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The puzzle thread
 

nownow's picture
ST
(Baboon, 113
 
Points)
 on 11/9/10 at 6:53am
mind boggling.JPG

NOTE: If you have seen a problem before, please don't solve it.

This is a thread for posting logic/probability ques, to help people prep for trading interviews. If you have an interesting question that you have heard or thought of, post it to give people practice. I'll post at least one per week.

Difficulty (1-5, 5 being hardest). If it is labeled 6...you should probably move on.

Question #1 (2)
You have a bag with 5 red balls, 3 blue balls. You pull balls out of the hat one at a time. If you get a blue ball, you put it back. If you get a red ball, you remove it. How many draws will you make in expectation until all RED balls have been removed.

SOLUTION: qtrade

Question #2 (4)
You have a digital clock which either displays a time A:BC:DE or AB:CD:EF, depending on the hour. Sometimes, the time displayed will be a palindrome. What is the shortest distance between two consecutive palindrome times? The longest distance between consecutive?

SOLUTION: First half, by zbb. Second half by marketimpact (prior exposure).

Question #3 (1)
Using all of the numbers 12 4 2 and 1, and addition/multiplication/subtraction/division, create a logical expression that equals 24.

SOLVED by Eddie. Note there are multiple solutions.

Question #4 (2)
Using all of the numbers 8 8 3 and 3, and addition/multiplication/subtraction/division, create a logical expression that equals 24.

SOLUTION: edtkh

Question #5 (2)
A couple have children until they have a son. Assume no twins are born. In expectation, how many children do they have?

SOLUTION: edtkh

Question #6 (3)
How many trailing zeros does 1000! have.

SOLUTION: LTV

Question #7 (3)
A spider starts at one corner of a cube. Each second, he randomly chooses one of the edges he is adjacent to and walks along it. What's the expected number of seconds before the spider reaches the furthest corner from his starting point.

SOLUTION: qtrade

Question #8 (5):
A spider starts at one corner of a cube. Each second, he randomly chooses one of the edges he is adjacent to and walks along it. What's the expected number of seconds before the spider reaches a corner that is the furthest corner from ANY corner that the spider has been to so far.

SOLUTION: vectorkid

Question #9 (5):
You start with a well shuffled deck (26 black cards, 26 red) and X dollars. Cards are turned face up one by one. Before every card is turned face up, you may bet any fraction of your money on the color of the card about to be turned over - you double your bet if you win, lose it otherwise. What is the maximum return you can guarantee.

SOLUTION: marketimpact (Note: previous exposure)

Question #10 (2)
Solve for x when x^x^x^x^.... = 3, where ^ is means to the power of, and the expression is nested and infinite.

SOLUTION: qtrade

Questions 11-16 (Difficulty 5)
These questions involve games:

Game A: You roll an n sided die, If the result is 1, you may re-roll or take the result. On the re-roll, you may re-roll a 2 or less, then a 3 or less, ect. On the nth re-roll, you may re-roll n or less (any result). Each re-roll is optional.

Game B: You roll an n sided die, and may re-roll an n or less (any result) or take the result. Subsequently, you may re-roll an n-1 or less, then n-2, ect. On the nth re-roll you may re-roll a 1 only. Each re-roll is optional.

#11. For what n is the ratio of the expected value of game A to the mean result of one die roll ((n+1)/2) greatest.

#12. For what n is the ratio of the expected value of game A to the maximum result of the die (n) the greatest.

#13. For what n is the ratio of the expected value of game B to the mean result of one die roll ((n+1)/2) greatest.

#14. For what n is the ratio of the expected value of game B to the maximum result of the die (n) the greatest.

#15. For what n is E(A)*E(B)/n greatest.

#16 For what n is E(A)*E(B)/(n+1) greatest.

&& means and
| means such that
x^x^x^... = x^(x^(x^(...

Question #17 (3):
Find the set {x1,...,xn | x1+x2+...xn = 16 && n>=1 && xi >=0}, for which x1*x2*...*xn is maximized.

SOLUTION: monkeysama

Question #18 (4):
Generalize the previous result for x1+...xn = k. At the very least give an asymptotic solution.

