## Saturday, December 3, 2011

### Quant Interview Questions

If you're looking for more questions of this kind to build up your quantitative muscle before tackling books like "Heard on the Street", here are a couple of good recommendations:
1. The basic book of quant questions (100 questions - mostly for basic review)
2. The big book of quant questions (1000 challenging questions - this is more advanced)

Have you seen this free app from ChiPrime that tests your basic math even before you try the puzzlers below? Or you can try this web-page (but the app puts all questions in one place at your finger-tips). If you find the below too easy, then try this other ChiPrime app.

New questions posted periodically with answers here.

NEW!!!! ChiPrime now on Reddit!

NEW!!!! Any of the questions in the medium to hard category from the ChiPrime adaptive math quiz should be fair game at a technical interview - I would expect a competent candidate to solve it in 2 minutes or less.

(c) ChiPrime 2014

Here, I capture some puzzlers I ask/was asked at interviews or by friends:
1. (warm up) a two digit number ab (a and b are the digits) when multiplied by 6 gives bbb. What is ab?
2. (warm up) a number N divides 17 and 30 leaving the same remainder. What is the largest possible value of N?
3. (warm up) what is the value of sqrt(7+sqrt(48)) - sqrt(3)?
4. (warm up) what is the value of ln(ln(i))?
5. What are the values of sqrt(2+sqrt(2+sqrt(2+.... ad inf? and 2+1/(2+1/(2+... ad inf?
6. Lattice points are points in space all of whose coordinates are integers. How many lattice points are on the surface of a sphere of radius 1? 2? 45? Can you determine how many lattice points exist on the surface of a sphere of radius 10^10?
7. What is the angle at the center of a circle such that the triangle formed by the two radii and the chord connecting them where they meet the circle, has the largest area?
8. Given a chess board with a corner square removed and a number of L-shaped pieces composed of 3 squares each, tell how you could tile the rest of the plane with them. How many such pieces would you need?
9. *** (cont'd) Given a full chess-board now, can you say which other square you can remove so that tiling the rest of the full plane is still possible? Can you prove a full-tiling is not possible if any of the other squares were to be removed instead?
10. Write a computer program that, given a number e.g. 2391, and a set of binary operations e.g.(+,-,*,/), returns the set of all unique numbers that can be generated by applying these operations to the digits of the number? Extra points if each number in the returned data structure also points to one or more calculations that result in that number.
11. (from MIT Technology Review) what is the smallest number such that if you take the digit in the units place and move it to the front moving all the digits back one, you are left with double the original number? i.e. if you start with 132, you get 213, you want the smallest number so this transformation doubles the original number. What is it?
12. What four points on the surface of a sphere are the furthest apart from each other? eight? three?
13. Which is greater e**pi or pi**e where e is the base of natural logarithms? You should be able to prove this without pen and paper, using simple calculus.
14. Given a random number generator that generates random integer values between 1 and 5 (both inclusive), how would you construct a random number generator that can generate integer values between 1 and 7 both inclusive?
15. Generate all combinations of 3 letters in the English alphabet. Randomly delete 90% of these "words". Words are friends if two of them differ in only one letter. A Social Network of a word x is the list of all its friends, their friends etc. so all connected nodes emanating from x are part of its Social Network. Write a program to compute the complete social network of any random word in the set. What is the size of the social network of the most popular words? the loneliest words?
16. Given six points in three dimensional space, what is the maximum number of triangles you can form with them as vertices? the minimum?
17. (from "Heard on the Street") what is the smallest number such that it leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by 3, ... a remainder of 9 when divided by 10?
18. given a fair coin, how many tosses do you need on average before you get your first tail?
19. what is the probability that given six throws of a fair die, every face shows up once?
20. Show how to solve using Monte Carlo simulation: given a stick of length L, what is the probability that if I break it into three pieces, the three pieces can form a triangle?
21. Walmart sells a simple calendar in which there are two small wooden cubes used to display the date - there is one number on each face. Given two cubes, decide how you would code the digits on the surfaces of the cubes such that using them together you can display all dates in any given month (essentially all numbers 1-31).
