Brain Teaser
1
A pile of 1000 stones, split it to two piles x and y, you get a number x*y. Split x and y further, you a number for each of them. of them. Repeat the process until there are 1000 piles of 1 stone, add all the numbers together, what is the sum? Why do you always get the same answer even your splitting is arbitrary? Answer: 1000*999/2
2
A string of length 1, cut it to two strings of length x and y, you get a number x*y. Then you do similar splitting and add all the numbers up. What is the limit of the of the sum? Answer: 1/2, use geometrical method, x*y is like area of a of a square.
3
某容器中有一种细胞，在下一时刻，它可能分裂成 1 个，2 个或 者 3 个，每个状态都是 0.25 的概率，还有 0.25 的概率死亡即变成 如此继 续，问，这个容器中最 这个容 器中最后没有 后没有存活的 存活的细胞的 细胞的概率是 概率是多大。 多大。 0. 如此继续， （其实就是分支过程，只是想尽量按照原题目的意思翻译）
4
My wife and I went to a dinner party with four other couples. At the beginning of the of the party, some people shook hands. (Obviously, no one shook his or her own hand or spouse's hand, and no one shook hands with the same person twice.) During the party I surveyed all the other people as to how many hands each one shook. I got
different answers from everyone. What did my wife say? Answer: 1) There are 10 individuals in the puzzle 2) There are only 8 people in each person’s "set" of possible handshakes 3) There must be 9 unique answers (the asker doesn't count), therefore the answers must be 0‐8 (9 unique answers), this also means if there must be a repetition it's the asker and his wife. 4) If one person (let’s call them A) must shake 8 hands, then for there to be a 0 answer it must be A's partner, B, (since the only person outside A's "set" list is their partner, and they must use everyone in their set to get 8) The score is: z
A=8
z
B=0
z
Everyone else =1
5) Since the 8 and 0 answers must be unique (no one else can now shake hands with them), the 8‐0 pair "exits" the puzzle, their "set" is removed limiting each subsequent pair's possible handshake partners to total set‐2=6. 6) For there to be a 7 answer, C, must now shake hands with everyone in their "set" (now 6). For there to be a 1 answer, it must be
C's partner, D (who already shook with A), the only person outside C's set. Now the score is z
A=8
z
B=0
z
C=7 (1 from shaking with A + 6 remaining set)
z
D=1 (from shaking with A)
z
Everyone else 2 (1 from shaking with A, 1 from shaking with C)
7) Again, 7 and 1 are unique so that pair is removed reducing the available hand shake set, set‐2=4. 8) Repeating those steps, each pair must represent the answers 8‐0, 7‐1, 6‐2, 5‐3 leaving the wife to have the number "4" along with the asker. I drew several diagrams to help come up with this logic; this is what they look like.
5
Answer this question in 10 seconds: a ball with a radius 2 weighs 80
ounces, how much does a ball with a radius 3 weigh? Answer: య
270 ounces, 6
ଶ
ଷయ
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଼ ୶
You are in a rowing boat on a lake. A large heavy rock is also in the boat. You heave the rock overboard. It sinks to the bottom of the lake. What happens to the water level in the lake? Does it rise, fall or stay the same?
7
You are provided 2 unequal lengths of rope and told that both take exactly 1 hour to burn from end to end, although they don’t necessarily do so in a uniform fashion. How would you go about measuring an exact 15 minute period?
Probability
1
A normal dice with 6 sides with numbers from 1 to 6, toss it at most three times. After each toss, you get a number x, you can either continue or get x dollars. What is the game worth? And what would you like to pay to play the game? Answer: You have to work backwards, like an American option, get asked about this question several times, this seems a must‐ask question for interviewers. If you are risk‐neutral, you get the same answer as in the first question, not otherwise.
