Concrete mathematics : a foundation for computer science

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A ANSWERS TO EXERCISES 485
1.13 Given n straight lines that define L, regions, we can replace them
by extremely narrow zig-zags with segments sufficiently long that there are
nine intersections between each pair of zig-zags. This shows that ZZ, =
ZZ, ’ +9n-8, for a’11 n > 0; consequently ZZ, = 9S, -8n+ 1 = ;n2 - In+ 1.
1.14 The number Iof new 3-dimensional regions defined by each new cut is
the number of 2-dimensional regions defined in the new plane by its intersections
with the previous planes. Hence P, = P, ’ + L, ~1, and it turns out
that P5 = 26. (Six cuts in a cubical piece of cheese can make 27 cubelets, or
up to P6 = 42 cuts of weirder shapes.)
Incidentally, the solution to this recurrence fits into a nice pattern if
we express it in terms of binomial coefficients (see Chapter 5):
x, = (;)i-(1;);
L,, = (;)i-(;)-(1);
pn = (‘3)+-(;)-(1)+(Y)
Here X, is the maximum number of l-dimensional regions definable by n
dimensions! points on a line.
1.15 The function I satisfies the same recurrence as J when n > 1, but I( 1)
is undefined. Since I(2) = 2 and I( 3) =I 1, there’s no value of I ( 1) = OL that
will allow us to use our general method; the “end game” of unfolding depends
on the two leading bits in n’s binary representation.
If n = 2” + 2mp1 +k,whereO~k 2. Another way to express
this, in terms of the representation n = 2” + 1, is to say that
I(n) = {
J(n) + 2 ml , ifO