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Cell[BoxData[{
    \(\(\("\<Putting stuff in quotes allows you to write comments. To compile \
a line, you need to press SHIFT-ENTER. Do not compile the instruction lines \
unless you want to see these lines printed again.\>"\)\(\[IndentingNewLine]\)
    \) (*\ 
      anything\ between\ these\ is\ commented\ out\ and\ not\ displayed\ *) \
\), "\[IndentingNewLine]", 
    \( (*\ WARNING : \ 
        THIS\ TEMPLATE\ WAS\ WRITTEN\ FOR\ MATHEMATICA\ VERSION\ 5\ AND\ \
EARLIER . \ SADLY, \ 
      MANY\ OF\ THE\ GRAPHICS\ COMMANDS\ NO\ LONGER\ WORK\ IN\ VERSION\ 6, \ 
      SUCH\ AS\ PLOT\ AND\ \(HISTOGRAM!!\)\ *) \)}], "Input"],

Cell[BoxData[
    \("Putting stuff in quotes allows you to write comments. To compile a \
line, you need to press SHIFT-ENTER. Do not compile the instruction lines \
unless you want to see these lines printed again."\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\(\(num\  = \ 4\ *5\) \)\(;\)\(\ \)\( (*\ 
      this\ multiplies\ 4\ and\ 5\ and\ stores\ the\ result\ in\ num; \ 
      not\ we\ do\ not\ need\ to\ have\ previously\ defined\ the\ variable\ \
num, \ it\ defines\ it\ immediately . \ 
            also\ it\ is\ okay\ for\ comments\ to\ go\ on\ for\ several\ \
lines . \ the\ semi - colon\ means\ calculate\ the\ answer, \ 
      but\ do\ not\ display\ it\ when\ you\ compile\ the\ line . \ 
          if\ we\ type\ num\ on\ the\ next\ line\ and\ compile, \ 
      we\ get\ 20. \ *) \)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(num\  = \ \(\(4\)\(\ \)\(*\)\(\ \)\(5\)\(\ \)\( (*\ 
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Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ 
      Mathematica\ automatically\ multiplies\ two\ quantities\ separated\ by\ \
a\ space . \ If\ we\ have\ a\ b\ /\ 
          c\ d\ then\ it\ does\ \((\((a\ *\ b)\)\ /\ c)\)\ *\ d; \ 
      thus\ it\ does\ multiplication\ and\ division\ in\ the\ order\ it\ \
reads\ them . \ 
          To\ be\ safe\ it\ is\ best\ to\ put\ in\ extra\ \
\(\(parentheses\)\(.\)\)\ *) \)\(\[IndentingNewLine]\)\(num\  = \ 
      4\ 5\ /\ 2\ 3\)\)\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ 
      Many\ standard\ functions\ start\ off\ with\ a\ capital\ letter\ and\ \
then\ the\ name; \ 
      the\ argument\ is\ in\ braces . \ 
          We\ can\ have\ many\ things\ in\ one\ input\ block, \ 
      just\ hit\ return\ after\ \(\(each\)\(.\)\)\ *) \)\(\[IndentingNewLine]\
\)\( (*\ Note\ E\ is\ 2.71828 ... , \ Pi\ is\ 3.14159 ... , \ 
      I\ is\ the\ sqrt \((\(-1\))\), \  ... \ *) \)\(\[IndentingNewLine]\)\(a\
\  = \ Cos[1.0]\[IndentingNewLine]
    b\  = \ Sin[1.0]\[IndentingNewLine]
    c\  = \ Sqrt[a - b]\[IndentingNewLine]
    d\  = \ Log[10, 100]\[IndentingNewLine]
    e\  = \ Exp[\(-10.0\)]\[IndentingNewLine]
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    Pi\[IndentingNewLine]
    I\)\)\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ 
      here\ is\ how\ to\ input\ a\ 3  
        x3\ matrix\ *) \)\(\[IndentingNewLine]\)\(M\  = \ {\ {1, 2, 3}, \ {4, 
          5, 6}, \ {0, 8, 9}}\)\)\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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}, Open  ]],

Cell[BoxData[
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quantities.\>"\)], "Input"],

Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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      Root[\(-21\) + 3\ #1 - 15\ #1\^2 + #1\^3 &, 3], 
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Cell[BoxData[
    \("\<On some versions of mathematica, when you compile the above it gives \
Root[blah]. This is because the answer is 
not exact. To get an exact answer, we use the Numerical Command, N. We first \
give an example. If we type Pi, we get the exact
answer (a symbol). If we type N[Pi], it gives us a few digits. If we need \
more digits, do N[Pi,numdigitsneed]\>"\)], "Input"],

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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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      0.0533787233371055987485182658`8.803268164257855 - 
        1.1862487783775078868856538435`10.150075759926084\ \[ImaginaryI]}\)], \
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Cell[BoxData[
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digits). These values are stored, and we can pull off entries
one at a time.\>"\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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}, Open  ]],

Cell[BoxData[
    \("\<Note how we were able to store the eigenvalues in a vector, and then \
read off the components. Sometimes we need single brackets, sometimes we need \
double brackets.
Using N[ ... ] gives numerical values.\>"\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
    \("\<The above generates a random 4x4 matrix with entries chosen from the \
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Cell[CellGroupData[{

