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So I'm an undergraduate first-year in NYC, and I'm looking to go into finance. Now a lot of these firms will ask you the cliche interview questions. "What can you bring to the team? What is your greatest weakness?" Blah blah blah, shit like that. However, a lot of these firms will also ask you more technical math questions.
Some institutions, like Jane Street Capital, are famous for these. They just ask you really hard math questions.
I was wondering if any of you had these. And they're not just so I can use them to see what they're like, but I think it'd be nice to get a compilation going.
For example, here are some I have:
1.) 25 horses, five lanes, no stopwatch. Find the 3 fastest horses in as few races as possible.
2.) 30 strings. Two parallel rows of 30 holes. Each of the 30 strings goes through one of the 30 holes in the first row, and then one of the 30 holes in the second row. What is the expected number of crossovers, where one string overlaps the other?
3.) 2 strings, a box of matches. Each of the strings takes an hour to burn from end to end. However, they don't burn uniformly, so one inch of the string could take 59 minutes to burn through, and the rest burns up in a minute. Measure 15 minutes of time.
Also, does anyone know what level of math they want? I made AIME in high school and got an 11, and went to a bunch of those national math competitions (ARML, HMMT, PuMAC), but there are definitely people who are better than me who have been rejected.
Anyone who actually works at Jane Street or DE Shaw or some other big and famous firm is welcome to post here. I could use literally any advice.
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On March 29 2011 17:54 DTK-m2 wrote:
So I'm an undergraduate first-year in NYC, and I'm looking to go into finance. Now a lot of these firms will ask you the cliche interview questions. "What can you bring to the team? What is your greatest weakness?" Blah blah blah, shit like that. However, a lot of these firms will also ask you more technical math questions.
Some institutions, like Jane Street Capital, are famous for these. They just ask you really hard math questions.
I was wondering if any of you had these. And they're not just so I can use them to see what they're like, but I think it'd be nice to get a compilation going.
For example, here are some I have:
1.) 25 horses, five lanes, no stopwatch. Find the 3 fastest horses in as few races as possible.
2.) 30 strings. Two parallel rows of 30 holes. Each of the 30 strings goes through one of the 30 holes in the first row, and then one of the 30 holes in the second row. What is the expected number of crossovers, where one string overlaps the other?
3.) 2 strings, a box of matches. Each of the strings takes an hour to burn from end to end. However, they don't burn uniformly, so one inch of the string could take 59 minutes to burn through, and the rest burns up in a minute. Measure 15 minutes of time.
Also, does anyone know what level of math they want? I made AIME in high school and got an 11, and went to a bunch of those national math competitions (ARML, HMMT, PuMAC), but there are definitely people who are better than me who have been rejected.
Anyone who actually works at Jane Street or DE Shaw or some other big and famous firm is welcome to post here. I could use literally any advice.
I think this is correct:
1) 7. Race them in groups of 5. Race the winners of those groups. Then race the only 5 that could potentially be 2nd or 3rd (you can work that out, it's easy to see)
2) Err, misread the problem. I'll do it later.
3) Start one burning at one end, the other at both. After the latter burns entirely light the other end of the first. From that point until it finishes burning is 15 minutes.
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I'm not planning on going into finance, but I have a lot of friends who interview for these sort of things. A couple of the questions they've been asked:
4) You're in the center of a circular lake. There's a wolf on the edge that runs 4 times faster than you swim. He will always take the shortest path to the closest point on the shore to your current location. How do you escape?
5) There are 100 people in separate rooms. They can talk as much as they want until they "begin." Once they begin, they are called one at a time, in no particular order, into another room. In this room is only a light switch that turns on and off a lightbulb in the room. Noone can see the lightbulb unless they are in the room, and the only thing you can do in the room is check the lightbulb, turn it on, or turn it off. How do the people know when all 100 have been called? (You can be called more than once before everyone else has gone once).
6) An explorer lands on an island run by a cult. In this cult, if you know your own eye color, you must kill yourself. There are 500 blue-eyed people on the island and 500 brown-eyed people, but obviously they don't know this. Every night there is a meeting, at the end of which, if you know your eye color you must kill yourself. Before leaving, the explorer attends one of these meetings, and says where everyone can hear "It's so nice to see another blue-eyed person" then escapes. What happens and when?
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Responding to PJ. How do you know you dont have a starting group with the 3 best horses. Your way doesnt work since you cant assume the best horses are in different groups
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edit: never mind, i think 7 is correct found the question and answer from another site.
+ Show Spoiler +
Race all 25 horses in groups of 5. Call the groups A, B, C, D and E. Call the winners from each of these races A1, B1, C1, D1, and E1. That's 5 races so far, and any horse who came in 4th or 5th in any race is out.
Now, race A1, B1, C1, D1, and E1 together. That's 6 races. Let's just assume that A1 wins, B1 is second, and C1 is third. That means:
(1) A1 is definitely the fastest horse overall.
