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This is the planet Rhombicosidodecahedria:

a rhombicosidodecahedron

This lovely planet has 62 countries, each with its own distinct history and culture. By an amazing coincidence, the countries all happen to coincide perfectly with the faces of a rhombicosidodecahedron. You, as a tourist, are naturally hoping to visit all of them.

How many different ways can you travel though each of the countries on this planet, visiting each exactly once?

Some rules:

  • Your tour may start in any country you wish.
  • You must visit each country exactly once. Once you have visited a country, you cannot visit it again. (The word "tour" should not be taken to imply a requirement that you return to your starting point.)
  • You may only move between countries that are adjacent. In other words, each new country you visit must share a border with the previous one. (If they just touch at the corners that doesn't count.)
  • For purposes of this puzzle, every country should be treated as unique, and reflections and rotations all count as different paths. (In other words, the orange pentagon counts as a different starting point from the purple pentagon.)
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  • $\begingroup$ matrix mapping part is harder than writing a code to find all possibilities :D $\endgroup$
    – Oray
    Mar 11, 2023 at 7:47
  • $\begingroup$ @Oray - rot13(V ubcr lbh qvqa'g jvaq hc fcraqvat gbb zhpu gvzr gelvat gb oehgr sbepr guvf. Vs lbh qvq, V ncbybtvmr. Guvf vf qrsvavgryl bar bs gubfr chmmyrf jurer oehgr sbepvat vf qbvat vg gur uneq jnl! V jnf grzcgrq gb tvir guvf gur "ab pbzchgref" gnt, ohg V jnf nsenvq gur zrer cerfrapr bs gung gnt jbhyq tvir njnl gur frperg.) $\endgroup$ Mar 11, 2023 at 19:44
  • $\begingroup$ haha don’t worry i did not :) $\endgroup$
    – Oray
    Mar 12, 2023 at 7:18
  • $\begingroup$ In graph theory (but also in common parlance), the word "tour" implies returning to your starting point. Is this your intention? It seems to contradict the requirement of not visiting any country twice. $\endgroup$
    – Oliphaunt
    Mar 21, 2023 at 0:29
  • 1
    $\begingroup$ @Oliphaunt - I edited the rules to clarify this a bit. $\endgroup$ Mar 21, 2023 at 17:17

1 Answer 1

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There are exactly

Zero ways to complete such a tour.

Proof:

There are 12 pentagon countries and 20 triangle countries. Of these 32 countries, no two share a border. Any path visiting each of these countries exactly once would necessarily pass through 31 other countries, but there are only 30 square countries. At least one square country must be visited twice.

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  • $\begingroup$ Well done! rot13[V fghzoyrq npebff guvf juvyr npghnyyl gelvat gb frdhragvnyyl cnvag gur snprf bs guvf bowrpg va n 3q zbqryyvat cebtenz. (V jnf oberq, BX?) V jnf irel fhecevfrq jura V pbhyqa'g qb vg ng nyy. Gur xrl vafvtug gung qvq vg sbe zr jnf gung rirel bgure fgrc zhfg or n fdhner, juvpu zrnaf guvf vfa'g ernyyl nobhg trbzrgel, vg'f nobhg cnevgl.] $\endgroup$ Mar 11, 2023 at 19:33
  • $\begingroup$ Another way to formulate it is that rot13 (lbh unir n ovcnegvgr tencu jvgu 32 pbhagevrf va bar cneg naq 30 va gur bgure. Lbh pna'g znxr n Unzvygbavna cngu orpnhfr lbh arrq gb nygreangr unyirf.) It is really the same argument restated. $\endgroup$ Mar 12, 2023 at 5:59

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