Cell[BoxData[
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.77808 .46279 .78884 .47292 .81521 .47453 .80457 .46375 Metetra
.404 .691 .978 r
.80457 .46375 .81521 .47453 .84207 .47715 .83154 .46566 Metetra
.395 .673 .971 r
.83154 .46566 .84207 .47715 .86942 .48065 .85899 .46838 Metetra
.393 .658 .965 r
.85899 .46838 .86942 .48065 .89729 .48485 .88695 .47175 Metetra
.397 .646 .957 r
.88695 .47175 .89729 .48485 .92567 .48955 .9154 .47559 Metetra
.407 .638 .95 r
.9154 .47559 .92567 .48955 .95456 .49455 .94436 .4797 Metetra
.781 .769 .842 r
.33385 .64264 .34784 .65485 .3708 .63666 .35693 .62527 Metetra
.784 .782 .851 r
.35693 .62527 .3708 .63666 .39379 .61864 .38005 .60805 Metetra
.785 .796 .861 r
.38005 .60805 .39379 .61864 .41681 .60098 .40322 .59114 Metetra
.785 .809 .872 r
.40322 .59114 .41681 .60098 .43987 .58385 .42644 .57472 Metetra
.782 .823 .885 r
.42644 .57472 .43987 .58385 .46297 .56742 .44971 .55893 Metetra
.776 .837 .899 r
.44971 .55893 .46297 .56742 .48612 .55185 .47304 .54393 Metetra
.766 .849 .914 r
.47304 .54393 .48612 .55185 .50935 .53728 .49645 .52984 Metetra
.751 .86 .929 r
.49645 .52984 .50935 .53728 .53267 .52383 .51997 .51677 Metetra
.732 .868 .944 r
.51997 .51677 .53267 .52383 .55612 .51159 .54362 .50482 Metetra
.708 .872 .959 r
.54362 .50482 .55612 .51159 .57972 .50065 .56742 .49405 Metetra
.678 .873 .972 r
.56742 .49405 .57972 .50065 .6035 .49108 .5914 .48454 Metetra
.644 .868 .983 r
.5914 .48454 .6035 .49108 .62751 .4829 .6156 .47631 Metetra
.606 .857 .991 r
.6156 .47631 .62751 .4829 .65176 .47614 .64005 .46939 Metetra
.566 .842 .997 r
.64005 .46939 .65176 .47614 .67631 .4708 .66478 .46377 Metetra
.527 .823 .999 r
.66478 .46377 .67631 .4708 .70118 .46685 .68983 .45943 Metetra
.491 .801 .999 r
.68983 .45943 .70118 .46685 .72641 .46424 .71523 .45633 Metetra
.46 .778 .998 r
.71523 .45633 .72641 .46424 .75203 .46292 .741 .4544 Metetra
.435 .754 .995 r
.741 .4544 .75203 .46292 .77808 .46279 .76719 .45356 Metetra
.418 .732 .991 r
.76719 .45356 .77808 .46279 .80457 .46375 .79381 .45371 Metetra
.408 .712 .985 r
.79381 .45371 .80457 .46375 .83154 .46566 .82088 .45473 Metetra
.405 .694 .979 r
.82088 .45473 .83154 .46566 .85899 .46838 .84843 .45649 Metetra
.408 .678 .972 r
.84843 .45649 .85899 .46838 .88695 .47175 .87646 .45883 Metetra
.416 .665 .963 r
.87646 .45883 .88695 .47175 .9154 .47559 .90497 .46159 Metetra
.429 .654 .954 r
.90497 .46159 .9154 .47559 .94436 .4797 .93397 .46458 Metetra
.772 .769 .85 r
.31963 .63024 .33385 .64264 .35693 .62527 .34282 .61388 Metetra
.776 .785 .86 r
.34282 .61388 .35693 .62527 .38005 .60805 .36607 .59764 Metetra
.778 .801 .872 r
.36607 .59764 .38005 .60805 .40322 .59114 .38937 .58167 Metetra
.777 .818 .885 r
.38937 .58167 .40322 .59114 .42644 .57472 .41274 .56612 Metetra
.774 .835 .899 r
.41274 .56612 .42644 .57472 .44971 .55893 .43618 .55113 Metetra
.767 .851 .914 r
.43618 .55113 .44971 .55893 .47304 .54393 .45969 .53684 Metetra
.756 .866 .929 r
.45969 .53684 .47304 .54393 .49645 .52984 .48329 .52335 Metetra
.74 .879 .944 r
.48329 .52335 .49645 .52984 .51997 .51677 .50701 .51076 Metetra
.719 .888 .958 r
.50701 .51076 .51997 .51677 .54362 .50482 .53086 .49917 Metetra
.691 .893 .971 r
.53086 .49917 .54362 .50482 .56742 .49405 .55487 .48865 Metetra
.659 .893 .981 r
.55487 .48865 .56742 .49405 .5914 .48454 .57907 .47924 Metetra
.622 .887 .989 r
.57907 .47924 .5914 .48454 .6156 .47631 .60348 .47098 Metetra
.583 .875 .994 r
.60348 .47098 .6156 .47631 .64005 .46939 .62813 .46388 Metetra
.545 .859 .997 r
.62813 .46388 .64005 .46939 .66478 .46377 .65307 .45795 Metetra
.508 .84 .998 r
.65307 .45795 .66478 .46377 .68983 .45943 .6783 .45315 Metetra
.476 .819 .998 r
.6783 .45315 .68983 .45943 .71523 .45633 .70388 .44946 Metetra
.451 .797 .998 r
.70388 .44946 .71523 .45633 .741 .4544 .72982 .44682 Metetra
.433 .775 .997 r
.72982 .44682 .741 .4544 .76719 .45356 .75615 .44514 Metetra
.422 .755 .995 r
.75615 .44514 .76719 .45356 .79381 .45371 .7829 .44434 Metetra
.419 .735 .991 r
.7829 .44434 .79381 .45371 .82088 .45473 .81009 .44431 Metetra
.422 .717 .986 r
.81009 .44431 .82088 .45473 .84843 .45649 .83772 .44493 Metetra
.43 .7 .978 r
.83772 .44493 .84843 .45649 .87646 .45883 .86582 .44605 Metetra
.441 .685 .968 r
.86582 .44605 .87646 .45883 .90497 .46159 .89438 .44753 Metetra
.455 .672 .957 r
.89438 .44753 .90497 .46159 .93397 .46458 .92341 .44921 Metetra
.761 .768 .859 r
.3052 .61764 .31963 .63024 .34282 .61388 .32847 .60246 Metetra
.765 .787 .871 r
.32847 .60246 .34282 .61388 .36607 .59764 .35182 .58737 Metetra
.767 .806 .884 r
.35182 .58737 .36607 .59764 .38937 .58167 .37525 .5725 Metetra
.766 .826 .899 r
.37525 .5725 .38937 .58167 .41274 .56612 .39876 .55799 Metetra
.762 .846 .914 r
.39876 .55799 .41274 .56612 .43618 .55113 .42236 .54394 Metetra
.755 .864 .929 r
.42236 .54394 .43618 .55113 .45969 .53684 .44605 .53048 Metetra
.742 .881 .944 r
.44605 .53048 .45969 .53684 .48329 .52335 .46985 .5177 Metetra
.723 .895 .957 r
.46985 .5177 .48329 .52335 .50701 .51076 .49377 .50571 Metetra
.699 .904 .969 r
.49377 .50571 .50701 .51076 .53086 .49917 .51784 .49456 Metetra
.669 .908 .978 r
.51784 .49456 .53086 .49917 .55487 .48865 .54207 .48433 Metetra
.634 .906 .983 r
.54207 .48433 .55487 .48865 .57907 .47924 .56649 .47505 Metetra
.596 .899 .987 r
.56649 .47505 .57907 .47924 .60348 .47098 .59112 .46677 Metetra
.558 .886 .988 r
.59112 .46677 .60348 .47098 .62813 .46388 .61599 .45948 Metetra
.521 .87 .989 r
.61599 .45948 .62813 .46388 .65307 .45795 .64114 .4532 Metetra
.49 .852 .99 r
.64114 .4532 .65307 .45795 .6783 .45315 .66658 .4479 Metetra
.464 .834 .992 r
.66658 .4479 .6783 .45315 .70388 .44946 .69234 .44354 Metetra
.447 .815 .994 r
.69234 .44354 .70388 .44946 .72982 .44682 .71846 .44008 Metetra
.437 .796 .996 r
.71846 .44008 .72982 .44682 .75615 .44514 .74495 .43745 Metetra
.434 .778 .997 r
.74495 .43745 .75615 .44514 .7829 .44434 .77183 .43556 Metetra
.437 .76 .996 r
.77183 .43556 .7829 .44434 .81009 .44431 .79912 .43433 Metetra
.446 .742 .991 r
.79912 .43433 .81009 .44431 .83772 .44493 .82685 .43364 Metetra
.457 .725 .983 r
.82685 .43364 .83772 .44493 .86582 .44605 .85501 .43337 Metetra
.471 .707 .972 r
.85501 .43337 .86582 .44605 .89438 .44753 .88361 .43339 Metetra
.486 .691 .959 r
.88361 .43339 .89438 .44753 .92341 .44921 .91265 .43355 Metetra
.747 .767 .869 r
.29053 .60484 .3052 .61764 .32847 .60246 .31387 .591 Metetra
.751 .788 .883 r
.31387 .591 .32847 .60246 .35182 .58737 .33731 .57721 Metetra
.752 .81 .898 r
.33731 .57721 .35182 .58737 .37525 .5725 .36085 .56359 Metetra
.751 .832 .914 r
.36085 .56359 .37525 .5725 .39876 .55799 .3845 .55023 Metetra
.746 .854 .93 r
.3845 .55023 .39876 .55799 .42236 .54394 .40825 .53726 Metetra
.737 .875 .945 r
.40825 .53726 .42236 .54394 .44605 .53048 .43212 .52475 Metetra
.722 .893 .958 r
.43212 .52475 .44605 .53048 .46985 .5177 .45611 .5128 Metetra
.7 .907 .968 r
.45611 .5128 .46985 .5177 .49377 .50571 .48023 .50147 Metetra
.673 .915 .976 r
.48023 .50147 .49377 .50571 .51784 .49456 .50452 .49084 Metetra
.64 .917 .979 r
.50452 .49084 .51784 .49456 .54207 .48433 .52898 .48095 Metetra
.604 .913 .979 r
.52898 .48095 .54207 .48433 .56649 .47505 .55363 .47184 Metetra
.566 .904 .978 r
.55363 .47184 .56649 .47505 .59112 .46677 .5785 .46354 Metetra
.53 .891 .976 r
.5785 .46354 .59112 .46677 .61599 .45948 .6036 .45606 Metetra
.499 .876 .975 r
.6036 .45606 .61599 .45948 .64114 .4532 .62897 .4494 Metetra
.475 .861 .977 r
.62897 .4494 .64114 .4532 .66658 .4479 .65463 .44353 Metetra
.458 .846 .981 r
.65463 .44353 .66658 .4479 .69234 .44354 .6806 .43844 Metetra
.449 .832 .987 r
.6806 .43844 .69234 .44354 .71846 .44008 .7069 .43407 Metetra
.447 .817 .993 r
.7069 .43407 .71846 .44008 .74495 .43745 .73355 .43037 Metetra
.453 .802 .998 r
.73355 .43037 .74495 .43745 .77183 .43556 .76057 .42727 Metetra
.463 .786 .999 r
.76057 .42727 .77183 .43556 .79912 .43433 .78798 .4247 Metetra
.475 .768 .995 r
.78798 .4247 .79912 .43433 .82685 .43364 .81579 .42255 Metetra
.49 .749 .987 r
.81579 .42255 .82685 .43364 .85501 .43337 .84402 .42073 Metetra
.505 .73 .974 r
.84402 .42073 .85501 .43337 .88361 .43339 .87265 .41913 Metetra
.519 .711 .959 r
.87265 .41913 .88361 .43339 .91265 .43355 .90171 .41762 Metetra
.73 .765 .881 r
.27563 .59183 .29053 .60484 .31387 .591 .29901 .57945 Metetra
.733 .788 .897 r
.29901 .57945 .31387 .591 .33731 .57721 .32253 .56709 Metetra
.734 .813 .913 r
.32253 .56709 .33731 .57721 .36085 .56359 .34617 .55484 Metetra
.732 .837 .93 r
.34617 .55484 .36085 .56359 .3845 .55023 .36993 .54278 Metetra
.725 .861 .945 r
.36993 .54278 .3845 .55023 .40825 .53726 .39383 .53099 Metetra
.713 .882 .959 r
