# 2013 UNCO Math Contest II Problems/Problem 10

## Problem

Dav designs a robot, which he calls FrankenCoder, to print nonsense text by scrambling eleven-letter messages. The robot always repeats the same scrambling rule. $\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline 1st: & E & N & I & G & M & A & C & R & U & S & H \\ \hline 2nd: & G & E & N & I & U & S & C & H & A & R & M \\ \hline 3rd: & I & G & E & N & A & R & C & M & S & H & U \\ \hline \end{tabular}$

FrankenCoder scrambles ENIGMACRUSH into GENIUSCHARM. Using the same rule, Franken- Coder then scrambles GENIUSCHARM into IGENARCMSHU, and so on. $[asy] path Sq=polygon(4); path He=polygon(6); draw(rotate(45)*Sq,black); draw(shift(3.5,0)*rotate(30)*He,black); draw(circle((1.5,-1.5),.5),black); MP("1",(0,1),dir(90)); MP("2",(1,0),dir(0)); MP("3",(0,-1),dir(270)); MP("4",(-1,0),dir(180)); MP("5",(3.5,1),dir(90)); MP("11",(3.5+sqrt(3)/2,1/2),dir(30)); MP("8",(3.5+sqrt(3)/2,-1/2),dir(-30)); MP("10",(3.5+0,-1),dir(-90)); MP("6",(3.5-sqrt(3)/2,-1/2),dir(-150)); MP("9",(3.5-sqrt(3)/2,1/2),dir(150)); MP("7",(1.5,-1.5),W); draw(arc((1.6,-1.5),.2,135,-135),arrow=ArcArrow(TeXHead)); path Ra1=((0,0)--(sqrt(2)/2,sqrt(2)/2)); draw(shift(-1,.4)*Ra1,arrow=Arrow(TeXHead)); draw(shift(0.4,1)*rotate(-90)*Ra1,arrow=Arrow(TeXHead)); draw(shift(1,-.4)*rotate(-180)*Ra1,arrow=Arrow(TeXHead)); draw(shift(-.4,-1)*rotate(90)*Ra1,arrow=Arrow(TeXHead)); path Ra2=((0,0)--(.85*sqrt(3)/2,.45)); draw(shift(3.7,1.2)*rotate(-60)*Ra2,arrow=Arrow(TeXHead)); draw(shift(3.7+1,.4)*rotate(-120)*Ra2,arrow=Arrow(TeXHead)); draw(shift(4.5,-.8)*rotate(-180)*Ra2,arrow=Arrow(TeXHead)); draw(shift(3.3,-1.2)*rotate(-240)*Ra2,arrow=Arrow(TeXHead)); draw(shift(2.3,-.4)*rotate(-300)*Ra2,arrow=Arrow(TeXHead)); draw(shift(2.5,.8)*rotate(0)*Ra2,arrow=Arrow(TeXHead)); [/asy]$

FrankenCoder’s internal wiring for scrambling letters is diagrammed at left, depicted as a collection of cycles. The arrows show how each of the eleven letters moves in a single scramble: letter $7$ stays in its place, the first four letters move in a cycle, and the other six letters also trade positions in a cycle. Dav sees that the messages printed by FrankenCoder repeat cyclically in paragraphs: eventually, the original message ENIGMACRUSH reappears as the start of a new paragraph identical to the first paragraph. (a) How many distinct messages does each paragraph contain? (b) Dav tries to improve the robot, to get an even longer paragraph of distinct messages, by drawing different wiring diagrams for the eleven letter positions. Experimenting with component cycles of various lengths, he perfects his ultimate robot: FrankenCoder-II, a robot that produces the longest possible paragraph of distinct eleven-letter messages. How many distinct messages does FrankenCoder-II produce? (c) Draw a wiring diagram that could describe FrankenCoder-II. There may be ties, since different wiring diagrams can make robots that print paragraphs that have the same length. Draw just one wiring diagram. (d) Dav realizes that because there are ties for the best wiring diagram, he can build an entire army of distinct robots that are as good as FrankenCoder-II at creating long paragraphs. How many distinct robots can he build that are as good as FrankenCoder-II? Include FrankenCoder-II in your count. (Two robots are regarded as distinct if they scramble the starting message ENIGMACRUSH into different messages.)

## Solution

(a) $12$ (b) $30$ (c) $2,661,120$