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Samwise's problem
In a mathematical Lord of the Rings riddle, Gandalf discovers that Sauron's hidden telephone line is running under ground in a straight line under Sam's garden. He tells Sam to dig for the phone line, in order to get find the traitors, whom conspires with Sauron.
Sam lives on a quadratic property where each side is one hobbit length unit long. As the phone line runs in a straight line, Sam decides to dig all the way around his property, but Gandalf advices against it, saying that that Sam can dig a lot less. Sam then decides to dig on three sides, as the line must cross at least two sides. Again Gandalf declares that the problem can be optimized. What is the shortest length Sam has to dig?

Solution to Samwise's problem
It is a fair assumption that the shortest solution is on the for scetched in th drawing below
Figur: Outline of shortest soultions to Samwise's problem. The thin white lines indicate, where Sam should dig. x is the only unknown parameter.
Image P1
We know calculate the length Sam was has to dig as a function of x
L=1/sqrt(2)+sqrt(2)x+2sqrt((1-x)^2+x^2)
We then derive with respect to x in order to find the minimum value of L
0=sqrt(2)+(4x-2)/sqrt((1-x)^2+x^2)
=>
2=(16x^2+4-16x)/((1-x)^2+x^2)
=>
4x^2+2-4x=16x^2+4-16x
=>
x=1/2+s/sqrt(12)
where s is eithe -1 or 1. Choosing s=-1 (there is no difference if you consider the geometry) we get the shortest length Sam has to dig

L=1/sqrt(2)+sqrt(2)x+2sqrt((1-x)^2+x^2)=2.638958434



Venuspassage 2012
Den sjette juni 2012 passerede planeten Venus ind foran solskiven set fra jorden. Dette skete klokken ca halv tre dansk tid, så vi måtte her i landet vente til solen stod op (klokken 04:35 i Aarhus) før vi kunne se dette fænomen. På trods af nogle skyer i horisonten var vejret overordnet fint, og jeg fik nogle gode billeder af passagen. Næste venuspassage er den 11.12.2117 og kan ikke ses fra Danmark.
Figur: Venuspassagen 6.6.12. Fotografiet er taget af undertegnede med mit Konika-Minolta Maxxum 5D med ISO 100 og en lukketid 1/500s. Den store plet øverst til højre er Venus; de mindre pletter er solpletter.
Image P1




Rasmus Udengaard Skannrup
2013-02-23
web (at) quarks.dk