SOLUTION: jhconnel

Question #19 (5):
Find the set {x1,...,xn | x1+x2+...xn = 16 && n>=1 && xi >=0}, for which x1^x2^...^xn is maximized.

Question #20 (6):
Generalize the previous result for x1+...xn = k. At the very least give an asymptotic solution.

Question #21 (4):
A generalization of #7. Let the spider be in a corner of a regular polyhedron with n sides. What is the expected number of seconds until the spider reaches the furthest corner.

Question #22 (6):
A generalization of #8. Let the spider be in a corner of a regular polyhedron with n sides. What is the expected number of seconds until the spider reaches a corner which is the furthest from a corner than the spider has already been to.

Question #23 (3):
Find the smallest positive x such that x mod k = k-1, for all 2<=k<=10. Mod means, remainder after dividing. For example, 9 mod 2 = 1.

SOLUTION: jhconnel

Question #24 (2):
What is the angle formed by the hands of the clock when it is 1:45?

SOLUTION: Thisdude

Question #25 (4):
Find the set {x1,...,xn | x1^x2^...^xn = 16 && n>=1 && xi >=0}, for which x1*x2*...*xn is maximized.

Question #26 (5):
Find the set {x1,...,xn | x1^x2^...^xn = k && n>=1 && xi >=0}, for which x1*x2*...*xn is maximized.

Question #27 (3):
Order every person in the world from youngest to oldest, accurate right down to the microsecond. Assume there are N people in the world, where N ~ 6.3-6.4 billion. Randomly assign every person in the world one of the integers between 1 and N (only assign each number once!), and call this number xi for the ith youngest person. What is the probability that for at least one 1 <= i <= N, xi = i.

SOLUTION: vectorkid

Question #28 (2):
Show that the sum of all numbers with 9 digits or less is divisible by 9.

SOLUTION: zbb

Question #29 (4):
You start with a well shuffled deck (26 black cards, 26 red) and X dollars. Cards are turned face up one by one. Every time a black card is turned up, you earn $1. Every time a red card is turned, you lose $1. What is your strategy for playing the game? How much would you pay to play this game?

Question #30 (4):
You have a dartboard that is 16 inches in diameter. The dartboard consists of 4 bands which are each 2 inches in width. In addition, each of these bands is divided into 20 equally sized segments. Everyone who is playing is very drunk, so they have an equal probability of hitting any place on the dartboard. How many darts must be thrown until the probability that each segment has been hit is at least 1/2.

Question #31 (2):
Roll two 6-sided die. What is the expected value of the product?

SOLUTION: jhconnel

Question #32 (2):
Roll two 6-sided die. What is the most likely product?

SOLUTION: jhconnel

Question #33 (2):
Roll two 6-sided die. Pick a number. If the product of the die roles is equal to your number, you win that much in cold hard cash. What number should you pick.

SOLUTION: jhconnel

Question #34 (2):
Roll j copies of an n-sided die. What is the expected value of the product?

SOLUTION: jhconnel

Question #35 (4):
Roll j copies of an n-sided die. What is the most likely product?

Question #36 (5):
Roll j copies of an n-sided die. Pick a number. If the product of the die roles is equal to your number, you win that much in cold hard cash. What number should you pick.

Question #37 (3):
Find {x1,...,x7 | for at least one i, xi > 0 && xi are integers && x1*...*x7 = x1+....+x7}.

Question #38 (3):
Find {x1,...,x7 | for at least one i, xi > 0 && xi are integers && xi^...^xj = x1+....+x7}, where i,...,j is an ordering of 1,...,7.

Question #39 (3):
Find {x1,...,x7 | for at least one i, xi > 0 && xi are integers && xi^...^xj = x1*....*x7}, where i,...,j is an ordering of 1,...,7.

Question #40 (5)
A spider starts at one corner of a cube. Each second, he randomly chooses one of the edges he is adjacent to and walks along it. What's the expected number of seconds before the spider has visited every corner of the cube.

Question #41 (6):
A generalization of #40. Let the spider be in a corner of a regular polyhedron with n sides. What is the expected number of seconds until the spider has visited every corner.

Question #42 (4):
What is the expected number of times an n-sided die must be rolled before two numbers repeat?

Question #43 (4):
A n-sided die is rolled until a number is repeated. What is the most likely number of die rolls?

