Challenge problems for tech candidates

Show us that you are smart and resourceful enough to come join our team. Listed below are a set of questions that we'd like you to take a look at if you feel like a challenge. When applying to one of our jobs, send these questions along with your amazing answers along with your resume.

You don't have to answer all of them or necessarily provide us with complete solutions -- we're mostly interested in how you approach problems and their solutions. We know you're busy and don't want to waste your time. Still, we believe that the kind of person who enjoys problems like these would enjoy working with us. They are not brain-teasers, but rather problems that require some basic mathematical concepts, a bit of research, and some simple scripting. Make sketches or use diagrams, take shortcuts if they're there, and have some fun with them.

Mathematical

1. Suppose that my "true love" decides to give me all the gifts described in the song "The 12 Days of Christmas"" (http://en.wikipedia.org/wiki/The_Twelve_Days_of_Christmas_%28song%29). The one hitch however, is that my true love can't find lords-a-leaping. How many gifts total will I have received on (and including) the 11th day assuming that I'm never given any lords-a-leaping? Hint: read the lyrics of the song carefully, the number is bigger than you might expect.

2. Most Western music is based on a 12-tone scale. Suppose that instead we wanted to work with a 19-tone scale. Instead of notes A-G, we'll have notes t0, t1, t2, ..., t18. Suppose that t0 resonates at 100hz. At what frequency does t16 resonate?

3. Suppose that you have a deck of four cards: the aces of each suit. Suppose that your friend has the same deck. If each of you turns over a card, one by one, until you go through all 4, what is the probability that both of you turn over the same card at once?

4. What is the angle between these two vectors? v1= [1, 1, 1, 1, 1, 10, 5, 0, -12, 122, 21, 1, -1, -1, 2, -1, 1, 3, -6, -11, 12, 1, 1, 1 ] v2=[-1, -1, 2, -1, 1, 3, -6, 99, 12, 1, 1, 1 , 1, 1, 1, 1, 1, 10, 5, 0, -12, 119, 24, 1]

5. What is the 211th Fibonacci number? Please don't use a search engine and please explain briefly how you figured it out.

Decoding

1. Consider the set {a, b, c, d, ..., z }. Suppose we define an operation, *, on this set. Suppose we know the following:

a * b = ab = a
bc = c
cd = g
ef = u
ee = q
wy = i
wn = a
pq = g
rm = w
zc = y

Also, note that the operation is associative: (ca)t=c(at), so cat=(c*a)*t. For those of you who are mathematically inclined, the set is a semi-group with an identity element (b), but it is not a group because it has no inverse.

We haven't given you all the information you need: you're going to have to guess at what the operation really is.

Using your best guess, what's the value of the following?

(a) faulkner
(b) oconnor
(c) welty

Just to be clear: the value of 'faulkner' is f * a * u * l * k * n * e * r. And remember it's an associative operator.

2. Below is a string of text. We'd like you to use a Caesar cipher to decode it and then provide the English language translation:

lfnwr hmtnyhmjfss fnw htnwnhmjfs

Note: in this cipher, spaces are mapped to spaces

3. The following string has had two separate vowels removed. The missing vowels are represented by _ and *. What language was the original string written in?

as h*nnick j__ yn s*ilsh_y dy r*w _h mi_

Programming

1. Write a recursive function, f, such that f(n) will enumerate all the permutations of the set (1 ... n). Write the function in the language of your choice.

2. Describe how you might write a program that generates palindromes for you.

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