22. given a ruler (straight edge) and a compass with which you can draw circles of any radius you choose, devise a means of dividing a segment of length "a" also given to you, into 7 equal parts.
23. two trains 60 mi apart, and tratveling 40 mph and 60 mph respectively, are headed towards each other on a track. a bird that flies 100 mph leaves the first train for the second, touches the second then comes back to the first, and repeats this trip till it is crushed between the trains when they crash. how far does the bird travel before it dies?
24. given an equilateral triangle ABC, and any point P inside it, what is the sum of the lengths of the perpendiculars from P to each of the three sides?
25. with a throw of a normal pair of dice, the probability of getting numbers between 1 and 12 is not equal. design a pair of dice (i.e. tell what positive integers on each face) so the probability of getting all numbers from 1 to 12 is the same. (Sicherman dice problem)
26. given a square ABCD and equilateral triangle ABP with P inside the square, what is the measure of angle ACP?
27. if you regress Y on X (i.e. you end up with Y=alpha1+beta1*X+epsilon1), and then you regress X on Y (yielding X=alpha2+beta2*X+epsilon2), is there a mathematical relationship between beta1 and beta2? what is it?
28. what is the angle at the center of a regular tetrahedron?
29. imagine a Rubik cube 10x10x10 floating in the air. if the outer layer of cubelets fall off, how many cubelets are on the ground?
30. how many squares on a chess board? (hint: the answer is not 64)
31. imagine a large cube that represents a room. At two opposite corners of the cube are a pot of honey and a spider. The spider can only move along the edges of the room-cube. How many moves on average must the spider make before it reaches the honey?
32. (same setup as last problem). what is the shortest distance the spider needs to travel, if it can move along the surfaces of the cube (i.e. not just along the edges) if it wants to reach the honey?
33. the interviewer shows you a wall with two holes: one a perfect circle, the second a square (assume whatever dimensions you want for the two). you are asked to present the simplest solid object that can plug both holes perfectly. what is it?
34. what is special about 1729? how do e, i and pi relate? what are quaternions?
35. compute the cube root of 8.01 to (say) 3 decimal places. calculate the cube of 33 verbally.
36. (***) given a compass (to draw circles of any radius you wish), and a straight edge (ruler) with only 3 marks on it for 0, 1 unit, and a units, devise a means of constructing a segment of length a^(3/2).
37. (***) four circles each with radius R and each centered on a vertex of a square ABCD of side R intersect inside it to form a curvilinear quadrilateral PQRS - this is the area common to all four circles. what is the area PQRS?
38. (***) "google bill-board puzzle". what is the first 10-digit prime number you can find in the digits of e?
39. (***) write a program to generate the first million digits in the square root of 2.
40. how would you solve the simultaneous equations: x^2+y=7, x+y^2=11?
41. prove that for any positive real numbers a, b, c:  (a+b+c)*(1/a+1/b+1/c) >= 9.
42. let x and y be real numbers such that x.sqrt(1-y^2) + y.sqrt(1-x^2)=1. prove that x^2+y^2=1.
43. given log(2)=0.6931, compute the approximate value of (2.1)^(2.1) verbally.
44. given a square matrix A, devise a way of computing e^A where e is the transcendental number that forms the base of natural logarithms. Under what conditions is this computation possible?
45. (goldman sachs) two friends agree to meet at the train station between 1 and 2 pm tomorrow. They also agree that if one of them shows up and the other doesn't, they will wait 15 mins and then leave. What is the probability that the meeting takes place?
46. (goldman sachs) given a linked list and two pointers, how will you determine if there is a loop in the linked list?
47. what is the sum of the first 500 digits after the decimal point in the square root of 2? how would you go about computing this value?
48. given two coins, one which is fair and a second where the probability of heads is 3/4, which coin was the more likely one flipped if you are told you got 6 heads out of 10 flips? 7 heads?
49. What is a partial fraction? a partial function? a lambda expression? a monad?
50. (C puzzle) assume the program includes stdio.h. What is the output of the following code? Error? Why?  int main() { char* p="this is a test"; printf("%c",3[p]); return 0;}
51. Prove the formula for the area of a circle using the fact that the circle is a limiting case of a polygon with infinite vertices.
52. (python puzzle)
x=4*[[]] # x is now [[],[],[],[]]
for i in range(len(x)): x[i]+=[i];
print x;
# this returns [[0,1,2,3],[0,1,2,3],[0,1,2,3],[0,1,2,3]]