Question #44 (4):
A n-sided die is rolled until all n numbers have appeared. What is the expected number of die rolls?

Question #45 (4):
A n-sided die is rolled until all n numbers have appeared. What is the most likely number of die rolls?

Question #46 (5):
A n-sided die is rolled until a number is repeated. Before the die is rolled, you may pick any number j. If the first repetition occurs in the jth roll, you win j dollars. What j should you pick.

Question #47 (5):
A n-sided die is rolled until a number is repeated. When a number is first repeated, you win j dollars where j is the number of rolls that have occurred. What is the expected value of your reward.

Question #48 (5):
A n-sided die is rolled until all n numbers have come up. Before the die is rolled, you may pick any number j. If the last number comes up on the jth roll, you win j dollars. What j should you pick.

Question #49 (5):
A n-sided die is rolled until all n numbers have come up. Once all numbers have come up, you win j dollars where j is the number of rolls of the die. What is the expected value of your reward.

Question #50 (4):
You have an n-sided die (n>=1). You must roll the die n times, and take the lowest value. What choice of n maximizes the result.

Tags:
Edmundo Braverman's picture

LOL. I'll take the easy

ST
(Human, 14,409
 
Points)
 on 11/9/10 at 6:26am

LOL. I'll take the easy one:

#3 (4*12)/(2*1)=24

zbb's picture

#6 211 If I haven't been

IB
 
Points)
 on 11/9/10 at 7:57am

#6 211

If I haven't been tricked:

if you count upwards, every ten numbers has
a no ending in 5
a no ending in 2

multiply these, and it'll and it'll create a number ending in 0

every ten also has a number ending in 0

except, 9 of those will end in two 0s
and one in three

therefore, each ten that you cunt will add two more 0s, but for the exceptions

there are 100 tens in 1000

plus nine and two more for the "exceptions"

nownow's picture

zbb wrote: #6 211 If I

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 8:25am
zbb:

#6 211

If I haven't been tricked:

if you count upwards, every ten numbers has
a no ending in 5
a no ending in 2

multiply these, and it'll and it'll create a number ending in 0

every ten also has a number ending in 0

except, 9 of those will end in two 0s
and one in three

therefore, each ten that you cunt will add two more 0s, but for the exceptions

there are 100 tens in 1000

plus nine and two more for the "exceptions"

nope, sorry.

LTV's picture

thisdude wrote: zbb

HF
(Gorilla, 538
 
Points)
 on 11/9/10 at 8:55am
thisdude:
zbb:

#6 211

If I haven't been tricked:

if you count upwards, every ten numbers has
a no ending in 5
a no ending in 2

multiply these, and it'll and it'll create a number ending in 0

every ten also has a number ending in 0

except, 9 of those will end in two 0s
and one in three

therefore, each ten that you cunt will add two more 0s, but for the exceptions

there are 100 tens in 1000

plus nine and two more for the "exceptions"

nope, sorry.

Number of factors that are multiples of 10. Since for every multiple of (5)^n (where n is a Natural number) there should be a corresponding even number multiple to make the product a multiple 10, the number of factors that are multiples of (5)^n should give me the answer. 1000/5 + 1000/25 + 1000/125 + 1000/625= 200+40+8+1.

249 trailing 0s.

zbb's picture

I never said I was smart. I

IB
 
Points)
 on 11/9/10 at 8:55am

I never said I was smart. I give up... for now at least

rafiki's picture

Question #2 (4) You have a

(Orangutan, 317
 
Points)
 on 11/9/10 at 8:58am

Question #2 (4)
You have a digital clock which either displays a time A:BC:DE or AB:CD:EF, depending on the hour. Sometimes, the time displayed will be a palindrome. What is the shortest distance between two palindrome times? The longest distance?

Assumptions if the format is A:BC:DE there is still a 0 where A would be in the character time.