.39383 .53099 .40825 .53726 .43212 .52475 .41786 .51954 Metetra
.695 .9 .97 r
.41786 .51954 .43212 .52475 .45611 .5128 .44204 .50849 Metetra
.671 .913 .976 r
.44204 .50849 .45611 .5128 .48023 .50147 .46638 .49792 Metetra
.641 .919 .978 r
.46638 .49792 .48023 .50147 .50452 .49084 .49089 .48786 Metetra
.607 .918 .976 r
.49089 .48786 .50452 .49084 .52898 .48095 .51558 .47836 Metetra
.571 .912 .97 r
.51558 .47836 .52898 .48095 .55363 .47184 .54047 .46945 Metetra
.536 .902 .965 r
.54047 .46945 .55363 .47184 .5785 .46354 .56558 .46114 Metetra
.505 .89 .96 r
.56558 .46114 .5785 .46354 .6036 .45606 .59093 .45345 Metetra
.481 .878 .96 r
.59093 .45345 .6036 .45606 .62897 .4494 .61654 .44638 Metetra
.465 .867 .963 r
.61654 .44638 .62897 .4494 .65463 .44353 .64243 .4399 Metetra
.458 .857 .97 r
.64243 .4399 .65463 .44353 .6806 .43844 .66861 .434 Metetra
.458 .848 .98 r
.66861 .434 .6806 .43844 .7069 .43407 .6951 .42864 Metetra
.466 .838 .99 r
.6951 .42864 .7069 .43407 .73355 .43037 .72193 .42378 Metetra
.478 .826 .997 r
.72193 .42378 .73355 .43037 .76057 .42727 .74911 .41937 Metetra
.494 .812 1 r
.74911 .41937 .76057 .42727 .78798 .4247 .77664 .41533 Metetra
.51 .794 .997 r
.77664 .41533 .78798 .4247 .81579 .42255 .80455 .41159 Metetra
.526 .774 .988 r
.80455 .41159 .81579 .42255 .84402 .42073 .83283 .40809 Metetra
.541 .752 .974 r
.83283 .40809 .84402 .42073 .87265 .41913 .8615 .40472 Metetra
.553 .73 .957 r
.8615 .40472 .87265 .41913 .90171 .41762 .89056 .40141 Metetra
.709 .762 .894 r
.26048 .57861 .27563 .59183 .29901 .57945 .28389 .5678 Metetra
.711 .787 .911 r
.28389 .5678 .29901 .57945 .32253 .56709 .30746 .55697 Metetra
.711 .813 .929 r
.30746 .55697 .32253 .56709 .34617 .55484 .33118 .54619 Metetra
.707 .839 .946 r
.33118 .54619 .34617 .55484 .36993 .54278 .35505 .53552 Metetra
.698 .864 .961 r
.35505 .53552 .36993 .54278 .39383 .53099 .37908 .525 Metetra
.684 .886 .972 r
.37908 .525 .39383 .53099 .41786 .51954 .40328 .51469 Metetra
.663 .902 .979 r
.40328 .51469 .41786 .51954 .44204 .50849 .42765 .50464 Metetra
.637 .913 .981 r
.42765 .50464 .44204 .50849 .46638 .49792 .45219 .49488 Metetra
.605 .917 .977 r
.45219 .49488 .46638 .49792 .49089 .48786 .47692 .48545 Metetra
.571 .914 .969 r
.47692 .48545 .49089 .48786 .51558 .47836 .50185 .47638 Metetra
.538 .907 .959 r
.50185 .47638 .51558 .47836 .54047 .46945 .52699 .46769 Metetra
.509 .897 .95 r
.52699 .46769 .54047 .46945 .56558 .46114 .55236 .45938 Metetra
.485 .888 .945 r
.55236 .45938 .56558 .46114 .59093 .45345 .57796 .45147 Metetra
.47 .879 .945 r
.57796 .45147 .59093 .45345 .61654 .44638 .60382 .44396 Metetra
.464 .873 .951 r
.60382 .44396 .61654 .44638 .64243 .4399 .62995 .43683 Metetra
.466 .869 .961 r
.62995 .43683 .64243 .4399 .66861 .434 .65636 .43007 Metetra
.476 .865 .974 r
.65636 .43007 .66861 .434 .6951 .42864 .68306 .42365 Metetra
.491 .859 .987 r
.68306 .42365 .6951 .42864 .72193 .42378 .71008 .41755 Metetra
.51 .85 .996 r
.71008 .41755 .72193 .42378 .74911 .41937 .73741 .41171 Metetra
.53 .836 .999 r
.73741 .41171 .74911 .41937 .77664 .41533 .76508 .40611 Metetra
.548 .818 .996 r
.76508 .40611 .77664 .41533 .80455 .41159 .79309 .40068 Metetra
.564 .796 .986 r
.79309 .40068 .80455 .41159 .83283 .40809 .82144 .39538 Metetra
.578 .772 .971 r
.82144 .39538 .83283 .40809 .8615 .40472 .85015 .39014 Metetra
.588 .748 .953 r
.85015 .39014 .8615 .40472 .89056 .40141 .87922 .3849 Metetra
.685 .758 .907 r
.24509 .56517 .26048 .57861 .28389 .5678 .26851 .556 Metetra
.685 .784 .926 r
.26851 .556 .28389 .5678 .30746 .55697 .2921 .54678 Metetra
.683 .811 .944 r
.2921 .54678 .30746 .55697 .33118 .54619 .31589 .53755 Metetra
.676 .838 .961 r
.31589 .53755 .33118 .54619 .35505 .53552 .33985 .52834 Metetra
.665 .863 .975 r
.33985 .52834 .35505 .53552 .37908 .525 .36401 .51918 Metetra
.648 .883 .983 r
.36401 .51918 .37908 .525 .40328 .51469 .38836 .51008 Metetra
.626 .899 .986 r
.38836 .51008 .40328 .51469 .42765 .50464 .4129 .50107 Metetra
.598 .907 .982 r
.4129 .50107 .42765 .50464 .45219 .49488 .43765 .49218 Metetra
.568 .909 .973 r
.43765 .49218 .45219 .49488 .47692 .48545 .4626 .48342 Metetra
.537 .905 .961 r
.4626 .48342 .47692 .48545 .50185 .47638 .48777 .47481 Metetra
.509 .899 .948 r
.48777 .47481 .50185 .47638 .52699 .46769 .51317 .46635 Metetra
.487 .891 .938 r
.51317 .46635 .52699 .46769 .55236 .45938 .5388 .45805 Metetra
.473 .885 .933 r
.5388 .45805 .55236 .45938 .57796 .45147 .56467 .44991 Metetra
.467 .882 .935 r
.56467 .44991 .57796 .45147 .60382 .44396 .59079 .44194 Metetra
.471 .881 .943 r
.59079 .44194 .60382 .44396 .62995 .43683 .61717 .43413 Metetra
.482 .881 .955 r
.61717 .43413 .62995 .43683 .65636 .43007 .64382 .42646 Metetra
.5 .881 .97 r
.64382 .42646 .65636 .43007 .68306 .42365 .67074 .41892 Metetra
.522 .878 .984 r
.67074 .41892 .68306 .42365 .71008 .41755 .69796 .4115 Metetra
.546 .871 .993 r
.69796 .4115 .71008 .41755 .73741 .41171 .72547 .40418 Metetra
.568 .858 .997 r
.72547 .40418 .73741 .41171 .76508 .40611 .75328 .39693 Metetra
.588 .839 .992 r
.75328 .39693 .76508 .40611 .79309 .40068 .7814 .38972 Metetra
.603 .816 .982 r
.7814 .38972 .79309 .40068 .82144 .39538 .80983 .38254 Metetra
.614 .79 .966 r
.80983 .38254 .82144 .39538 .85015 .39014 .83859 .37533 Metetra
.622 .764 .948 r
.83859 .37533 .85015 .39014 .87922 .3849 .86766 .36808 Metetra
.656 .751 .92 r
.22944 .55152 .24509 .56517 .26851 .556 .25284 .54402 Metetra
.654 .778 .94 r
.25284 .54402 .26851 .556 .2921 .54678 .27646 .53646 Metetra
.649 .805 .958 r
.27646 .53646 .2921 .54678 .31589 .53755 .30029 .52884 Metetra
.64 .832 .974 r
.30029 .52884 .31589 .53755 .33985 .52834 .32433 .52114 Metetra
.626 .856 .986 r
.32433 .52114 .33985 .52834 .36401 .51918 .3486 .51337 Metetra
.608 .876 .992 r
.3486 .51337 .36401 .51918 .38836 .51008 .37309 .50553 Metetra
.584 .889 .99 r
.37309 .50553 .38836 .51008 .4129 .50107 .3978 .49761 Metetra
.558 .896 .982 r
.3978 .49761 .4129 .50107 .43765 .49218 .42275 .48963 Metetra
.531 .897 .969 r
.42275 .48963 .43765 .49218 .4626 .48342 .44792 .48157 Metetra
.506 .894 .954 r
.44792 .48157 .4626 .48342 .48777 .47481 .47334 .47343 Metetra
.486 .89 .94 r
.47334 .47343 .48777 .47481 .51317 .46635 .49899 .46522 Metetra
.473 .886 .931 r
.49899 .46522 .51317 .46635 .5388 .45805 .52489 .45693 Metetra
.469 .885 .927 r
.52489 .45693 .5388 .45805 .56467 .44991 .55103 .44856 Metetra
.473 .886 .931 r
.55103 .44856 .56467 .44991 .59079 .44194 .57742 .44011 Metetra
.486 .89 .94 r
.57742 .44011 .59079 .44194 .61717 .43413 .60407 .43158 Metetra
.506 .894 .954 r
.60407 .43158 .61717 .43413 .64382 .42646 .63097 .42297 Metetra
.531 .897 .969 r
.63097 .42297 .64382 .42646 .67074 .41892 .65813 .41427 Metetra
.558 .896 .982 r
.65813 .41427 .67074 .41892 .69796 .4115 .68556 .40549 Metetra
.584 .889 .99 r
.68556 .40549 .69796 .4115 .72547 .40418 .71325 .39662 Metetra
.608 .876 .992 r
.71325 .39662 .72547 .40418 .75328 .39693 .74122 .38767 Metetra
.626 .856 .986 r
.74122 .38767 .75328 .39693 .7814 .38972 .76946 .37863 Metetra
.64 .832 .974 r
.76946 .37863 .7814 .38972 .80983 .38254 .79799 .36949 Metetra
.649 .805 .958 r
.79799 .36949 .80983 .38254 .83859 .37533 .8268 .36027 Metetra
.654 .778 .94 r
.8268 .36027 .83859 .37533 .86766 .36808 .8559 .35095 Metetra
.622 .742 .933 r
.21353 .53763 .22944 .55152 .25284 .54402 .2369 .53184 Metetra
.617 .769 .953 r
.2369 .53184 .25284 .54402 .27646 .53646 .26051 .52595 Metetra
.609 .796 .971 r
.26051 .52595 .27646 .53646 .30029 .52884 .28437 .51994 Metetra
.597 .821 .985 r
.28437 .51994 .30029 .52884 .32433 .52114 .30848 .51378 Metetra
.582 .844 .994 r
.30848 .51378 .32433 .52114 .3486 .51337 .33284 .50743 Metetra
.563 .862 .996 r
.33284 .50743 .3486 .51337 .37309 .50553 .35746 .50087 Metetra
.541 .874 .991 r
.35746 .50087 .37309 .50553 .3978 .49761 .38234 .49407 Metetra
.519 .881 .98 r
.38234 .49407 .3978 .49761 .42275 .48963 .40748 .48701 Metetra
.498 .883 .965 r
.40748 .48701 .42275 .48963 .44792 .48157 .43287 .47966 Metetra
.481 .883 .95 r
.43287 .47966 .44792 .48157 .47334 .47343 .45853 .47202 Metetra
.471 .883 .937 r
.45853 .47202 .47334 .47343 .49899 .46522 .48444 .46406 Metetra
.468 .884 .93 r
.48444 .46406 .49899 .46522 .52489 .45693 .51061 .45578 Metetra
.474 .887 .928 r
.51061 .45578 .52489 .45693 .55103 .44856 .53703 .44717 Metetra
.488 .893 .933 r
.53703 .44717 .55103 .44856 .57742 .44011 .5637 .43823 Metetra
.51 .901 .943 r
.5637 .43823 .57742 .44011 .60407 .43158 .59062 .42896 Metetra
.536 .908 .956 r
.59062 .42896 .60407 .43158 .63097 .42297 .61779 .41938 Metetra
.566 .912 .969 r
.61779 .41938 .63097 .42297 .65813 .41427 .6452 .4095 Metetra
.596 .911 .98 r
.6452 .4095 .65813 .41427 .68556 .40549 .67285 .39933 Metetra
