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\large{ {\bf {\large First lecture worksheet \hfill Math21a, Fall 2011, Oliver Knill  }} \\

1) Can you figure out, what the set of all points in space which satisfy $x^2-2x+y^2 = 3$ is? \\

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2) Find the equation of all points in space which have the same distance to $(1,1,1)$ and $(0,0,0)$. \\ 

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3) An {\bf Euler brick} is a cuboid of dimensions a,b,c such that the
face diagonals are integers.
Verify (possibly with a calculator on your phone or by hand) 
that a cuboid with side length $(a,b,c) = (240,117,44)$ is an Euler brick. \\

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\footnote{
If also the space diagonal is an integer, an Euler brick is called a {\bf perfect cuboid}.
It is an open mathematical problem, whether a perfect cuboid exists. 
}

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