Puzzle – the dinner introduction
May 3, 2007 pm31 2:54 pm
At a dinner party, Alec, Barney, Cathy, Delia, Edgar, Francine, Greta, and Harold, 8 people who have not previously met, sit at one square table.
“Let’s shake hands” says one, and all eight reach out, grasp another hand, and shake.
“Four handshakes, and no arms crossing! Can we do that another way?” And they did. Again, and again, and again…. until there were no more new patterns.
Each time all eight shook hands. While some individual handshakes repeated, the overall pattern was different each time. How many times did they shake? If they are sitting in alphabetical order, how many times does Francine shake the hand of each other guest?
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JD2718 
I’m familiar with the famous-as-sequences-of-positive-integers-go answer to the first part of this question. I really like the second question — it shows you exactly how to build a recursion to calculate small values of the sequence. (“The sequence,” meaning what you get if you replace 4 couples successively by 1 couple, 2 couples, and so on.)
rot13’d solution:
gurer ner guerr pnfrf: gurer znl or mreb, bar be gjb abanqwnprag unaqfunxrf.
va gur gur svefg pnfr gurer ner pyrneyl gjb cbffvovyvgvrf. bapr nal bs gur crbcyr qrgrezvarf gurve cnegare (yrsg be evtug), rirelbar’f cnegare vf qrgrezvarq.
va gur frpbaq pnfr, bapr gur abanqwnprag cnve vf qrgrezvarq, gur ragver unaqfunxr vf qrgrezvarq. n zvahgr bs gubhtug fubhyq pbaivapr lbh gung:
* gur abanqwnprag cnve zhfg fcyvg gur tebhc vagb gjb naq sbhe
* gurer ner rvtug jnlf bs qbvat guvf.
va gur guveq pnfr, gurer ner sbhe cbffvovyvgvrf, ol fvzcyr rahzrengvba.
fb svany nafjre: sbhegrra.
vf guvf evtug?
decrypt – http://www.retards.org/projects/rot13/
First off, that encrypt/decrypt stuff is cool.
Second, you are entirely correct. There are a whole bunch of ways of breaking up this question to do the counting, yours is perfectly acceptable, and you have produced the correct number.
As JBL pointed out, this is part of a famous sequence (my Barcelona Barber problem, somewhere in the archives, has the same answer). The fun part is to show why the answers belong to this sequence, or how the answers to the two problems can be made to correspond.
Hmm. That decrypt is not as cool as I thought: (Don’t translate this unless you want the answer) Gurfr ner nyy Pngnyna ahzoref
Jrveq. V npghnyyl gubhtug nobhg pngnyna ahzoref, ohg gura V erpnyyrq gung vg jnf gur ahzore bs jnlf bs qvivqvat vagb gevnatyrf, naq qrpvqrq gurer’f ab ernfba sbe vg gb or gur fnzr nf gur ahzore bs unaqfunxrf. Npghnyyl V fgvyy pna’g frr n 1-1 pbeerfcbaqrapr.
Yup, they show up in a bunch of seemingly unrelated problems. A good challenge for people who quickly get the right number is to find the correspondence. Why does the number of triangulations of a hexagon have the same answer as the dinner party with 4 couples?