.623 .903 .985 r
.67285 .39933 .68556 .40549 .71325 .39662 .70075 .38888 Metetra
.645 .889 .984 r
.70075 .38888 .71325 .39662 .74122 .38767 .72888 .37819 Metetra
.662 .869 .977 r
.72888 .37819 .74122 .38767 .76946 .37863 .75727 .36728 Metetra
.674 .844 .965 r
.75727 .36728 .76946 .37863 .79799 .36949 .7859 .35618 Metetra
.68 .817 .949 r
.7859 .35618 .79799 .36949 .8268 .36027 .81478 .34491 Metetra
.683 .789 .931 r
.81478 .34491 .8268 .36027 .8559 .35095 .84391 .33351 Metetra
.583 .731 .945 r
.19735 .52351 .21353 .53763 .2369 .53184 .22067 .5194 Metetra
.574 .756 .964 r
.22067 .5194 .2369 .53184 .26051 .52595 .24427 .51517 Metetra
.563 .781 .98 r
.24427 .51517 .26051 .52595 .28437 .51994 .26814 .51077 Metetra
.55 .805 .993 r
.26814 .51077 .28437 .51994 .30848 .51378 .29229 .50614 Metetra
.534 .826 .999 r
.29229 .50614 .30848 .51378 .33284 .50743 .31674 .50121 Metetra
.516 .843 .998 r
.31674 .50121 .33284 .50743 .35746 .50087 .34148 .49593 Metetra
.498 .855 .991 r
.34148 .49593 .35746 .50087 .38234 .49407 .36651 .49025 Metetra
.482 .863 .979 r
.36651 .49025 .38234 .49407 .40748 .48701 .39183 .48411 Metetra
.47 .868 .965 r
.39183 .48411 .40748 .48701 .43287 .47966 .41744 .47748 Metetra
.463 .873 .951 r
.41744 .47748 .43287 .47966 .45853 .47202 .44333 .47033 Metetra
.463 .877 .941 r
.44333 .47033 .45853 .47202 .48444 .46406 .4695 .46262 Metetra
.472 .884 .935 r
.4695 .46262 .48444 .46406 .51061 .45578 .49595 .45434 Metetra
.488 .892 .935 r
.49595 .45434 .51061 .45578 .53703 .44717 .52265 .44549 Metetra
.511 .902 .941 r
.52265 .44549 .53703 .44717 .5637 .43823 .54961 .43605 Metetra
.539 .912 .95 r
.54961 .43605 .5637 .43823 .59062 .42896 .57681 .42605 Metetra
.57 .919 .961 r
.57681 .42605 .59062 .42896 .61779 .41938 .60425 .41549 Metetra
.603 .923 .97 r
.60425 .41549 .61779 .41938 .6452 .4095 .63192 .40441 Metetra
.633 .921 .977 r
.63192 .40441 .6452 .4095 .67285 .39933 .65982 .39283 Metetra
.66 .912 .979 r
.65982 .39283 .67285 .39933 .70075 .38888 .68793 .3808 Metetra
.68 .897 .975 r
.68793 .3808 .70075 .38888 .72888 .37819 .71626 .36837 Metetra
.695 .876 .966 r
.71626 .36837 .72888 .37819 .75727 .36728 .74479 .35559 Metetra
.704 .852 .953 r
.74479 .35559 .75727 .36728 .7859 .35618 .77354 .34252 Metetra
.709 .826 .937 r
.77354 .34252 .7859 .35618 .81478 .34491 .80251 .32921 Metetra
.709 .798 .92 r
.80251 .32921 .81478 .34491 .84391 .33351 .8317 .31573 Metetra
.538 .715 .954 r
.1809 .50916 .19735 .52351 .22067 .5194 .20416 .50667 Metetra
.526 .738 .972 r
.20416 .50667 .22067 .5194 .24427 .51517 .22772 .50406 Metetra
.513 .761 .986 r
.22772 .50406 .24427 .51517 .26814 .51077 .25159 .50123 Metetra
.499 .783 .996 r
.25159 .50123 .26814 .51077 .29229 .50614 .27578 .49809 Metetra
.484 .803 1 r
.27578 .49809 .29229 .50614 .31674 .50121 .30029 .49455 Metetra
.47 .82 .997 r
.30029 .49455 .31674 .50121 .34148 .49593 .32514 .49053 Metetra
.458 .833 .99 r
.32514 .49053 .34148 .49593 .36651 .49025 .35031 .48594 Metetra
.451 .845 .979 r
.35031 .48594 .36651 .49025 .39183 .48411 .37581 .48071 Metetra
.448 .854 .967 r
.37581 .48071 .39183 .48411 .41744 .47748 .40162 .47478 Metetra
.453 .864 .956 r
.40162 .47478 .41744 .47748 .44333 .47033 .42775 .46811 Metetra
.464 .874 .949 r
.42775 .46811 .44333 .47033 .4695 .46262 .45418 .46065 Metetra
.483 .886 .946 r
.45418 .46065 .4695 .46262 .49595 .45434 .48089 .45237 Metetra
.508 .898 .947 r
.48089 .45237 .49595 .45434 .52265 .44549 .50788 .44327 Metetra
.538 .91 .952 r
.50788 .44327 .52265 .44549 .54961 .43605 .53514 .43333 Metetra
.572 .921 .959 r
.53514 .43333 .54961 .43605 .57681 .42605 .56263 .42259 Metetra
.606 .928 .966 r
.56263 .42259 .57681 .42605 .60425 .41549 .59036 .41106 Metetra
.639 .93 .971 r
.59036 .41106 .60425 .41549 .63192 .40441 .6183 .39879 Metetra
.669 .926 .973 r
.6183 .39879 .63192 .40441 .65982 .39283 .64645 .38582 Metetra
.693 .916 .97 r
.64645 .38582 .65982 .39283 .68793 .3808 .67479 .37222 Metetra
.711 .9 .964 r
.67479 .37222 .68793 .3808 .71626 .36837 .70332 .35807 Metetra
.724 .88 .953 r
.70332 .35807 .71626 .36837 .74479 .35559 .73203 .34345 Metetra
.731 .856 .94 r
.73203 .34345 .74479 .35559 .77354 .34252 .76092 .32843 Metetra
.733 .831 .925 r
.76092 .32843 .77354 .34252 .80251 .32921 .78999 .31313 Metetra
.732 .805 .909 r
.78999 .31313 .80251 .32921 .8317 .31573 .81925 .29762 Metetra
.49 .697 .96 r
.16417 .49455 .1809 .50916 .20416 .50667 .18735 .49362 Metetra
.475 .717 .976 r
.18735 .49362 .20416 .50667 .22772 .50406 .21086 .49255 Metetra
.46 .737 .988 r
.21086 .49255 .22772 .50406 .25159 .50123 .23472 .49122 Metetra
.447 .758 .995 r
.23472 .49122 .25159 .50123 .27578 .49809 .25893 .48951 Metetra
.435 .777 .997 r
.25893 .48951 .27578 .49809 .30029 .49455 .2835 .4873 Metetra
.427 .794 .995 r
.2835 .4873 .30029 .49455 .32514 .49053 .30844 .48448 Metetra
.423 .81 .989 r
.30844 .48448 .32514 .49053 .35031 .48594 .33374 .48093 Metetra
.425 .826 .981 r
.33374 .48093 .35031 .48594 .37581 .48071 .35941 .47658 Metetra
.434 .841 .972 r
.35941 .47658 .37581 .48071 .40162 .47478 .38542 .47133 Metetra
.45 .856 .966 r
.38542 .47133 .40162 .47478 .42775 .46811 .41178 .46511 Metetra
.472 .872 .961 r
.41178 .46511 .42775 .46811 .45418 .46065 .43846 .45788 Metetra
.5 .888 .96 r
.43846 .45788 .45418 .46065 .48089 .45237 .46545 .44959 Metetra
.533 .903 .961 r
.46545 .44959 .48089 .45237 .50788 .44327 .49273 .44024 Metetra
.569 .916 .964 r
.49273 .44024 .50788 .44327 .53514 .43333 .52027 .42981 Metetra
.605 .926 .967 r
.52027 .42981 .53514 .43333 .56263 .42259 .54806 .41834 Metetra
.64 .932 .969 r
.54806 .41834 .56263 .42259 .59036 .41106 .57608 .40586 Metetra
.672 .932 .969 r
.57608 .40586 .59036 .41106 .6183 .39879 .60431 .39242 Metetra
.7 .926 .966 r
.60431 .39242 .6183 .39879 .64645 .38582 .63272 .37809 Metetra
.722 .915 .96 r
.63272 .37809 .64645 .38582 .67479 .37222 .66131 .36297 Metetra
.738 .899 .951 r
.66131 .36297 .67479 .37222 .70332 .35807 .69005 .34715 Metetra
.748 .88 .94 r
.69005 .34715 .70332 .35807 .73203 .34345 .71896 .33074 Metetra
.753 .858 .927 r
.71896 .33074 .73203 .34345 .76092 .32843 .74801 .31385 Metetra
.754 .834 .913 r
.74801 .31385 .76092 .32843 .78999 .31313 .77721 .29662 Metetra
.752 .81 .899 r
.77721 .29662 .78999 .31313 .81925 .29762 .80656 .27915 Metetra
.439 .675 .963 r
.14715 .4797 .16417 .49455 .18735 .49362 .17025 .4802 Metetra
.422 .692 .976 r
.17025 .4802 .18735 .49362 .21086 .49255 .19371 .48057 Metetra
.407 .71 .985 r
.19371 .48057 .21086 .49255 .23472 .49122 .21754 .48063 Metetra
.396 .729 .99 r
.21754 .48063 .23472 .49122 .25893 .48951 .24176 .48026 Metetra
.389 .748 .992 r
.24176 .48026 .25893 .48951 .2835 .4873 .26638 .47929 Metetra
.389 .768 .991 r
.26638 .47929 .2835 .4873 .30844 .48448 .2914 .4776 Metetra
.395 .787 .988 r
.2914 .4776 .30844 .48448 .33374 .48093 .31682 .47504 Metetra
.407 .808 .984 r
.31682 .47504 .33374 .48093 .35941 .47658 .34264 .4715 Metetra
.427 .828 .98 r
.34264 .4715 .35941 .47658 .38542 .47133 .36885 .46687 Metetra
.454 .849 .977 r
.36885 .46687 .38542 .47133 .41178 .46511 .39542 .46108 Metetra
.486 .869 .975 r
.39542 .46108 .41178 .46511 .43846 .45788 .42235 .45405 Metetra
.522 .888 .975 r
.42235 .45405 .43846 .45788 .46545 .44959 .44961 .44575 Metetra
.56 .905 .975 r
.44961 .44575 .46545 .44959 .49273 .44024 .47717 .43614 Metetra
.599 .918 .975 r
.47717 .43614 .49273 .44024 .52027 .42981 .50501 .42523 Metetra
.637 .926 .973 r
.50501 .42523 .52027 .42981 .54806 .41834 .5331 .41306 Metetra
.671 .93 .971 r
.5331 .41306 .54806 .41834 .57608 .40586 .56142 .39965 Metetra
.701 .928 .966 r
.56142 .39965 .57608 .40586 .60431 .39242 .58994 .38509 Metetra
.726 .921 .958 r
.58994 .38509 .60431 .39242 .63272 .37809 .61863 .36947 Metetra
.745 .91 .949 r
.61863 .36947 .63272 .37809 .66131 .36297 .64747 .35289 Metetra
.759 .894 .938 r
.64747 .35289 .66131 .36297 .69005 .34715 .67646 .33548 Metetra
.767 .876 .926 r
.67646 .33548 .69005 .34715 .71896 .33074 .70557 .31737 Metetra
.771 .856 .913 r
.70557 .31737 .71896 .33074 .74801 .31385 .7348 .29871 Metetra
.772 .835 .901 r
.7348 .29871 .74801 .31385 .77721 .29662 .76415 .27964 Metetra
.769 .813 .888 r
.76415 .27964 .77721 .29662 .80656 .27915 .79363 .26033 Metetra
.386 .651 .962 r
.12983 .46458 .14715 .4797 .17025 .4802 .15285 .46639 Metetra
.369 .665 .972 r
.15285 .46639 .17025 .4802 .19371 .48057 .17625 .46805 Metetra
.356 .681 .979 r
.17625 .46805 .19371 .48057 .21754 .48063 .20005 .46939 Metetra
.35 .699 .983 r
.20005 .46939 .21754 .48063 .24176 .48026 .22427 .47022 Metetra
.35 .719 .986 r
.22427 .47022 .24176 .48026 .26638 .47929 .24893 .47039 Metetra
.358 .741 .987 r
.24893 .47039 .26638 .47929 .2914 .4776 .27402 .46971 Metetra
