Hint for the coin puzzle (rot13): As mentioned in the article, gur fbyhgvba trarenyvmrf gb nal cbjre bs guerr, gubhtu gur ahzore bs jrvtuvatf jvyy bs pbhefr inel.
Solution (rot13): Fcyvg gur pbvaf vagb guerr tebhcf bs guerr. Chg gjb tebhcf ba gur gjb fvqrf bs gur fpnyr, yrnivat gur guveq bss. Vs gurl ner rdhny va jrvtug, gur snxr pbva vf va gur guveq tebhc. Bgurejvfr vg vf va gur yvtugre bs gur gjb.
Qvfpneq gur tebhcf gur snxr vfa'g va. Chg gjb bs gur erznvavat pbvaf ba gur gjb fvqrf bs gur fpnyr. Ntnva, vs gurl ner rdhny va jrvtug, gur snxr vf gur guveq; bgurejvfr vg vf gur yvtugre bs gur gjb.
Guvf vf n grkgobbx qvivqr-naq-pbadhre nytbevguz. Gur xrl vafvtug vf gung lbh pna qvivqr ol n snpgbe bs guerr, abg whfg gjb.
The insight for size of groups can be shown by constructing the weighing that maximizes entropy.
With lg(9) bits of information needed to identify the odd coin, and a scale with only three possible outcomes (< > =), we can show the best sequence of weighings must reveal lg(3) + lg(3) bits of information by yielding a uniform distribution on the outcomes.
Solution (rot13): Fcyvg gur pbvaf vagb guerr tebhcf bs guerr. Chg gjb tebhcf ba gur gjb fvqrf bs gur fpnyr, yrnivat gur guveq bss. Vs gurl ner rdhny va jrvtug, gur snxr pbva vf va gur guveq tebhc. Bgurejvfr vg vf va gur yvtugre bs gur gjb.
Qvfpneq gur tebhcf gur snxr vfa'g va. Chg gjb bs gur erznvavat pbvaf ba gur gjb fvqrf bs gur fpnyr. Ntnva, vs gurl ner rdhny va jrvtug, gur snxr vf gur guveq; bgurejvfr vg vf gur yvtugre bs gur gjb.
Guvf vf n grkgobbx qvivqr-naq-pbadhre nytbevguz. Gur xrl vafvtug vf gung lbh pna qvivqr ol n snpgbe bs guerr, abg whfg gjb.