1.
Solve this cryptic equation, realizing of course that values for M and E could
be interchanged. No leading zeros are allowed.

WWWDOT
- GOOGLE = DOTCOM

This
can be solved through systematic application of logic. For
example, cannot be equal to 0, since . That would make , but , which is not possible.

Here
is a slow brute-force method of solution that takes a few minutes on a
relatively fast machine:

This
gives the two solutions

777589
- 188106 == 589483

777589 - 188103 == 589486

Here
is another solution using Mathematica's Reduce command:

A
faster (but slightly more obscure) piece of code is the following:

Faster
still using the same approach (and requiring ~300 MB of memory):

Even
faster using the same approach (that does not exclude leading zeros in the
solution, but that can easily be weeded out at the end):

Here
is an independent solution method that uses branch-and-prune techniques:

And
the winner for overall fastest:

2.
Write a haiku describing possible methods for predicting search traffic
seasonality.

MathWorld's
search engine

seemed slowed this May. Undergrads

prepping for finals.

3.

1

1 1

2 1

1 2 1 1

1 1 1 2 2 1

What's the next line?

312211.
This is the "look and say" sequence in which each term after the
first describes the previous term: one 1 (11); two 1s (21); one 2 and one 1
(1211); one 1, one 2, and two 1's (111221); and so on. See the look and say
sequence entry on MathWorld for a complete write-up and the algebraic form of a
fascinating related quantity known as Conway's constant.

4.
You are in a maze of twisty little passages, all alike. There is a dusty laptop here with a weak wireless connection.
There are dull, lifeless gnomes strolling around. What dost thou do?

A)
Wander aimlessly, bumping into obstacles until you are eaten by a grue.

B) Use the laptop as a digging device to tunnel to the next level.

C) Play MPoRPG until the battery dies along with your hopes.

D) Use the computer to map the nodes of the maze and discover an exit path.

E) Email your resume to Google,
tell the lead gnome you quit and find yourself in whole different world [sic].

In
general, make a state diagram . However, this method would not work in certain pathological cases
such as, say, a fractal maze. For an example of this and commentary, see Ed
Pegg's column about state diagrams and mazes .

5.
What's broken with Unix?

Their
reproductive capabilities.

How would you fix it?

[This
exercise is left to the
reader.]

6.
On your first day at Google, you
discover that your cubicle mate wrote the textbook you used as a primary
resource in your first year of graduate school. Do you:

A)
Fawn obsequiously and ask if you can have an autograph.

B) Sit perfectly still and use only soft keystrokes to avoid disturbing her
concentration

C) Leave her daily offerings of granola and English toffee from the food bins.

D) Quote your favorite formula from the textbook and explain how it's now your
mantra.

E) Show her how example 17b could have been solved with 34 fewer lines of code.

[This
exercise is left to the reader.]

7.
Which of the following expresses Google's
over-arching philosophy?

A)
"I'm feeling lucky"

B) "Don't be evil"

C) "Oh, I already fixed that"

D) "You should never be more than 50 feet from food"

E) All of
the above

[This
exercise is left to the reader.]

8.
How many different ways can you color an icosahedron with one of three colors
on each face?

For
an asymmetric 20-sided solid, there are possible 3-colorings . For a symmetric
20-sided object, the Polya enumeration theorem can be used to obtain the number
of distinct colorings. Here is a concise Mathematica implementation:

What
colors would you choose?

[This
exercise is left to the reader.]

9.
This space left intentionally blank. Please fill it with something that
improves upon emptiness.

For
nearly 10,000 images of mathematical functions, see The Wolfram Functions Site
visualization gallery .

10.
On an infinite, two-dimensional, rectangular lattice of 1-ohm resistors, what is
the resistance
between two nodes that are a knight's move away?

This
problem is discussed in J. Cserti's 1999 arXiv preprint . It is also discussed
in The Mathematica GuideBook for Symbolics, the forthcoming fourth volume in
Michael Trott's GuideBook series, the first two of which were published just
last week by Springer-Verlag. The contents for all four GuideBooks, including
the two not yet published, are available on the DVD distributed with the first
two GuideBooks.

11.
It's 2PM on a sunny Sunday afternoon in the Bay Area. You're minutes from the
Pacific Ocean, redwood forest hiking trails and world class cultural
attractions. What do you do?

[This
exercise is left to the reader.]