.373 .765 .988 r
.27402 .46971 .2914 .4776 .31682 .47504 .29955 .46805 Metetra
.397 .79 .988 r
.29955 .46805 .31682 .47504 .34264 .4715 .32552 .46524 Metetra
.427 .816 .988 r
.32552 .46524 .34264 .4715 .36885 .46687 .3519 .46118 Metetra
.463 .841 .988 r
.3519 .46118 .36885 .46687 .39542 .46108 .37869 .45577 Metetra
.503 .864 .987 r
.37869 .45577 .39542 .46108 .42235 .45405 .40586 .44891 Metetra
.545 .884 .986 r
.40586 .44891 .42235 .45405 .44961 .44575 .43338 .44057 Metetra
.587 .901 .984 r
.43338 .44057 .44961 .44575 .47717 .43614 .46122 .43071 Metetra
.627 .913 .981 r
.46122 .43071 .47717 .43614 .50501 .42523 .48936 .41935 Metetra
.665 .92 .975 r
.48936 .41935 .50501 .42523 .5331 .41306 .51775 .4065 Metetra
.697 .922 .968 r
.51775 .4065 .5331 .41306 .56142 .39965 .54637 .39222 Metetra
.725 .919 .959 r
.54637 .39222 .56142 .39965 .58994 .38509 .57518 .37661 Metetra
.747 .912 .949 r
.57518 .37661 .58994 .38509 .61863 .36947 .60416 .35977 Metetra
.764 .9 .937 r
.60416 .35977 .61863 .36947 .64747 .35289 .63328 .34183 Metetra
.775 .886 .925 r
.63328 .34183 .64747 .35289 .67646 .33548 .66251 .32293 Metetra
.782 .87 .913 r
.66251 .32293 .67646 .33548 .70557 .31737 .69185 .30323 Metetra
.786 .852 .901 r
.69185 .30323 .70557 .31737 .7348 .29871 .72129 .28292 Metetra
.785 .834 .889 r
.72129 .28292 .7348 .29871 .76415 .27964 .75082 .26216 Metetra
.782 .815 .878 r
.75082 .26216 .76415 .27964 .79363 .26033 .78045 .24114 Metetra
.336 .626 .958 r
.11221 .44921 .12983 .46458 .15285 .46639 .13515 .45215 Metetra
.32 .638 .965 r
.13515 .45215 .15285 .46639 .17625 .46805 .15848 .45494 Metetra
.311 .652 .97 r
.15848 .45494 .17625 .46805 .20005 .46939 .18225 .45739 Metetra
.31 .67 .975 r
.18225 .45739 .20005 .46939 .22427 .47022 .20647 .45929 Metetra
.318 .692 .979 r
.20647 .45929 .22427 .47022 .24893 .47039 .23116 .46044 Metetra
.335 .717 .983 r
.23116 .46044 .24893 .47039 .27402 .46971 .25632 .46066 Metetra
.36 .745 .987 r
.25632 .46066 .27402 .46971 .29955 .46805 .28195 .45977 Metetra
.393 .773 .991 r
.28195 .45977 .29955 .46805 .32552 .46524 .30805 .45761 Metetra
.432 .802 .994 r
.30805 .45761 .32552 .46524 .3519 .46118 .33461 .45404 Metetra
.476 .83 .996 r
.33461 .45404 .3519 .46118 .37869 .45577 .3616 .44894 Metetra
.522 .855 .996 r
.3616 .44894 .37869 .45577 .40586 .44891 .38899 .44222 Metetra
.567 .876 .994 r
.38899 .44222 .40586 .44891 .43338 .44057 .41677 .43382 Metetra
.611 .892 .989 r
.41677 .43382 .43338 .44057 .46122 .43071 .44489 .42372 Metetra
.651 .903 .982 r
.44489 .42372 .46122 .43071 .48936 .41935 .47332 .41191 Metetra
.687 .908 .973 r
.47332 .41191 .48936 .41935 .51775 .4065 .50201 .39843 Metetra
.717 .909 .962 r
.50201 .39843 .51775 .4065 .54637 .39222 .53093 .38336 Metetra
.742 .905 .95 r
.53093 .38336 .54637 .39222 .57518 .37661 .56004 .36678 Metetra
.762 .898 .937 r
.56004 .36678 .57518 .37661 .60416 .35977 .58931 .34882 Metetra
.777 .888 .925 r
.58931 .34882 .60416 .35977 .63328 .34183 .61871 .32962 Metetra
.787 .876 .912 r
.61871 .32962 .63328 .34183 .66251 .32293 .64822 .30937 Metetra
.793 .862 .9 r
.64822 .30937 .66251 .32293 .69185 .30323 .6778 .28824 Metetra
.796 .847 .889 r
.6778 .28824 .69185 .30323 .72129 .28292 .70746 .26643 Metetra
.796 .831 .879 r
.70746 .26643 .72129 .28292 .75082 .26216 .73719 .24414 Metetra
.793 .815 .869 r
.73719 .24414 .75082 .26216 .78045 .24114 .767 .22157 Metetra
.29 .602 .952 r
.09428 .43355 .11221 .44921 .13515 .45215 .11714 .43744 Metetra
.277 .612 .956 r
.11714 .43744 .13515 .45215 .15848 .45494 .14042 .44119 Metetra
.273 .626 .961 r
.14042 .44119 .15848 .45494 .18225 .45739 .16415 .44456 Metetra
.279 .645 .967 r
.16415 .44456 .18225 .45739 .20647 .45929 .18837 .44734 Metetra
.295 .668 .973 r
.18837 .44734 .20647 .45929 .23116 .46044 .21308 .44932 Metetra
.32 .696 .98 r
.21308 .44932 .23116 .46044 .25632 .46066 .2383 .45029 Metetra
.355 .726 .987 r
.2383 .45029 .25632 .46066 .28195 .45977 .26403 .45004 Metetra
.396 .757 .993 r
.26403 .45004 .28195 .45977 .30805 .45761 .29026 .44841 Metetra
.442 .788 .998 r
.29026 .44841 .30805 .45761 .33461 .45404 .31698 .44523 Metetra
.491 .816 1 r
.31698 .44523 .33461 .45404 .3616 .44894 .34416 .44038 Metetra
.54 .841 .999 r
.34416 .44038 .3616 .44894 .38899 .44222 .37178 .43375 Metetra
.587 .861 .995 r
.37178 .43375 .38899 .44222 .41677 .43382 .3998 .42527 Metetra
.63 .876 .987 r
.3998 .42527 .41677 .43382 .44489 .42372 .42818 .41492 Metetra
.669 .886 .978 r
.42818 .41492 .44489 .42372 .47332 .41191 .45689 .40271 Metetra
.703 .891 .966 r
.45689 .40271 .47332 .41191 .50201 .39843 .48588 .38866 Metetra
.731 .891 .953 r
.48588 .38866 .50201 .39843 .53093 .38336 .5151 .37285 Metetra
.754 .888 .939 r
.5151 .37285 .53093 .38336 .56004 .36678 .54452 .3554 Metetra
.772 .882 .925 r
.54452 .3554 .56004 .36678 .58931 .34882 .57409 .33645 Metetra
.786 .874 .912 r
.57409 .33645 .58931 .34882 .61871 .32962 .60378 .31615 Metetra
.795 .864 .9 r
.60378 .31615 .61871 .32962 .64822 .30937 .63356 .2947 Metetra
.801 .853 .889 r
.63356 .2947 .64822 .30937 .6778 .28824 .66341 .2723 Metetra
.804 .84 .878 r
.66341 .2723 .6778 .28824 .70746 .26643 .69332 .24917 Metetra
.804 .828 .869 r
.69332 .24917 .70746 .26643 .73719 .24414 .72327 .22553 Metetra
.802 .815 .862 r
.72327 .22553 .73719 .24414 .767 .22157 .75328 .20161 Metetra
.251 .581 .944 r
.07603 .41762 .09428 .43355 .11714 .43744 .09882 .42225 Metetra
.243 .589 .948 r
.09882 .42225 .11714 .43744 .14042 .44119 .12205 .42673 Metetra
.245 .604 .953 r
.12205 .42673 .14042 .44119 .16415 .44456 .14577 .43082 Metetra
.258 .623 .959 r
.14577 .43082 .16415 .44456 .18837 .44734 .16998 .4343 Metetra
.282 .648 .968 r
.16998 .4343 .18837 .44734 .21308 .44932 .19472 .43692 Metetra
.315 .677 .977 r
.19472 .43692 .21308 .44932 .2383 .45029 .22 .43846 Metetra
.356 .709 .985 r
.22 .43846 .2383 .45029 .26403 .45004 .24582 .4387 Metetra
.403 .741 .993 r
.24582 .4387 .26403 .45004 .29026 .44841 .27217 .43746 Metetra
.454 .772 .997 r
.27217 .43746 .29026 .44841 .31698 .44523 .29903 .43457 Metetra
.506 .799 .998 r
.29903 .43457 .31698 .44523 .34416 .44038 .32639 .42988 Metetra
.556 .823 .996 r
.32639 .42988 .34416 .44038 .37178 .43375 .35422 .42329 Metetra
.603 .842 .989 r
.35422 .42329 .37178 .43375 .3998 .42527 .38248 .41472 Metetra
.645 .856 .98 r
.38248 .41472 .3998 .42527 .42818 .41492 .41112 .40413 Metetra
.682 .865 .968 r
.41112 .40413 .42818 .41492 .45689 .40271 .4401 .39154 Metetra
.714 .869 .955 r
.4401 .39154 .45689 .40271 .48588 .38866 .46938 .37698 Metetra
.74 .87 .941 r
.46938 .37698 .48588 .38866 .5151 .37285 .49889 .36054 Metetra
.761 .868 .927 r
.49889 .36054 .5151 .37285 .54452 .3554 .52861 .34234 Metetra
.778 .864 .913 r
.52861 .34234 .54452 .3554 .57409 .33645 .55848 .32253 Metetra
.79 .858 .9 r
.55848 .32253 .57409 .33645 .60378 .31615 .58847 .30128 Metetra
.799 .851 .889 r
.58847 .30128 .60378 .31615 .63356 .2947 .61854 .27881 Metetra
.805 .842 .878 r
.61854 .27881 .63356 .2947 .66341 .2723 .64867 .25533 Metetra
.809 .833 .869 r
.64867 .25533 .66341 .2723 .69332 .24917 .67884 .23108 Metetra
.809 .823 .861 r
.67884 .23108 .69332 .24917 .72327 .22553 .70905 .20631 Metetra
.808 .813 .855 r
.70905 .20631 .72327 .22553 .75328 .20161 .73929 .18125 Metetra
.221 .565 .938 r
.05745 .40141 .07603 .41762 .09882 .42225 .08019 .40654 Metetra
.219 .572 .941 r
.08019 .40654 .09882 .42225 .12205 .42673 .10339 .41153 Metetra
.228 .587 .946 r
.10339 .41153 .12205 .42673 .14577 .43082 .12709 .41612 Metetra
.248 .607 .954 r
.12709 .41612 .14577 .43082 .16998 .4343 .15132 .42007 Metetra
.279 .633 .963 r
.15132 .42007 .16998 .4343 .19472 .43692 .17609 .42312 Metetra
.318 .662 .973 r
.17609 .42312 .19472 .43692 .22 .43846 .20143 .42505 Metetra
.364 .693 .982 r
.20143 .42505 .22 .43846 .24582 .4387 .22733 .42562 Metetra
.415 .724 .989 r
.22733 .42562 .24582 .4387 .27217 .43746 .25379 .42463 Metetra
.468 .754 .992 r
.25379 .42463 .27217 .43746 .29903 .43457 .2808 .42191 Metetra
.52 .78 .991 r
.2808 .42191 .29903 .43457 .32639 .42988 .30833 .41729 Metetra
.569 .802 .987 r
.30833 .41729 .32639 .42988 .35422 .42329 .33636 .41067 Metetra
.614 .819 .979 r
.33636 .41067 .35422 .42329 .38248 .41472 .36483 .40197 Metetra
.655 .832 .968 r
.36483 .40197 .38248 .41472 .41112 .40413 .39372 .39116 Metetra
.689 .841 .956 r
.39372 .39116 .41112 .40413 .4401 .39154 .42296 .37823 Metetra
.719 .846 .942 r
.42296 .37823 .4401 .39154 .46938 .37698 .45251 .36323 Metetra
.743 .848 .928 r
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      Editable->False]], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
    \(Integrate[\ t\ Exp[\(-t^2\)], \ {t, 0, x}]\)], "Input"],

