6.042/18.062J
Mathematics
for
Computer
Science
September
14,
2010
Tom
Leighton
and
Marten
van
Dijk
Problem
Set
2
Problem
1.
[12
points]
DeFne
a
3chain
to
be
a
(not
necessarily
contiguous)
subsequence
of
three
integers,
which
is
either
monotonically
increasing
or
monotonically
decreasing.
We
will
show
here
that
any
sequence
of
Fve
distinct
integers
will
contain
a
3chain
.
Write
the
sequence
as
a
1
,a
2
,a
3
,a
4
,a
5
.
Note
that
a
monotonically
increasing
sequences
is
one
in
which
each
term
is
greater
than
or
equal
to
the
previous
term.
Similarly,
a
monotonically
decreasing
sequence
is
one
in
which
each
term
is
less
than
or
equal
to
the
previous
term.
Lastly,
a
subsequence
is
a
sequence
derived
from
the
original
sequence
by
deleting
some
elements
without
changing
the
location
of
the
remaining
elements.
(a)
[4
pts]
Assume
that
a
1
< a
2
.
Show
that
if
there
is
no
3chain
in
our
sequence,
then
a
3
must
be
less
than
a
1
.
(Hint:
consider
a
4
!)
(b)
[2
pts]
Using
the
previous
part,
show
that
if
a
1
< a
2
and
there
is
no
3chain
in
our
sequence,
then
a
3
< a
4
< a
2
.
(c)
[2
pts]
Assuming
that
a
1
< a
2
and
a
3
< a
4
< a
2
,
show
that
any
value
of
a
5
must
result
in
a
3chain
.
(d)
[4
pts]
Using
the
previous
parts,
prove
by
contradiction
that
any
sequence
of
Fve
distinct
integers
must
contain
a
3chain
.
Problem
2.
[8
points]
Prove
by
either
the
Well
Ordering
Principle
or
induction
that
for
all
nonnegative
integers,
n
:
n
(
)
2
∑
i
3
=
n
(
n
+
1)
.
(1)
2
i
=0
Problem
3.
[25
points]
The
following
problem
is
fairly
tough
until
you
hear
a
certain
oneword
clue.
The
solution
is
elegant
but
is
slightly
tricky,
so
don’t
hesitate
to
ask
for
hints!
During
6.042,
the
students
are
sitting
in
an
n
×
n
grid.
A
sudden
outbreak
of
beaver
ﬂu
(a
rare
variant
of
bird
ﬂu
that
lasts
forever;
symptoms
include
yearning
for
problem
sets
and
craving
for
ice
cream
study
sessions)
causes
some
students
to
get
infected.
Here
is
an
example
where
n
=
6
and
infected
students
are
marked
×
.
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Problem
Set
2
×
×
×
×
×
×
×
×
Now
the
infection
begins
to
spread
every
minute
(in
discrete
timesteps).
Two
students
are
considered
adjacent
if
they
share
an
edge
(i.e.,
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Natural number, B F J O C D G H K L N A B C D E F G H I J K L M N O, C D E F G H I J K L M O

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