Nov. 11, 2019, John Æonid — VERY ROUGH DRAFT, YET TO BE PROOFED

Four people in a room all shake hands with each other exactly once.  How many times did they shake hands?

It relates to the square of the number but is less than that, because no one shakes hands with themselves.  But, each person does shake hands with every other person, and every other person is one less than the number of people.    But, the shaking of hands is non-directional.  So, you can't just take n(n-1); you have to take half of that.  That gives us n(n-1)/2.  If you fill a table with the pairs of people that shake hands, what you find that the cells of the table that are filled once for each pair of people that shake hands, what you see is that the filled cells form a triangle.

And, as it happens with the two terms n and n-1, exactly one of them will always be an even number.  So, these can often be easy to work out in one's head.  The number four yields four and three, and four is even, so divide four in half, and multiply that by three.  That makes two times three. 

$ perl -E '
say "<table>";
@fw = qw(
  2 5
  5 5 5 5
  8 8 8 8
);
say "  <thead>";
$fmt = join "", map { "<th>%" . $_ . "s</th>" } @fw;
printf "     <tr>$fmt</tr>\n", qw(
  n sq
  trip htrip trim htrim
  tripPrd htripPrd trimPrd htrimPrd
);
say "  </thead>";
say "  <tbody>";
$fmt = join "", map { "<td>%" . $_ . "i</td>" } @fw;
for $ni (2..24) {
  $sq = $ni ** 2;
  $trip = $ni * ( $ni + 1 );
  $trim = $ni * ( $ni - 1 );
  $htrim = $trim / 2;
  $htrip = $trip / 2;
  $tripPrd = $tripPrv * $trip;
  $htripPrd = $htripPrv * $htrip;
  $trimPrd = $trimPrv * $trim;
  $htrimPrd = $htrimPrv * $htrim;
  printf "    <tr>$fmt</tr>\n",
    $ni, $sq,
    $trip, $htrip, $trim, $htrim,
    $tripPrd, $htripPrd, $trimPrd, $htrimPrd;
  $tripPrv = $trip;
  $htripPrv = $htrip;
  $trimPrv = $trim;
  $htrimPrv = $htrim;
  $li{$sq}++;
  $li{$trip}++;
  $li{%htrip}++;
  $li{$trim}++;
  $li{$htrim}++;
  $li{$tripPrd}++;
  $li{$htripPrd}++;
  $li{$trimPrd}++;
  $li{$htrimPrd}++;
}
say "  </tbody>";
say "</table>";
say "";
say "<table>";
for $ky (sort {$a <=> $b} keys %li) {
  printf "  <tr><td>%7i</td><td>%2i</td></tr>\n", $ky, $li{$ky};
}
say "</table>";'