Cell[BoxData[
    \(1\/2 - \[ExponentialE]\^\(-x\^2\)\/2\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(NIntegrate[\ t\ Exp[\(-t^2\)], \ {t, 0, 2}]\)], "Input"],

Cell[BoxData[
    \(0.490842180555633`\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\)\)], \
"Input"],

Cell[BoxData[
    \(\(\( (*\ 
      we\ now\ show\ how\ to\ define\ functions . \ 
          We\ use\ underscore\ by\ the\ variable\ names\ *) \)\(\
\[IndentingNewLine]\)\(\(\(nthprime[n_]\  := \ Prime[n]\) \)\(;\)\(\ \)\( (*\ 
      this\ gives\ the\ nth\ prime\ *) \)\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(nthprime[10]\), "\[IndentingNewLine]", 
    \(nthprime[9]\)}], "Input"],

Cell[BoxData[
    \(29\)], "Output"],

Cell[BoxData[
    \(23\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\(\(diffnthprime[n_]\  := \ 
        nthprime[n]\  - \ nthprime[n - 1]\) \)\(;\)\(\ \)\( (*\ 
      this\ gives\ the\ difference\ b/w\ the\ nth\ and\ \((n - 1)\) 
        st\ prime\ *) \)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(diffnthprime[10]\)], "Input"],

Cell[BoxData[
    \(6\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(powersum[n_, k_]\  := \ Sum[\ i^k, \ {i, 0, n}]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(powersum[100, 2]\)], "Input"],

Cell[BoxData[
    \(338350\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\
\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)\)\)], "Input"],

Cell[BoxData[
    \(\(\( (*\ 
      we\ now\ show\ how\ to\ define\ functions . \ 
          We\ use\ underscore\ by\ the\ variable\ names, \ 
      and\ then\ Module\ followed\ by\ braces . \ 
          Whatever\ is\ inside\ the\ braces\ are\ local\ variables, \ 
      known\ only\ to\ this\ procedure\ or\ function . \ 
          If\ we\ write\ something\ equals\ {}\ then\ we' 
        ve\ initialized\ it\ to\ be\ the\ empty\ \(\(array\)\(.\)\)\ *) \)\(\
\[IndentingNewLine]\)\( (*\ 
      this\ function\ counts\ the\ number\ of\ solutions\ to\ x^2\  + \ 
          y^2\  = \ a\ modulo\ n; \ it\ has\ two\ inputs, \ 
      n\ and\ \(\(a\)\(.\)\)\ *) \)\(\[IndentingNewLine]\)\( (*\ 
      we\ find\ all\ solutions\ and\ store\ it\ in\ an\ array; \ 
      as\ we\ want\ to\ be\ able\ to\ access\ this\ array\ afterwards, \ 
      we\ do\ not\ have\ it\ as\ a\ local\ but\ rather\ global\ variable, \ 
      ie, \ not\ inside\ the\ braces\ inside\ the\ \(\(module\)\(.\)\)\ *) \)\
\(\[IndentingNewLine]\)\(NumSolns[n_, a_]\  := \ 
        Module[{x, y, count}, \  (*\ 
            we\ put\ the\ local\ variables\ in\ braces\ after\ module\ \
*) \[IndentingNewLine]count\  = \ 0; \  (*\ we\ initialize\ count, \ 
            the\ number\ of\ solns, \ to\ zero\ *) \[IndentingNewLine]Clear[
            ListOfSolns]; \  (*\ if\ we\ had\ used\ ListOfSolns\ before, \ 
            this\ clears\ it\ *) \[IndentingNewLine]ListOfSolns\  = \ {}; \  \
(*\ we\ set\ ListOfSolns\ to\ the\ empty\ array\ *) \[IndentingNewLine]For\ [
            x\  = \ 0, \ 
            x\  \[LessEqual] \ n - 1, \ \(x++\), \  (*\ 
              how\ we\ do\ a\ for\ loop\ *) \[IndentingNewLine]{\ \  (*\ 
                starting\ brace\ of\ the\ for\ loop\ \
*) \[IndentingNewLine]For\ [y\  = \ 0, \ 
                y \[LessEqual] \ 
                  n - 1, \ \(y++\), \
\[IndentingNewLine]{\[IndentingNewLine]If[\ 
                    Mod[x^2\  + \ y^2\  - \ a, 
                        n]\  \[Equal] \ \ 0, \ \[IndentingNewLine]{\ \
\[IndentingNewLine]Print["\<We found a solution: (x,y) = (\>", \ 
                        x, \ "\<, \>", \ y, \ "\<).\>"]; \  (*\ 
                        prints\ the\ found\ solution\ *) \[IndentingNewLine]\ 
                      count\  = \ count + 1; \  (*\ 
                        
                        increases\ count\ by\ 1\ to\ indicate\ found\ a\ soln\
\ *) \[IndentingNewLine]ListOfSolns\  = \ 
                        Append[ListOfSolns, \ {x, y}]; \  (*\ 
                        adds\ to\ list\ of\ solns\ *) \[IndentingNewLine]}\
\[IndentingNewLine]]; \  (*\ 
                    end\ of\ If\ loop\ *) \[IndentingNewLine]}]; \  (*\ 
                end\ of\ y\ For\ loop\ *) \[IndentingNewLine]}\ ];  (*\ 
            end\ of\ x\ For\ loop\ *) \[IndentingNewLine]Print["\<The number \
of solutions when the modulus = \>", \ n, \ "\<, a = \>", \ 
            a, \ "\< is equal to \>", \ 
            count];\[IndentingNewLine]];\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ we\ compile\ the\ above\ when\ n = \(5\ and\ a = 1\); \ 
      since\ count\ is\ a\ local\ variable, \ 
      when\ we\ print\ count\ AFTER\ running\ the\ program\ it\ is\ not\ \
related\ to\ the\ value\ from\ the\ program; \ however, \ 
      the\ value\ of\ ListOfSolns\ IS\ related\ to\ its\ value\ from\ running\
\ the\ \(\(program\)\(.\)\)\ *) \)\(\[IndentingNewLine]\)\(NumSolns[5, 
      1]\[IndentingNewLine]
    ListOfSolns\[IndentingNewLine]
    count\)\)\)], "Input"],

Cell[BoxData[
    InterpretationBox[\("We found a solution: (x,y) = ("\[InvisibleSpace]0\
\[InvisibleSpace]", "\[InvisibleSpace]1\[InvisibleSpace]")."\),
      SequenceForm[ "We found a solution: (x,y) = (", 0, ", ", 1, ")."],
      Editable->False]], "Print"],

Cell[BoxData[
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\[InvisibleSpace]", "\[InvisibleSpace]4\[InvisibleSpace]")."\),
      SequenceForm[ "We found a solution: (x,y) = (", 0, ", ", 4, ")."],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox[\("We found a solution: (x,y) = ("\[InvisibleSpace]1\
\[InvisibleSpace]", "\[InvisibleSpace]0\[InvisibleSpace]")."\),
      SequenceForm[ "We found a solution: (x,y) = (", 1, ", ", 0, ")."],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox[\("We found a solution: (x,y) = ("\[InvisibleSpace]4\
\[InvisibleSpace]", "\[InvisibleSpace]0\[InvisibleSpace]")."\),
      SequenceForm[ "We found a solution: (x,y) = (", 4, ", ", 0, ")."],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox[\("The number of solutions when the modulus = "\
\[InvisibleSpace]5\[InvisibleSpace]", a = \
"\[InvisibleSpace]1\[InvisibleSpace]" is equal to "\[InvisibleSpace]4\),
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      "The number of solutions when the modulus = ", 5, ", a = ", 1, 
        " is equal to ", 4],
      Editable->False]], "Print"],

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(ListOfSolns[\([1]\)]\  (*\ 
      this\ access\ the\ first\ entry\ of\ the\ soln, \ 
      a\ pair\ *) \[IndentingNewLine]
    ListOfSolns[\([1, 1]\)]\  (*\ 
      accesses\ the\ first\ entry\ of\ the\ first\ entry\ of\ the\ solns\ \
*) \[IndentingNewLine]
    \(ListOfSolns[\([1]\)]\)[\([1]\)]\)\(\ \)\( (*\ same\ as\ above, \ 
      alternate\ way\ to\ access\ it\ *) \)\)\)], "Input"],

Cell[BoxData[
    \({0, 1}\)], "Output"],

Cell[BoxData[
    \(0\)], "Output"],

Cell[BoxData[
    \(0\)], "Output"]
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Cell[BoxData[
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\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\)\)], \
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Cell[BoxData[{
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    \(<< Statistics`DataManipulation`\), "\[IndentingNewLine]", 
    \(\(\(<< Statistics`NormalDistribution`\)\(\ \)\( (*\ 
      loads\ in\ some\ packages\ *) \)\(\[IndentingNewLine]\)\( (*\ 
      this\ function\ calculates\ n\ numbers\ chosen\ uniformly\ in\ [0, 
          1], \ say\ x1, \  ... , \ 
      x_numdo . \ 
          It\ then\ orders\ them\ in\ increasing\ order, \ \(\(\(say\ y1\)\(\ \
\)\(\[LessEqual]\)\)\  ... \)\  \[LessEqual] \ 
        ynumdo . \ As\ there\ are\ numdo\ elements, \ 
      the\ average\ spacing\ between\ ordered\ elements\ is\ 1/numdo; \ 
      thus\ if\ we\ multiply\ by\ numdo\ then\ on\ average\ the\ differences\ \
are\ of\ size\ 1. \[IndentingNewLine]The\ program\ then\ calculates\ these\ \
normalized\ differences, \ 
      stores\ them\ in\ a\ file . \ \[IndentingNewLine]We\ then\ make\ a\ \
histogram\ of\ the\ \(\(differences\)\(.\)\)\ \ *) \)\(\[IndentingNewLine]\)
    \) (*\ we\ have\ only\ one\ local\ dummy\ variable, \ i; \ 
      it\ is\ in\ braces\ \(\(below\)\(.\)\)\ *) \), "\[IndentingNewLine]", 
    \(\(UniformDiff[numdo_]\  := \ 
        Module[{i\ }, \[IndentingNewLine]ValueList\  = \ {}; \  (*\ 
            initializes\ to\ the\ empty\ list\ *) \[IndentingNewLine]DiffList\
\  = \ {}; \  (*\ 
            initializes\ to\ the\ empty\ list\ *) \[IndentingNewLine]For[
            i\  = \ 1, \ 
            i\  \[LessEqual] \ 
              numdo, \ \(i++\), \[IndentingNewLine]\(ValueList\  = \ 
                Append[ValueList, \ Random[]];\)\[IndentingNewLine]]; \  (*\ 
            end\ of\ i\ For\ loop\ \
*) \[IndentingNewLine]\[IndentingNewLine]ValueList\  = \ 
            Sort[ValueList]; \  (*\ 
            this\ sorts\ the\ list\ into\ increasing\ order\ \
*) \[IndentingNewLine]\[IndentingNewLine]For[i\  = \ 1, \ 
            i\  \[LessEqual] \ 
              numdo\  - \ 
                1, \ \(i++\), \
\[IndentingNewLine]{\[IndentingNewLine]\(DiffList\  = \ 
                  Append[DiffList, \ \((ValueList[\([i + 1]\)]\  - \ 
                          ValueList[\([i]\)])\)\ *\ 
                      numdo];\)\[IndentingNewLine]}\  (*\ 
              we\ use\ braces\ to\ begin\ and\ end\ statements\ in\ the\ for\ \
loop; \ there\ is\ no\ need\ here\ as\ there\ is\ only\ one\ statement\ in\ \
the\ for\ \(\(loop\)\(.\)\)\ *) \[IndentingNewLine]]; \  (*\ 
            end\ of\ i\ For\ loop\ *) \[IndentingNewLine]\[IndentingNewLine]\ 
          Histogram[\ DiffList]; \ \  (*\ 
            this\ prints\ a\ histogram\ *) \[IndentingNewLine] (*\ 
            the\ second\ line\ scales\ the\ histogram\ so\ that\ we\ have\ \
percents\ and\ not\ total\ numbers\ *) \[IndentingNewLine]Histogram[
            DiffList, \ HistogramCategories\  \[Rule] \ bincat, \ 
            HistogramScale\  \[Rule] \ 
              1];\[IndentingNewLine]\[IndentingNewLine]];\)\)}], "Input"],

