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 have 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.

This formula has become a favorite of mine. It is the formula for counting how many non-directional connections there can be among some number of things. Computer Science calls this a complete graph. As a study of these numbers and related numbers as well, I've generated the first 24 of these.

My motivation for this was the realization that the number 12 has factors that are two triangular numbers, 2 and 3, as well as the product of triangular number, 3 and the square that follows it, 4. And, the number 12 has proven to be very useful in creating schemes that are conveniently divisible by these numbers.

n | number of entities to connect |

sq | n^{2} |

trip | n(n+1) |

htrip | n(n+1)/2 |

trim | n(n-1) |

htrim | n(n-1)/2 |

tripPrd | n(n+1) * trip_{i-1} |

htripPrd | n(n-1) * htrip_{i-1} |

trimPrd | n(n-1) * trim_{i-1} |

htrimPrd | n(n-1)/2 * htrim_{i-1} |

n | sq | trip | htrip | trim | htrim | tripPrd | htripPrd | trimPrd | htrimPrd |
---|---|---|---|---|---|---|---|---|---|

2 | 4 | 6 | 3 | 2 | 1 | 0 | 0 | 0 | 0 |

3 | 9 | 12 | 6 | 6 | 3 | 72 | 18 | 12 | 3 |

4 | 16 | 20 | 10 | 12 | 6 | 240 | 60 | 72 | 18 |

5 | 25 | 30 | 15 | 20 | 10 | 600 | 150 | 240 | 60 |

6 | 36 | 42 | 21 | 30 | 15 | 1260 | 315 | 600 | 150 |

7 | 49 | 56 | 28 | 42 | 21 | 2352 | 588 | 1260 | 315 |

8 | 64 | 72 | 36 | 56 | 28 | 4032 | 1008 | 2352 | 588 |

9 | 81 | 90 | 45 | 72 | 36 | 6480 | 1620 | 4032 | 1008 |

10 | 100 | 110 | 55 | 90 | 45 | 9900 | 2475 | 6480 | 1620 |

11 | 121 | 132 | 66 | 110 | 55 | 14520 | 3630 | 9900 | 2475 |

12 | 144 | 156 | 78 | 132 | 66 | 20592 | 5148 | 14520 | 3630 |

13 | 169 | 182 | 91 | 156 | 78 | 28392 | 7098 | 20592 | 5148 |

14 | 196 | 210 | 105 | 182 | 91 | 38220 | 9555 | 28392 | 7098 |

15 | 225 | 240 | 120 | 210 | 105 | 50400 | 12600 | 38220 | 9555 |

16 | 256 | 272 | 136 | 240 | 120 | 65280 | 16320 | 50400 | 12600 |

17 | 289 | 306 | 153 | 272 | 136 | 83232 | 20808 | 65280 | 16320 |

18 | 324 | 342 | 171 | 306 | 153 | 104652 | 26163 | 83232 | 20808 |

19 | 361 | 380 | 190 | 342 | 171 | 129960 | 32490 | 104652 | 26163 |

20 | 400 | 420 | 210 | 380 | 190 | 159600 | 39900 | 129960 | 32490 |

21 | 441 | 462 | 231 | 420 | 210 | 194040 | 48510 | 159600 | 39900 |

22 | 484 | 506 | 253 | 462 | 231 | 233772 | 58443 | 194040 | 48510 |

23 | 529 | 552 | 276 | 506 | 253 | 279312 | 69828 | 233772 | 58443 |

24 | 576 | 600 | 300 | 552 | 276 | 331200 | 82800 | 279312 | 69828 |

0 | 27 |

1 | 1 |

2 | 1 |

3 | 2 |

4 | 1 |

6 | 3 |

9 | 1 |

10 | 1 |

12 | 3 |

15 | 1 |

16 | 1 |

18 | 2 |

20 | 2 |

21 | 1 |

25 | 1 |

28 | 1 |

30 | 2 |

36 | 2 |

42 | 2 |

45 | 1 |

49 | 1 |

55 | 1 |

56 | 2 |

60 | 2 |

64 | 1 |

66 | 1 |

72 | 4 |

78 | 1 |

81 | 1 |

90 | 2 |

91 | 1 |

100 | 1 |

105 | 1 |

110 | 2 |

120 | 1 |

121 | 1 |

132 | 2 |

136 | 1 |

144 | 1 |

150 | 2 |

153 | 1 |

156 | 2 |

169 | 1 |

171 | 1 |

182 | 2 |

190 | 1 |

196 | 1 |

210 | 3 |

225 | 1 |

231 | 1 |

240 | 4 |

253 | 1 |

256 | 1 |

272 | 2 |

276 | 1 |

289 | 1 |

306 | 2 |

315 | 2 |

324 | 1 |

342 | 2 |

361 | 1 |

380 | 2 |

400 | 1 |

420 | 2 |

441 | 1 |

462 | 2 |

484 | 1 |

506 | 2 |

529 | 1 |

552 | 2 |

576 | 1 |

588 | 2 |

600 | 3 |

1008 | 2 |

1260 | 2 |

1620 | 2 |

2352 | 2 |

2475 | 2 |

3630 | 2 |

4032 | 2 |

5148 | 2 |

6480 | 2 |

7098 | 2 |

9555 | 2 |

9900 | 2 |

12600 | 2 |

14520 | 2 |

16320 | 2 |

20592 | 2 |

20808 | 2 |

26163 | 2 |

28392 | 2 |

32490 | 2 |

38220 | 2 |

39900 | 2 |

48510 | 2 |

50400 | 2 |

58443 | 2 |

65280 | 2 |

69828 | 2 |

82800 | 1 |

83232 | 2 |

104652 | 2 |

129960 | 2 |

159600 | 2 |

194040 | 2 |

233772 | 2 |

279312 | 2 |

331200 | 1 |

```
$ 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>";'
```