3.14159892 4637/1476
0.00000629 3.14158636 4238/1349
0.00000642 3.14159907 13556/4315
0.00000643 3.14158622 16597/5283
0.00000648 3.14158617 12359/3934
0.00000650 3.14159915 8919/2839
0.00000658 3.14158607 8121/2585
0.00000658 3.14159924 13201/4202
0.00000668 3.14158597 12004/3821
0.00000673 3.14158592 15887/5057
0.00000676 3.14159941 4282/1363
0.00000689 3.14158576 3883/1236
0.00000694 3.14159960 12491/3976
0.00000703 3.14158563 19060/6067
0.00000704 3.14159969 8209/2613
0.00000706 3.14158559 15177/4831
0.00000712 3.14158554 11294/3595
0.00000714 3.14159979 12136/3863
0.00000716 3.14158549 18705/5954
0.00000719 3.14159984 16063/5113
0.00000724 3.14158542 7411/2359
0.00000731 3.14158534 18350/5841
0.00000735 3.14160000 3927/1250
0.00000736 3.14158530 10939/3482
0.00000742 3.14158523 14467/4605
0.00000746 3.14158520 17995/5728
0.00000751 3.14160016 15353/4887
0.00000757 3.14160022 11426/3637
0.00000761 3.14158504 3528/1123
0.00000768 3.14160034 7499/2387
0.00000778 3.14158488 17285/5502
0.00000780 3.14160045 11071/3524
0.00000782 3.14158484 13757/4379
0.00000786 3.14160051 14643/4661
0.00000789 3.14158477 10229/3256
0.00000794 3.14158471 16930/5389
0.00000803 3.14158462 6701/2133
0.00000805 3.14160070 3572/1137
0.00000812 3.14158453 16575/5276
0.00000818 3.14158447 9874/3143
0.00000825 3.14160090 13933/4435
0.00000826 3.14158440 13047/4153
0.00000830 3.14158435 16220/5163
0.00000832 3.14160097 10361/3298
0.00000833 3.14158432 19393/6173
0.00000846 3.14160111 6789/2161
0.00000850 3.14158416 3173/1010
0.00000860 3.14160126 10006/3185
0.00000866 3.14158399 18683/5947
0.00000868 3.14160133 13223/4209
0.00000870 3.14158396 15510/4937
0.00000875 3.14158391 12337/3927
0.00000883 3.14158382 9164/2917
0.00000891 3.14158375 15155/4824
0.00000891 3.14160156 3217/1024
0.00000901 3.14158364 5991/1907
0.00000910 3.14160176 15730/5007
0.00000913 3.14158353 14800/4711
0.00000915 3.14160181 12513/3983
0.00000920 3.14158345 8809/2804
0.00000924 3.14160189 9296/2959
0.00000930 3.14158336 11627/3701
0.00000931 3.14160196 15375/4894
0.00000936 3.14158330 14445/4598
0.00000940 3.14158326 17263/5495
0.00000941 3.14160207 6079/1935
0.00000942 3.14158323 20081/6392
0.00000952 3.14160218 15020/4781
0.00000960 3.14158305 2818/897
0.00000960 3.14160225 8941/2846
0.00000969 3.14160234 11803/3757
0.00000975 3.14160240 14665/4668
0.00000978 3.14158287 19371/6166
0.00000981 3.14158284 16553/5269
0.00000985 3.14158280 13735/4372
0.00000992 3.14158273 10917/3475
0.00000997 3.14158269 19016/6053
0.00000998 3.14160263 2862/911
DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
pi = 4 * Atn(1)
n1 = 333: d1 = 106
n2 = 355: d2 = 113
For frst = -50 To 50
For scond = -50 To 50
DoEvents
If frst <> 0 And scond <> 0 Then
If gcd(frst, scond) = 1 Then
num = frst * n1 + scond * n2
den = frst * d1 + scond * d2
If num > 0 And den > 0 And Abs(pi - num / den) < 0.00001 Then
g = gcd(num, den)
num = num / g
den = den / g
Text1.Text = Text1.Text & mform(Abs(pi - num / den), "0.00000000") & mform(num / den, "###0.00000000") & " " & num & "/" & den & crlf
End If
End If
End If
Next
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function gcd(a, b)
x = a: y = b
Do
q = Int(x / y)
z = x - q * y
x = y: y = z
Loop Until z = 0
gcd = x
End Function
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
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Posted by Charlie
on 2016-03-14 09:05:33 |
computer-generated solving aid
