Q: It's 12:00 now. What time would it be when the minute hand crosses the hour hand again?

A: Between 1200 and 2400, this will happen for 11 times. So divide 12 hr by 11 and add to 1200, i.e.

13:05:27
Q: A cell can do 4 things: 1) Dies; 2) Nothing; 3) Duplicate; or 4) Triplicate. What is P(colony dies out) if we start with one single cell?

A: Solve the polynomial

$ p = \frac {1} {4} + \frac {1}{4} p + \frac {1}{4} p^2 + \frac {1}{4} p^3$
Q: A 6x6 matrix has entries of either +1 or -1. How many unique combinations are there such that all column products and row products are +1?

A: Consider the top left 5x5 sub-matrix. It can take any permutation (as long as each row/column contains no more than 3 same number) because we can rely on the rest of the 6x6 matrix (an "L" shape) to "fix" it. So the answer is

$ 2^{25} $
Q: There is an integer

*X*. The product of the digits is 96. What is the largest/smallest possible value of

*X*?

A:

smallest = 268, largest = 322222
Q: What is the sum of all

**digits** of all the numbers between 1 and 1000000?

A: Convince yourself that all digits appear for an equal number of times. The answer is

27000001
Q: A bus makes a stop, and 3/4 of all passengers get off, while 7 get on. This repeats for two more times. What is the smallest number of initial passengers?

A:

52
Q: Paint the outside of a cube, then dice it into 27 equal smaller cubes. Pick one at random and roll it. What is P(no paint visible after the roll)?

A:

$ 6/27 \times 1/6 + 1/27 $
Q: Two equally talented teams are playing a series of games in which the first team to win 4 games wins the series. What is P(7th game has to be played)?

A:

$ \frac {6!}{3!3!} \frac {1}{2^6} = \frac {5}{16}$
Q: I roll a die, spin a roulette and draw a card from a deck of 52. What is the probability that all three give the same number?

A:

$ 5 \times \frac 1 6 \times \frac 1 {13} \times 1 {38} $
Q: What is the last digit of 3 to the power 33?

A:

Some experimentation would indicate that $3^0=1$, $3^1=3$, $

3^2=9$, $3^3=27$, $3^4=81$, ..., hence $3^33

mod

10 = 3$
Q:

*N* couples shake hand with others. It is known that 1) They don't shake hand with their own partners; and 2) The first (2

*N* - 1) people surveyed all have different number of handshakes. What is the number of handshake the remaining person made?

A:

*N* - 1 (Can draw graphs to deduce; any analytic solution?)

Q: What is P(3 random points on a circle are within an 180 degree arc)?

A: There are two approaches: 1) Area on an abstract

__x-y__ plane,

*x* and

*y* are distances from the first point pinned down (call it

*z*, which is arbitrary); or 2) $ N \times \frac {1}{2^{N-1}} $, with $ N=3$. Both give the answer of

3/4.

Q: How do you simulate a 6-sided die using a fair coin? What's the expected number of toss to produce the number?

A: Use a binary scheme. With 3 tosses, there are 8 possible outcomes, which is more than enough to simulate 6 sides. The trick is to know which 2 to exclude. {HHH,TTT} would be a bad choice because you always have to wait until the 3rd toss to know if you failed. So we should choose something like {HHH,HHT} or {THH,THT}. The expected number of toss is given by solving

$E[N]=\frac 6 8 \times 3 + \frac 2 8 (2+E[N])$
Q: Place 3 points randomly on the perimeter of a square. What is P(they don't form a triangle) and P(all lie on different edges)?

A:

$P(they_don't`_`

form`_`

a_triangle) = 4/{4^3} = 1/16$; $P(all_lie_on_different_edges) =\frac {4 \times _3P_3}{4^3} = 3/8$
Q: 10 light bulbs are lined up in a row, and it is known that no two of the adjacent light bulbs can be both on. What is the possible number of arrangements?

A: Fibonacci sequence,

144
Q: You have a large number of red and blue balls (equally many) in a large container. What is P(odd number of red) if you draw a) 3 balls from the container? b) 10 balls from the container?

A:

0.5 regardless of how many you draw. For example if 3 are drawn, the binomial distribution is 1(odd) 3(even) 3(odd) 1(even); if 4 are drawn, the binomial distribution is 1(even) 4(odd) 6(even) 4(odd) 1(even).

Q: Transport as many apples as possible to a destination 1000 km away starting with 3000 apples. For each km of transportation, you lose 1 apple.

A: The best tactic is to always start with full load so as to maximize number of "apple km" traveled per apple lost. So the 1st stop is optimally at 3000 - 3X = 2000 <=> X = 333; the 2nd stop is optimally at 2000 - 2X = 1000 <=> X = 500. After 833 km, we are left with 1000 apples. Hence we will have

833 apples at the destination.

Q: You are playing against two players A and B individually in a tournament, with A being the better player. If you have to win at least 2

**consecutive** games out of 3 in order to be the champion, would you prefer the order ABA or BAB?

A: Out of the 8 outcomes {LLL,LLW,LWL,LWW,WLL,WLW,WWL,WWW}, only {WWW,WWL,LWW} are desirable. If q = P(winning against A) and p = P(winning against B) so that p > q,

P(champion with the order ABA) = pqp + pq(1-p) + (1-p)qp = 2pq - ppq

P(champion with the order BAB) = qpq + qp(1-q) + (1-q)pq = 2pq - pqq

So

P(champion with the order BAB) > P(champion with the order ABA)