From c69bcaba4969b3122abf40ed05279518cb318cc9 Mon Sep 17 00:00:00 2001 From: Jonathan Chan Date: Sun, 2 Jul 2017 13:44:17 -0700 Subject: [PATCH] Problem 68 (explanation; no computation required) --- 68.txt | 16 ++++++++++++++++ 1 file changed, 16 insertions(+) create mode 100644 68.txt diff --git a/68.txt b/68.txt new file mode 100644 index 0000000..8201ecd --- /dev/null +++ b/68.txt @@ -0,0 +1,16 @@ +I solved this by hand so I'll only give a brief outline of how. + +For a magic n-gon ring, its magic number is given by + (2*(sum of inside nodes) + (sum of outside nodes))/n +The maximal smallest outside node is (n+1) with the outside nodes as [n+1, n+2, ..., 2n-1, 2n]. + +For n = 5, the magic number is 14. +The outside nodes are [6, 7, 8, 9, 10] and the inside nodes are [1, 2, 3, 4, 5]. +To obtain a maximal 16-digit string, 6 must be connected to 5, so the first triple is 6-5-3. +The only way for a triple to contain 10 and add up to 14 is using a permutation of 10-3-1. +Therefore, the next triple going clockwise must be 10-3-1. +Following that, a maximal 16-digit string could contain 9, 8, 7 as the next outside nodes. +Using these values indeed yields a correct magic 5-gon ring. +The maximal 16-digit string is then constructed from the solution set + 6, 5, 3; 10, 3, 1; 9, 1, 4; 8, 4, 2; 7, 2, 5 +or 6531031914842725.