Shortest:
12:00:21 from 11:55:11 etc. = 5:10
unless we can use
01:00:10 from 12:55:21 = 4:49

Longest:
12:55:21 to 10:00:01 = 9:04:40
unless we can use
05:55:50 to 10:00:01 = 4:04:11

nownow's picture

LTV wrote: thisdude

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 8:58am
LTV:
thisdude:
zbb:

#6 211

If I haven't been tricked:

if you count upwards, every ten numbers has
a no ending in 5
a no ending in 2

multiply these, and it'll and it'll create a number ending in 0

every ten also has a number ending in 0

except, 9 of those will end in two 0s
and one in three

therefore, each ten that you cunt will add two more 0s, but for the exceptions

there are 100 tens in 1000

plus nine and two more for the "exceptions"

nope, sorry.

Number of factors that are multiples of 10. Since for every multiple of (5)^n (where n is a Natural number) there should be a corresponding even number multiple to make the product a multiple 10, the number of factors that are multiples of (5)^n should give me the answer. 1000/5 + 1000/25 + 1000/125 + 1000/625= 200+40+8+1.

249 trailing 0s.

Good job.

nownow's picture

Magua wrote: Question #2

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 9:05am
Magua:

Question #2 (4)
You have a digital clock which either displays a time A:BC:DE or AB:CD:EF, depending on the hour. Sometimes, the time displayed will be a palindrome. What is the shortest distance between two palindrome times? The longest distance?

Assumptions if the format is A:BC:DE there is still a 0 where A would be in the character time.

Shortest:
12:00:21 from 11:55:11 etc. = 5:10
unless we can use
01:00:10 from 12:55:21 = 4:49

Longest:
12:55:21 to 10:00:01 = 9:04:40
unless we can use
05:55:50 to 10:00:01 = 4:04:11

Wrong, note these are consecutive times so that might change your answer for the longest. Your shortest is definitely wrong. Also, to clarify, 05:55:50 is NOT palidromic since it will display as 5:55:50. Also, 0:30:30 still displays the 0.

zbb's picture

thisdude wrote: LTV

IB
 
Points)
 on 11/9/10 at 9:04am
thisdude:
LTV:
thisdude:
zbb:

#6 211

If I haven't been tricked:

if you count upwards, every ten numbers has
a no ending in 5
a no ending in 2

multiply these, and it'll and it'll create a number ending in 0

every ten also has a number ending in 0

except, 9 of those will end in two 0s
and one in three

therefore, each ten that you cunt will add two more 0s, but for the exceptions

there are 100 tens in 1000

plus nine and two more for the "exceptions"

nope, sorry.

Number of factors that are multiples of 10. Since for every multiple of (5)^n (where n is a Natural number) there should be a corresponding even number multiple to make the product a multiple 10, the number of factors that are multiples of (5)^n should give me the answer. 1000/5 + 1000/25 + 1000/125 + 1000/625= 200+40+8+1.

249 trailing 0s.

Good job.

Nicely done, LTV. I just PMed you sir

zbb's picture

Isn't the shortest 9:59:59 to

IB
 
Points)
 on 11/9/10 at 9:05am

Isn't the shortest
9:59:59 to 10:00:01?

nownow's picture

zbb wrote: Isn't the

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 9:06am
zbb:

Isn't the shortest
9:59:59 to 10:00:01?

Correct, now find the longest. :D

LTV's picture

zbb wrote: I just PMed you

HF
(Gorilla, 538
 
Points)
 on 11/9/10 at 9:20am
zbb:

I just PMed you sir

Responded

zbb's picture

#9 Let's hope I don't get

IB
 
Points)
 on 11/9/10 at 9:38am

#9 Let's hope I don't get this wrong

We're talking about guarantee

So, if by fluke chance the first 26 are of the same suit, the rest will be of the other
You can double your money 26x

Which would be 2^26-1 return

zbb's picture

But at the point of entering

IB
 
Points)
 on 11/9/10 at 9:45am

But at the point of entering the game
Off the top of my head
It would be 100%

zbb's picture

#2 If the answer's not 11

IB
 
Points)
 on 11/9/10 at 9:52am

#2 If the answer's not 11 minutes, it would have to be longer

I'd love to do a little more, but I gotta get going

edtkh's picture

#5: This is akin to the coin

ST
(Baboon, 174
 
Points)
 on 11/9/10 at 10:34am
shorttheworld's picture

you go to a top tier ivy

EN
 
Points)
 on 11/9/10 at 10:43am

I need a crib, a big estate, I need a boat and that need a lake, I need some salmon that need a plate, that need a chef so I feed my safe

From trading equities to slanging wine in Latin America

A ship is safe in harbor, but that is not what a ship i

nownow's picture

@ Zbb #9 is wrong and #2 is

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 11:13am
nownow's picture

edtkh wrote: #5: This is akin

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 10:50am
rufiolove's picture

Edmundo Braverman wrote: LOL.

