I always was intrigued by my original question which you may recall went like this:

"What's Hottie Hostess' partner's number if Hottie plus her invited guests total 1000?"

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It turns out, when there are 1000 dancers (including Hottie) there is no solution to the problem. Recall the requirements: Hottie wears a 1. Everybody else wears a number from 2 to n, where n is the total number of dancers. People are paired up in such a way that the sum of the numbers on every couple is a "perfect square." Example: In the 18 dancer problem, Hottie can pair up with the person wearing 15 because 1 + 15 = 16 = 4 * 4.

I spiffed up the tiny-c program I introduced way back when and used it to determine all solutions to the problem for dancer counts between 18 and 1032! There are 25 solutions. Hottie pairs up with dancer number 528 in both the 976 dancer and 1032 dancer situations. Note that 1 + 528 = 529 = 23 * 23.

I wrote a page which "graphs" these solutions. Visit Dance Graph. It simply plots a dot for each dancer. It gives you some sense of how unpredictable the gap is between one solution and the next. Kind of reminds me how prime numbers sometimes come close together and then there's a big gap before the next one shows up.

In the list below, please note that the solution for 540 dancers is hyperlinked to a page which shows the actual run of the program and thus all the pairings, not just Hottie's.

  4*4          18 dancers  1 + 15 = 16  

               26 dancers  1 + 15 = 16  

  5*5          no solutions       = 25

  6*6          50 dancers  1 + 35 = 36

               60 dancers  1 + 35 = 36    

  7*7          68 dancers  1 + 48 = 49

               86 dancers  1 + 48 = 49

  8*8          no solutions       = 64

  9*9         126 dancers  1 + 80 = 81                          

              138 dancers  1 + 80 = 81                          

 10*10        180 dancers  1 + 99 = 100 

 11*11        232 dancers  1 + 120 = 121

 12*12        248 dancers  1 + 143 = 144

 13*13        308 dancers  1 + 168 = 169

 14*14        370 dancers  1 + 195 = 196

 15*15        390 dancers  1 + 224 = 225

 16*16        470 dancers  1 + 255 = 256

 17*17        540 dancers  1 + 288 = 289

              570 dancers  1 + 288 = 289

 18*18        no solutions         = 324

 19*19        666 dancers  1 + 360 = 361

              714 dancers  1 + 360 = 361

 20*20        742 dancers  1 + 399 = 400

              784 dancers  1 + 399 = 400 

 21*21        no solutions         = 441

 22*22        896 dancers  1 + 483 = 484
            
              944 dancers  1 + 483 = 484

 23*23        976 dancers  1 + 528 = 529

             1032 dancers  1 + 528 = 529