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Clash Royale CLAN TAG #URR8PPP up vote 16 down vote favorite 8 This question is a follow-up to Maximum number of intersections between a quadrilateral and a pentagon, where it is shown that the boundaries $partial Q,partial P$ of a quadrilateral and a pentagon in the plane cannot intersect at more than $16$ points, since each side of $partial Q$ meets $partial P$ at an even number of points. Q: Given the boundaries $partial P_1, partial P_2$ of two pentagons in the plane, is it possible that $$ left|partial P_1 cap partial P_2 right| = 20?$$ Each side of $partial P_1$ meets $partial P_2$ at an even number of points, so equality is attained iff there is some configuration such that each side of $partial P_1$ meets each side of $partial P_2$ except one. $left|partial P_1 cap partial P_2right| = 18$ is possible, as shown below, and I believe that $left|partial P_1 cap partial P_2right| = 20$ is im possible, but I am failing to prove it. combinatorics curves intersectio...

Clash Royale CLAN TAG #URR8PPP up vote 552 down vote favorite 204 I came across the following image: Which states: $pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string if digits is the name of every person you will ever love, the date, time and manner of your death, and the answers to all the great questions of the universe. Is this true? Does it make absolutely any sense ? number-theory irrational-numbers pi share | cite | improve this question edited Aug 7 at 19:09 PerpetualJ 130 6 asked Oct 18 '12 at 14:35 Chani 2,922 3 11 12 61 This is unknown. All that is known about $pi$ is that it is transcendental. askamathematician.com/2009/11/… – picakhu Oct 18 '12 at 14:38 15 This is the assertion that $pi$ is base $8$ normal....