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Cell[BoxData[
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Cell[BoxData[
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Distributions. It has a lot of nice distributions,
for example, the Gaussian distribution. we will save the Gaussian \
distribution to the name gdist.\>"\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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with entries from a normal distribution. Watch what happens to the entries as \
we change the standard deviation.\>"\)], "Input"],

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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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\(\[IndentingNewLine]\)\)\)], "Input"],

Cell[BoxData[
    \("\<We now include a quick program to calculate eigenvalues of Toeplitz \
Matrices. This code can easily be 
modified for other investigations.\>"\)], "Input"],

Cell[BoxData[
    \("\<First we make sure we have loaded in the packages we need.\>"\)], \
"Input"],

Cell[BoxData[
    \(<< Graphics`Graphics`\)], "Input"],

Cell[BoxData[{
    \(<< Statistics`DataManipulation`\), "\[IndentingNewLine]", 
    \(<< Statistics`NormalDistribution`\)}], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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error: new symbol name \\\"\\!\\(ndist\\)\\\" is similar to existing symbol \
\\\"\\!\\(gdist\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
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Cell[BoxData[
    \("\<In the program below, we calculate num matrices, each is n x n. we \
choose the entries randomly from the standard normal
distribution subject to being a real symmetric toeplitz matrix with zeros on \
the main diagonal. remember a toeplitz matrix is constant along diagonals. we \
normalize the eigenvalues by
sqrt[n]. we also have some if statements that print based on where we are -- \
the point is to know how much longer the program
will have to run. we then create a histogram.\>"\)], "Input"],

Cell[BoxData[
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        Module[\ {aa, \ b}, \ \[IndentingNewLine]\ \ \ \ \ \ \ \ Clear[
            temp]; \[IndentingNewLine]\ \ \ \ \ \ \ \ temp\  = \ {}; \ \
\[IndentingNewLine]\ \ \ \ \ \ \ \ For[aa\  = \ 1, \ 
            aa\  \[LessEqual] \ 
              num, \ \(aa++\), \[IndentingNewLine]\ \ \ \ \ \ \ \ \ {\
\[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ Clear[
                a]; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ For[i\  = \ 1, \ 
                i\  \[LessEqual] \ n, \ \(i++\), \ 
                a[i, i]\  = \ 0]; \ \ \  (*\ initializes\ a, \ 
                a\ two - dimensional\ array, \ 
                to\ all\ zeros\ *) \[IndentingNewLine]\t\t\[IndentingNewLine]\
\ \ \ \ \ \ \ \ \ For[k\  = \ 2, \ 
                k\  \[LessEqual] \ 
                  n, \ \(k++\), \[IndentingNewLine]\t{\[IndentingNewLine]\ \ \
\ \ \ \ \ \ \ \ ran1\  = \ Random[ndist]\ ; \[IndentingNewLine]\t\ \ For[
                    i = 1, \ 
                    i\  \[LessEqual] \ 
                      n\  - \ k\  + \ 
                        1, \ \(i++\), \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ {\ \(a[\ i, \ 
                            k\  - 1\  + \ i]\  = \ \ \(a[
                              k\ \  - \ 1\  + \ i, \ \ i]\  = \ 
                            ran1\);\)\ \ \ }\ \ \[IndentingNewLine]\t\ \ \ \ \
\ \ \ ]; \  (*\ end\ of\ i\ for\ loop\ *) \[IndentingNewLine]\t}]; \  (*\ 
                end\ of\ k\ for\ loop\ *) \[IndentingNewLine] (*\ 
                this\ makes\ a, \ a\ 2 - dim\ array, \ 
                toeplitz\ *) \[IndentingNewLine]\[IndentingNewLine]\ \ \ \ \ \
\ \ \ \ \ \ \ mat\  = \ Array[\ a, \ {n, n}]; \  (*\ 
                this\ makes\ a\ matrix\ from\ a\ *) \[IndentingNewLine]\ \ \ \
\ \ \ \ \ \ \ \ \ tempev\  = \ \ N[
                  Eigenvalues[mat]\ /\ 
                    Sqrt[1.0*
                        n]]\ ; \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ If[\
\ Mod[aa, \ 20]\  \[Equal] \ 0, \ Print["\<We're at aa = \>", \ aa]]; \  (*\ 
                this\ prints\ every\ 20\ lines; \ 
                the\ advantage\ of\ such\ a\ line\ of\ code\ is\ that\ for\ \
long\ programs, \ 
                you\ get\ a\ sense\ of\ how\ long\ it\ has\ run\ \
*) \[IndentingNewLine]\t\ \ \ For[b\  = \ 1, \ 
                b \[LessEqual] \ 
                  n, \ \(b++\), \ \(temp\  = \ 
                    Append[\ temp, \ 
                      tempev[\([b]\)]\ ]\ ;\)\ \ ];\[IndentingNewLine]\ \ \ \ \
\ \ \ \ \ }\ ]; \  (*\ 
            end\ of\ aa\ loop\ *) \[IndentingNewLine]\[IndentingNewLine]\ 
          Histogram[\ temp\ ]; \ \[IndentingNewLine] (*\ 
            the\ second\ line\ scales\ the\ histogram\ so\ that\ we\ have\ \
percents\ and\ not\ total\ numbers\ *) \[IndentingNewLine]Histogram[temp, \ 
            HistogramCategories\  \[Rule] \ bincat, \ 
            HistogramScale\  \[Rule] \ 
              1];\[IndentingNewLine]\ \ ];\)\)\)], "Input"],

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Cell[BoxData[
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\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)\)\)], "Input"],

Cell[BoxData[
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Cell[BoxData[
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\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\
\[IndentingNewLine]\)\(\[IndentingNewLine]\)\)\)], "Input"],

Cell[BoxData[
    \("\<Below is some code for investigating 3x+1 and its generalizations\>"\
\)], "Input"],

Cell[BoxData[
    \(RemoveTwo[x_]\  := \ 
      Module[{i}, \ i\  = \ x; \ 
        While[\ \ Mod[i, 2]\  \[Equal] \ 0, \ i\  = \ i\ /\ 2]; \ 
        Return[i]]\)], "Input"],

Cell[BoxData[{
    \(\(F3[x_]\  := \ 
        RemoveTwo[\ 3*RemoveTwo[x]\  + \ 1];\)\), "\[IndentingNewLine]", 
    \(\(F5[x_]\  := \ RemoveTwo[\ 5*RemoveTwo[x]\  + \ 1];\)\)}], "Input"],

Cell[BoxData[
    \("\<We now give a code to check the first digits of the 3x+1 \
problem\>"\)], "Input"],

Cell[BoxData[
    \(\(ben3[start_, \ num_]\  := \ 
        Module[\ {d, m, j, x}, \[IndentingNewLine]For[m = 1, \ 
            m \[LessEqual] \ 9, \ \(m++\), \ 
            numdig[m]\  = \ 0]; \[IndentingNewLine]x\  = \ 
            start; \[IndentingNewLine]count\  = \ 0; \[IndentingNewLine]For[
            j\  = \ 1, \ 
            j\  \[LessEqual] \ 
              num, \ \(j++\), \[IndentingNewLine]{\[IndentingNewLine] (*\ \
\(Print["\<x = \>", \ x, \ "\< at j = \>", \ 
                    j];\)\ *) \[IndentingNewLine]\(numdig[\ \ \ Mod[\ \ Floor[
                      x*1.0\ /\ 10^Floor[Log\ [10, \ x]]\ ], \ 
                    10\ ]\ ]++\); \[IndentingNewLine]count\  = \ 
                count + 1; \[IndentingNewLine]x\  = \ 
                F3[x]; \[IndentingNewLine]If[x\  \[Equal] \ 1, \ 
                j\  = \ num\  + \ 
                    5];\[IndentingNewLine]}]; \[IndentingNewLine]bF[
              y_]\  := \ numdig[Floor[y\  + \  .5]]\ /
              count; \[IndentingNewLine]ben[y_]\  := \ 
            Log[10, \ 
              1\  + \ \((1.0/y)\)]; \[IndentingNewLine]Plot[\ {bF[y], \ 
              ben[y]}, \ {y,  .5, 
              9.5}]; \[IndentingNewLine]Print["\<Count (the number of terms \
before we hit one) = \>", \ 
            count]; \[IndentingNewLine]Print["\<The number is approximately \
10^\>", \ Floor[Log[10, 1.0\ start]]];\[IndentingNewLine]];\)\)], "Input"],

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Cell[BoxData[
    InterpretationBox[\("Count (the number of terms before we hit one) = "\
\[InvisibleSpace]529\),
      SequenceForm[ "Count (the number of terms before we hit one) = ", 529],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox[\("The number is approximately \
10^"\[InvisibleSpace]75\),
      SequenceForm[ "The number is approximately 10^", 75],
      Editable->False]], "Print"]
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Cell[BoxData[
    \("\<The following compares the terms of 5x+1 to a geometric brownian \
motion and the first digits of the
terms to Benford's law.\>"\)], "Input"],