12.
In your opinion, what is the most beautiful math equation ever
derived?

There
are obviously many candidates. The following list gives ten of the authors'
favorites:

1.
Archimedes' recurrence formula : , , ,

2. Euler formula :

3. Euler-Mascheroni constant :

4. Riemann hypothesis: and implies

5. Gaussian integral :

6. Ramanujan's prime product formula:

7. Zeta-regularized product :

8. Mandelbrot set recursion:

9. BBP formula :

10. Cauchy integral formula:

An
excellent paper discussing the most beautiful equations in physics is Daniel Z.
Freedman's " Some beautiful equations of mathematical physics ." Note
that the physics view on beauty in equations is less uniform than the
mathematical one. To quote the not-necessarily-standard view of theoretical
physicist P.A.M. Dirac, "It is more important to have beauty in one's
equations than to have them fit experiment."

13.
Which of the following is NOT an actual interest group formed by Google employees?

A.
Women's basketball

B. Buffy fans

C. Cricketeers

D. Nobel winners

E. Wine club

[This
exercise is left to the reader.]

14.
What will be the next great improvement in search
technology?

Semantic
searching of mathematical formulas. See
http://functions.wolfram.com/About/ourvision.html for work currently underway
at Wolfram Research that will be made available in the near future.

15. What is the optimal size of a project team, above
which additional members do not contribute productivity equivalent to the
percentage increase in the staff size?

A)
1

B) 3

C) 5

D) 11

E) 24

[This
exercise is left to the reader.]

16.
Given a triangle ABC, how would you use only a compass and straight
edge to find a point P such that triangles ABP, ACP and BCP have equal
perimeters? (Assume that ABC is constructed so that a solution does exist.)

This is the isoperimetric point , which is at the center of the larger Soddy
circle. It is related to Apollonius' problem . The three tangent circles are
easy to construct: The circle around has diameter , which gives the other two
circles. A summary of compass and straightedge constructions for the outer
Soddy circle can be found in " Apollonius' Problem: A Study of Solutions
and Their Connections" by David Gisch and Jason M. Ribando.

17.
Consider a function which, for a given whole number n, returns the number of
ones required when writing out all numbers between 0 and n. For example, f(13)=6. Notice that
f(1)=1. What is the next largest n such that f(n)=n?

The
following Mathematica code computes the
difference between [the
cumulative number of 1s in the positive integers up to n] and [the value of n
itself] as n ranges from 1 to 500,000:

The
solution to the problem is then
the first position greater than the first at which data equals 0:

which
are the first few terms of sequence A014778 in the On-Line Encyclopedia of
Integer Sequences.

Checking
by hand confirms that the numbers from 1 to 199981 contain a total of 199981
1s:

18. What is the coolest hack you've ever written?

While
there is no "correct" answer, a nice hack for solving the first
problem in the SIAM hundred-dollar, hundred-digit challenge can be achieved by
converting the limit into the strongly divergent series:

and
then using Mathematica's numerical function SequenceLimit to trivially get the
correct answer (to six digits),

You
must tweak parameters a bit or write your own sequence limit to get all 10
digits.

[Other
hacks are left to the reader.]

19.
'Tis known in refined company, that choosing K things out of N can be done in
ways as many as choosing N minus K from N: I pick K, you the remaining.

This
simply states the binomial coefficient identity .

Find
though a cooler bijection, where you show a knack uncanny, of making your
choices contain all K of mine. Oh, for pedantry: let K be no more than half N.

'Tis
more problematic to disentangle semantic meaning precise from the this
paragraph of verbiage peculiar.

20.
What number comes next in the sequence: 10, 9, 60, 90, 70, 66, ?

A)
96

B) 1000000000000000000000000000000000\

0000000000000000000000000000000000\

000000000000000000000000000000000

C) Either of the above

D) None of the above

This
can be looked up and found to be sequence A052196 in the On-Line Encyclopedia
of Integer Sequences, which gives the largest positive integer whose English
name has n letters. For example, the first few terms are ten, nine, sixty,
ninety, seventy, sixty-six, ninety-six, â€¦. A more correct sequence might be
ten, nine, sixty, googol, seventy, sixty-six, ninety-six, googolplex. And also
note, incidentally, that the correct spelling of the mathematical term "
googol" differs from the name of the company that made up this aptitude
test.

The
first few can be computed using the NumberName function in Eric Weisstein's
MathWorld packages:

A
mathematical solution could also be found by fitting a Lagrange interpolating
polynomial to the six known terms and extrapolating:

21.
In 29 words or fewer, describe what you would strive to accomplish if you
worked at Google Labs.

[This
exercise is left to the reader.]