IB
(Neanderthal, 3,415
 
Points)
 on 11/9/10 at 11:06am

1/2 of the WSO Bash Brothers
"Licensed to Ill It"
We all know Bro J did it...

edtkh's picture

#4: 8/(3-(8/3)) = 24. PS: I

ST
(Baboon, 174
 
Points)
 on 11/9/10 at 12:00pm
nownow's picture

edtkh wrote: #4: 8/(3-(8/3))

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 12:14pm
edtkh's picture

The trailing 0's for the

ST
(Baboon, 174
 
Points)
 on 11/9/10 at 12:07pm
nownow's picture

edtkh wrote: The trailing 0's

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 12:09pm
edtkh's picture

thisdude wrote: edtkh

ST
(Baboon, 174
 
Points)
 on 11/9/10 at 12:12pm
Erebus's picture

What is the angle formed by

 
Points)
 on 11/9/10 at 12:24pm
nownow's picture

Erebus wrote: What is the

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 12:27pm
ivoteforthatguy's picture

Question 9: My solution is:

O
(Neanderthal, 2,210
 
Points)
 on 11/9/10 at 12:41pm
nownow's picture

ivoteforthatguy

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 12:45pm
marketimpact's picture

Q9: this can't be done by

(Monkey, 49
 
Points)
 on 11/9/10 at 1:06pm

"I came, I saw, I networked"

ivoteforthatguy's picture

For q9: Maybe I am getting

O
(Neanderthal, 2,210
 
Points)
 on 11/9/10 at 1:20pm
nownow's picture

marketimpact wrote: Q9: this

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 1:50pm
marketimpact's picture

thisdude wrote: You have the

(Monkey, 49
 
Points)
 on 11/9/10 at 2:26pm

"I came, I saw, I networked"

nownow's picture

marketimpact wrote: thisdude

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 2:27pm
qtrade's picture

#1. Expected turns: 11.85 =

ST
(Chimp, 2
 
Points)
 on 11/9/10 at 5:59pm
LetsGoSailing's picture

#24: 135 degrees... oops,

(Monkey, 60
 
Points)
 on 11/9/10 at 6:27pm

A good friend will come and bail you out of jail...but a true friend will be sitting next to you saying, "Damn...that was fun!"

nownow's picture

qtrade wrote: #1. Expected

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 7:09pm
nownow's picture

LetsGoSailing wrote: #24: 135

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 6:34pm
physconomist's picture

#19 (3,3,3,3,3,1), maybe

CO
(Monkey, 58
 
Points)
 on 11/9/10 at 7:05pm
nownow's picture

physconomist wrote: #19

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 7:08pm
qtrade's picture

#17. All the x should be

ST
(Chimp, 2
 
Points)
 on 11/9/10 at 7:18pm
nownow's picture

qtrade wrote: #17. All the x

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 7:26pm
monkeysama's picture

#17 A: 2^8. The proof is left

O
(King Kong, 1,622
 
Points)
 on 11/9/10 at 7:26pm
nownow's picture

monkeysama wrote: #17 A: 2^8.

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 7:29pm
monkeysama's picture

thisdude wrote: monkeysama

O
(King Kong, 1,622
 
Points)
 on 11/9/10 at 7:46pm
nownow's picture

monkeysama wrote: thisdude

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 7:51pm
monkeysama's picture

OOOoohhhh....now he tells me.

O
(King Kong, 1,622
 
Points)
 on 11/9/10 at 7:49pm
monkeysama's picture

qtrade wrote: #17. All the x

O
(King Kong, 1,622
 
Points)
 on 11/9/10 at 8:37pm
nownow's picture

monkeysama wrote: qtrade

ST
(Baboon, 113
 
Points)
 on 11/9/10 at 8:46pm
monkeysama's picture

Huzzah!

O
(King Kong, 1,622
 
Points)
 on 11/9/10 at 8:48pm

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