2
A coin, toss a number times until you get a head, say N. You will get
N
paid with 2 , what is the game worth? And what would you like to pay to play the game? Answer: Infinity? If you are risk‐neutral, you get the same answer as in 3, not otherwise. 3
Three pieces of pizzas, one with both sides burned, the other one with 1 side burned, and the third one with 0 side burned. Stack them together blindly, you see the top surface is burned, what is the probability for the other side of the top pizza being burned? Answer: 2/3
4
A normal dice, two players, X and Y. X starts tossing the coin first, if a 6 is shown, the game stops and X wins, otherwise, Y tosses and so on. What is the probability for x to win? Answer: 6/11
5
Given a coin with probability p of landing on heads after a flip, what is the probability that the number of heads will ever equal the number of tails assuming an infinite number of flips?
6
Suppose there are n types of coupons and that the type of each new coupon obtained is independent of past selections and is likely to be any of the n types. Suppose one continues collecting until a complete
set is obtained. Find the probability that there is exactly one type i in the final collection. Answer: 1/n * (1+1/2+1/3+...+1/n) Mathematics
1
A random variable, W(t), follows Brownian motion, W(t)
n
is a
martingale, what're values of n? 2
What are AR, MA, ARCH and GARC, why different specifications? How to test non‐constant volatility?
3
Three random variables have the same pair wise correlation function, \rho, what's rho?
4
What is the distance from the origin to the surface 2x+3y+4z=12? Answer: get the normal vector to the surface or minimize x^2+y^2+z^2
5
i
What is i ? Answer: write the base i as e^(2N*Pi+1/2*pi)
6
积分 sec(x)从 0 到 pi/6
7
If X is a Gaussian variable N (0, 1), what is E [X ]?
n
Answer: use moment‐generating function and do Taylor expansion 8
What is type I and II errors? What is the power of test? Central limit
theorem, estimator theory. Answer: Statistic Book 9
What makes a valid correlation matrix? Answer: Symmetrical semi‐definite
10 How to generate correlated random walk series? Answer: Chelosky decomposition or diagnalization 11 What schemes to solve PDEs? 12 What is Quasi‐random number and why? 13 Do you know how to find minimum value of a function? Any other search methods besides Newton? 14 What is Monte Carlo simulation/integration? Logic reasoning
1
th
There are 100 statements, the n
statement says: among the 100
statement, at most (n‐1) are true. How many are true? Answer: th
If n>50, then if the n statement is false, Algorithm
1
What is the difference between thread and process? Answer:
1‐4 can be answered by reading Stroustrup's book 2
What are virtual function / inheritance, public / private / protected member / inheritance, static member / function, dynamic / static cast, and template?
3
STL questions: how to use sort, map? How to implement map idea with vector?
4
What is an initialization list and in what situation such an list must be provided?
5
What is design pattern, any examples? Answer: google
6
How to sort an array? What is the order of complexity? Answer: Know at least 3 sorting algorithm and say how it works
7
How to define a tree and visit all elements of the tree? Answer: Pre‐order, post‐order
Finance
1
A stock, $10 per share, a year later, $12 with probability 60%, $8 with probability 40%, what's the theoretical price of an at‐the‐money call option (underlying the stock) if risk free rate is 0?
2
A vanilla bond has a positive convexity, what security may have a
negative convexity? Answer: Callable bond 3
What is the distribution of correlation coefficient of two random stocks like? Answer: Positive skewed
4
How many stocks should be appropriate in a portfolio? Answer? 200?
5
How to ensure that your portfolio has a 60% probability of having a return >10%
6
两个股票，收益率相等，S1 的波动率 20%，S2 的波动率 30%，两 股票相关系数 50%，问如何分配一笔固定数目的本金到这两股， 使得投资的风险最小。
7
你有一盎司 gold， 过去半年的历史发现黄金的 return rate 是 10%， 而且一直在涨，而 risk free interest rate 只有 1%。你和你朋友决 定签订一个 forward contract ，你要在三个月后卖出去，怎么定价 呢？
8
At the money call option, 现价 100，明天有可能 120, 100, 90, 概率
都是 1/3，请问如何定价 Answer:
Market is incomplete, between (0, 20/3) 9
At the money call option, 现价 100，明天有可能 120, 90, 如果定价
为 7，请问，你的策略是什么？ Answer: Short the call; borrow cash and long stock to hedge to call. There is arbitrary opp. Behavioral
1
Give an example to illustrate how to deal with approaching deadline / bad collaborator / bad project / mistake and so on.