Cell[BoxData[
    \(glog[x_, \ end_]\  := \ 
      Module[\ {numat, \ tempplot, \ numdig, \ lognumdig, \ m, y, \ 
          temp}, \[IndentingNewLine]Clear[tempplot]; \ 
        Clear[temp]; \[IndentingNewLine]For[m\  = \ 1, \ 
          m\  \[LessEqual] \ 9, \ \(m++\), \ {\ lognumdig[m]\  = \ 0; \ 
            numdig[m]\  = \ 
              0;\ }\ ]; \[IndentingNewLine]tempplot\  = \ {\ {0, 
              0}\ }; \[IndentingNewLine]numat\  = \ 
          RemoveTwo[x]; \ \[IndentingNewLine]\ \ \ \ For[j = \ 1, \ 
            j\  \[LessEqual] \ 
              end, \ \ \(j++\), \ \[IndentingNewLine]\ \ \ \ \ \ {\ \ \
\[IndentingNewLine]If[\ 
                Mod[j, \ 1000]\  \[Equal] \ 
                  0, \ \(Print["\<j = \>", \ j, \ "\< x = \>", \ 
                    numat, \ "\< and goes to \>", \ 
                    F[numat]];\)\ ]; \[IndentingNewLine]AppendTo[\ 
                tempplot, \ {j, \ 
                  Log[E, numat]}\ ]; \[IndentingNewLine]AppendTo[\ 
                tempplot, \ \ {j, \ 
                  Log[E, \ 1.25]\ *\ j\  + \ 
                    Log[E, x]}]; \[IndentingNewLine]\(numdig[\ \ \ Mod[\ \ \
Floor[numat*1.0\ /\ 10^Floor[Log\ [10, \ numat]]\ ], \ 
                    10\ ]\ ]++\); \[IndentingNewLine]temp\  = \ 
                Exp[Log[10, \ numat]\  - \ 
                    j*Log[10, \ 
                        1.25]]; \[IndentingNewLine]\(lognumdig[\ \ \ Mod[\ \ \
Floor[temp*1.0\ /\ 10^Floor[Log\ [10, \ temp]]\ ], \ 
                    10\ ]\ ]++\); \[IndentingNewLine]numat\  = \ 
                5\ *\ numat + \ 1; \ \  (*\ 
                doing\ the\ 5  x + 
                  1\ problem\ *) \[IndentingNewLine]\ }\ \[IndentingNewLine]]\
\ \[IndentingNewLine]ListPlot[\ tempplot, \ 
            PlotJoined\  \[Rule] False]\ ; \[IndentingNewLine]Clear[
          tempplot]; \ 
        tempplot\  = \ {\ {0, 0}\ }; \[IndentingNewLine]For[\ m\  = \ 1, \ 
          m\  \[LessEqual] \ 9, \ \(m++\), \ 
          AppendTo[
            tempplot, \ {m, \ 
              1.0*numdig[m]\ /\ end}]]; \[IndentingNewLine]ListPlot[\ 
          tempplot]; \[IndentingNewLine]Clear[
          tempplot]; \[IndentingNewLine]\[IndentingNewLine]bF[y_]\  := \ 
          numdig[Floor[y\  + \  .5]]\ /\ end; \[IndentingNewLine]ben[
            y_]\  := \ Log[10, \ 
            1\  + \ \((1.0/y)\)]; \[IndentingNewLine]Plot[\ {bF[y], \ 
            ben[y]}, \ {y,  .5, 
            9.5}]; \[IndentingNewLine]\[IndentingNewLine]\ 
        tempplot\  = \ {\ {0, 0}\ }; \[IndentingNewLine]For[\ m\  = \ 1, \ 
          m\  \[LessEqual] \ 9, \ \(m++\), \ 
          AppendTo[
            tempplot, \ {m, \ 
              1.0*lognumdig[m]\ /\ end}]]; \[IndentingNewLine] (*\ \(ListPlot[
              tempplot];\)\ *) \[IndentingNewLine]\[IndentingNewLine]logbF[
            y_]\  := \ lognumdig[Floor[y\  + \  .5]]\ /\ 
            end; \[IndentingNewLine]Plot[\ {logbF[y], \ ben[y]}, \ {y,  .5, 
            9.5}];\[IndentingNewLine]\ ]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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22425926298695459752243641481049947700512903203673431562492623925209045410156\
\[InvisibleSpace]" and goes to "\[InvisibleSpace]F[
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      "j = ", 1000, " x = ", 
        7862745985889619055491504414298159654023995435681640107437943312781021\
224970909720278659755799648287379587790162981744109015482367407722926539812884\
943116621274909862215389391058408515181222930481021634958533578021176317937155\
742187533626934231622249605133841848543103431533184876468581445498810020480576\
993227606874217405498453316504213554591530916588741833532670313983639967891841\
409143476330361596300298501053692659403463302721459631650555988785588443008078\
030943487380620893416334403768034267699514055514077676641772301347225782533814\
594643007821389831802176662884612218451297844468886131281678183561327147273398\
378209362242592629869545975224364148104994770051290320367343156249262392520904\
5410156, " and goes to ", 
        F[ 
        7862745985889619055491504414298159654023995435681640107437943312781021\
224970909720278659755799648287379587790162981744109015482367407722926539812884\
943116621274909862215389391058408515181222930481021634958533578021176317937155\
742187533626934231622249605133841848543103431533184876468581445498810020480576\
993227606874217405498453316504213554591530916588741833532670313983639967891841\
409143476330361596300298501053692659403463302721459631650555988785588443008078\
030943487380620893416334403768034267699514055514077676641772301347225782533814\
594643007821389831802176662884612218451297844468886131281678183561327147273398\
378209362242592629869545975224364148104994770051290320367343156249262392520904\
5410156]],
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Cell[CellGroupData[{

Cell[BoxData[
    \(\(\( (*\ 
      does\ the\ josephus\ problem\ for\ k\  <= \ 
        4\ *) \)\(\[IndentingNewLine]\)\( (*\ 
      you\ have\ n\ people\ in\ a\ circle, \ 
      person\ one\ starts\ with\ a\ gun\ and\ kills\ the\ k - 
        1  st\ person\ from\ him\ and\ then\ passes\ the\ gun\ to\ the\ next\ \
person\ on\ the\ circle . \ 
            this\ continues\ until\ only\ 1\ person\ is\ \(\(left\)\(.\)\)\ \
*) \)\(\[IndentingNewLine]\)\(josp[n_, \ k_]\  := \ 
        Module[{}, \[IndentingNewLine]nlist\  = \ {}; \
\[IndentingNewLine]count\  = \ 0; \[IndentingNewLine]For[i\  = \ 1, \ 
            i\  \[LessEqual] \ n, \ \(i++\), \ 
            nlist\  = \ 
              AppendTo[nlist, i]]; \[IndentingNewLine]listlength\  = \ 
            Length[nlist]; \[IndentingNewLine]Print[
            nlist]; \[IndentingNewLine]While[
            listlength\  \[GreaterEqual] \ k\  && \ 
              count\  \[LessEqual] \ 
                n, \[IndentingNewLine]{\[IndentingNewLine]count\  = \ 
                count + 1; \  (*\ 
                makes\ sure\ coding\ error\ doesn' t\ have\ program\ run\ 4  
                  ever*) \[IndentingNewLine]newlist\  = \ {}; \
\[IndentingNewLine]tempnewlist\  = \ {}; \[IndentingNewLine]last\  = \ 
                0; \[IndentingNewLine]If[
                Mod[listlength, k]\  == \ 
                  0, \[IndentingNewLine]{\[IndentingNewLine]\(For[
                      i\  = \ 1, \ 
                      i\  \[LessEqual] \ 
                        listlength, \ \(i++\), \[IndentingNewLine]\(If[
                          Mod[i, k]\  \[NotEqual] \ 
                            0, \ \(newlist\  = \ 
                              AppendTo[newlist, 
                                nlist[\([i]\)]];\)];\)];\)\[IndentingNewLine]}\
]; \[IndentingNewLine]\[IndentingNewLine]If[
                Mod[listlength, k]\  != \ 
                  0, \ \[IndentingNewLine]{\[IndentingNewLine]tempnewlist\  = \
\ {}; \[IndentingNewLine]For[i\  = \ 1, \ 
                    i\  \[LessEqual] \ 
                      k*Floor[
                          listlength/k], \ \(i++\), \[IndentingNewLine]\(If[
                        Mod[i, k]\  \[NotEqual] \ 
                          0, \ \(tempnewlist\  = \ 
                            AppendTo[tempnewlist, 
                              nlist[\([i]\)]];\)];\)]; \
\[IndentingNewLine]newlist\  = \ {}; \[IndentingNewLine]remainder\  = \ 
                    listlength\  - \ 
                      k*\ Floor[listlength/k]; \[IndentingNewLine]For[
                    i\  = \ k*\ Floor[listlength/k] + 1, \ 
                    i\  <= \ listlength, \ \(i++\), \ 
                    newlist\  = \ 
                      AppendTo[newlist, \ 
                        nlist[\([i]\)]]]; \[IndentingNewLine]For[i\  = \ 1, \ 
                    i\  \[LessEqual] \ 
                      Length[tempnewlist], \ \(i++\), \
\[IndentingNewLine]newlist\  = \ 
                      AppendTo[newlist, 
                        tempnewlist[\([i]\)]]];\[IndentingNewLine]}]; \
\[IndentingNewLine]\[IndentingNewLine]nlist\  = \ 
                newlist; \[IndentingNewLine]Print[
                nlist]; \[IndentingNewLine]listlength\  = \ 
                Length[nlist];\[IndentingNewLine]\[IndentingNewLine]}]; \
\[IndentingNewLine]\[IndentingNewLine]If[k\  \[Equal] \ 3, \ 
            Print["\<{\>", nlist[\([2]\)], "\<}\>"]]; \[IndentingNewLine]If[
            k\ \  == \ 4\  && \ Length[nlist]\  \[Equal] \ 2, \ 
            Print["\<{\>", nlist[\([1]\)], "\<}\>"]]; \[IndentingNewLine]If[
            k\  == \ 3\  && \ Length[nlist]\  \[Equal] \ 3, 
            Print["\<{\>", 
              nlist[\([2]\)], \
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Cell[BoxData[
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\"\\!\\(Last\\)\\\". \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \
ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \
ButtonData:>\\\"General::spell1\\\"]\\)\"\>"}]], "Message"]
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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Cell[BoxData[
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\(\(ninelist\  = \ {1, 2, 3, 4, 5, 6, 7, 8, 9};\)\[IndentingNewLine]
    \(permninelist\  = \ Permutations[ninelist];\)\[IndentingNewLine]
    \(For[j\  = \ 1, \ 
        j\  \[LessEqual] \ \(9!\), \ \(j++\), \[IndentingNewLine]{\
\[IndentingNewLine]For[i\  = \ 1, \ 
            i\  \[LessEqual] \ 
              9, \ \(i++\), \[IndentingNewLine]{\[IndentingNewLine]\(matrix[
                    i]\  = \ \(permninelist[\([j]\)]\)[\([i]\)];\)\
\[IndentingNewLine]}]; \[IndentingNewLine]rows\  = \ 
            0; \[IndentingNewLine]columns\  = \ 
            0; \[IndentingNewLine]diag\  = \ 0; \[IndentingNewLine]If[\ 
            matrix[1]\  + \ matrix[2]\  + \ matrix[3]\  \[Equal] \ 15\  && \ 
              matrix[4]\  + \ matrix[5]\  + \ matrix[6]\  \[Equal] \ 
                15\  && \ matrix[7]\  + \ matrix[8]\  + \ 
                  matrix[9]\  \[Equal] \ 15, \ 
            rows\  = \ 1]; \[IndentingNewLine]\[IndentingNewLine]If[\ 
            matrix[1]\  + \ matrix[4]\  + \ matrix[7]\  \[Equal] \ 15\  && \ 
              matrix[2]\  + \ matrix[5]\  + \ matrix[8]\  \[Equal] \ 
                15\  && \ matrix[3]\  + \ matrix[6]\  + \ 
                  matrix[9]\  \[Equal] \ 15, \ 
            columns\  = \ 1]; \[IndentingNewLine]\[IndentingNewLine]If[\ 
            matrix[1]\  + \ matrix[4]\  + \ matrix[9]\  \[Equal] \ 
              matrix[3]\  + \ matrix[4]\  + \ matrix[7], \ 
            diag\  = \ 1]; \[IndentingNewLine]\[IndentingNewLine]If[
            rows\  == \ 1\  && \ columns\  \[Equal] \ 1\  && \ 
              diag\  \[Equal] \ 1, \[IndentingNewLine]\(Print[
                MatrixForm[{{matrix[1], matrix[2], matrix[3]}, {matrix[4], 
                      matrix[5], matrix[6]}, {matrix[7], matrix[8], 
                      matrix[9]}}]];\)];\[IndentingNewLine]\[IndentingNewLine]\
}];\)\)\)\)], "Input"]
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(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)