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Bugmaster comments on Open Thread, Apr. 13 - Apr. 19, 2015 - Less Wrong Discussion
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In keeping with the "puzzle" theme:
You are given a rectangular piece of paper (such as the placemat at a fast-food restaurant). Without using any measuring tools (such as a ruler, a tape measure, some clever length-measuring app on your smartphone, etc.), divide the paper into five equal parts.
Could you confirm or correct some guesses about exactly what problem is intended?
Folding is allowed, yes.
Parts can have different shapes (if you want), but must have the same area.
You cannot use compasses, but you can use an unmarked straightedge if you want to make precise creases, or to avoid ripping the paper in an untidy fashion. You are not allowed to mark the straightedge, of course.
If the procedure were carried out with infinite precision, then it would indeed produce exact fifths.
First, recruit five paper-maximizing ideal economic agents, and give them a picking order one through five. Let the last-picking agent divide up the rectangular piece of paper into five parts. Downsides: will cost you the piece of paper.
I believe I have it. rot13:
Sbyq naq hasbyq gur cncre ubevmbagnyyl, gura qb gur fnzr iregvpnyyl, gb znex gur zvqcbvag bs rnpu fvqr. Arkg, sbyq naq hasbyq gb znex sbhe yvarf: vs gur pbearef bs n cncre ner N, O, P, Q va beqre nebhaq gur crevzrgre, gura gur yvarf tb sebz N gb gur zvqcbvag bs O naq P, sebz O gb gur zvqcbvag bs P naq Q, sebz P gb gur zvqcbvag bs N naq Q, naq sebz Q gb gur zvqcbvag bs N naq O.
Gurfr cnegvgvba gur erpgnatyr vagb avar cvrprf: sbhe gevnatyrf, sbhe gencrmbvqf, naq bar cnenyyrybtenz. Yrg gur cnenyyrybtenz or bar cneg, naq tebhc rnpu gencrmbvq jvgu vgf bja nqwnprag gevnatyr gb znxr gur sbhe bgure cnegf.
Obahf: vs jr phg bhg nyy avar cvrprf, n gencrmbvq naq n gevnatyr pna or chg onpx gbtrgure va gur rknpg funcr bs gur cnenyyrybtenz.
No bending the paper?
Bending is allowed; see above.