(2) Any horse from group D or group E is out. (3) Any horse from group C, other than C1, is definitely out -- they lost to C1, who was third in the winners' race; thus for example, C2 is fourth fastest at best.
(4) B3 is out. He came in third to B1, thus he's at best 4th fastest overall.
Thus the only remaining horses vying for contention are A1, A2, A3, B1, B2, and C1. We know A1 is definitely the fastest. Thus race A2, A3, B1, B2, and C1 together. That's 7 races. The winner and second-place finisher of this race are the second and third-fastest overall.
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On March 29 2011 19:00 Marradron wrote:
Responding to PJ. How do you know you dont have a starting group with the 3 best horses. Your way doesnt work since you cant assume the best horses are in different groups
Picture this:
Columns are groups of horses in fives. They are conveniently sorted from fastest to slowest from the top left to the bottom right because... The first five races give you the top runners from each group. Throw the top runners from each group in a sixth race. You now have your guaranteed fastest horse at the most top left position. The seventh race has the leftover five horses marked by the orange circles (excluding the fastest horse at the upper left corner). The two winners are the remainder of your three fastest horses.
That's just how I see it though ;p
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OP. You created an awesome thread. I didn't even know firms asked these questions and I'm a finance major =X.
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United States24805 Posts
On March 29 2011 18:35 eluv wrote:
4) You're in the center of a circular lake. There's a wolf on the edge that runs 4 times faster than you swim. He will always take the shortest path to the closest point on the shore to your current location. How do you escape?
What is the answer to this one? If you start 1 inch to the left side of the middle, wait for the wolf to go to the left side, then make a break towards the right side he will beat you there since .5C = pi*r < 4r.
Is it some type of spiral shape taking advantage of the wolf's position? I don't know how to show that mathematically.
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On March 29 2011 19:37 micronesia wrote:Show nested quote +On March 29 2011 18:35 eluv wrote:
4) You're in the center of a circular lake. There's a wolf on the edge that runs 4 times faster than you swim. He will always take the shortest path to the closest point on the shore to your current location. How do you escape?
What is the answer to this one? If you start 1 inch to the left side of the middle, wait for the wolf to go to the left side, then make a break towards the right side he will beat you there since .5C = pi*r < 4r. Is it some type of spiral shape taking advantage of the wolf's position? I don't know how to show that mathematically.
IIRC if you spiral your path (or just row away from the ogre) you'll end up on the opposite side of the lake or at some point where you can just row straight toward shore and get away ;o
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On March 29 2011 19:37 micronesia wrote:Show nested quote +On March 29 2011 18:35 eluv wrote:
4) You're in the center of a circular lake. There's a wolf on the edge that runs 4 times faster than you swim. He will always take the shortest path to the closest point on the shore to your current location. How do you escape?
What is the answer to this one? If you start 1 inch to the left side of the middle, wait for the wolf to go to the left side, then make a break towards the right side he will beat you there since .5C = pi*r < 4r. Is it some type of spiral shape taking advantage of the wolf's position? I don't know how to show that mathematically.
You just start swimming directly away from the wolf, and then you constantly turn your path to swim directly away from the wolf. You will end up spiraling, but you will slowly get closer to the edge while the wolf will never make any progress around the circle towards you.
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On March 29 2011 19:46 Chriamon wrote:Show nested quote +On March 29 2011 19:37 micronesia wrote:On March 29 2011 18:35 eluv wrote:
4) You're in the center of a circular lake. There's a wolf on the edge that runs 4 times faster than you swim. He will always take the shortest path to the closest point on the shore to your current location. How do you escape?
What is the answer to this one? If you start 1 inch to the left side of the middle, wait for the wolf to go to the left side, then make a break towards the right side he will beat you there since .5C = pi*r < 4r. Is it some type of spiral shape taking advantage of the wolf's position? I don't know how to show that mathematically. You just start swimming directly away from the wolf, and then you constantly turn your path to swim directly away from the wolf. You will end up spiraling, but you will slowly get closer to the edge while the wolf will never make any progress around the circle towards you. (or maybe he will make progress, i havent done the math)
I still don't get it 
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On March 29 2011 19:50 endy wrote:Show nested quote +On March 29 2011 19:46 Chriamon wrote:On March 29 2011 19:37 micronesia wrote:On March 29 2011 18:35 eluv wrote:
4) You're in the center of a circular lake. There's a wolf on the edge that runs 4 times faster than you swim. He will always take the shortest path to the closest point on the shore to your current location. How do you escape?
What is the answer to this one? If you start 1 inch to the left side of the middle, wait for the wolf to go to the left side, then make a break towards the right side he will beat you there since .5C = pi*r < 4r. Is it some type of spiral shape taking advantage of the wolf's position? I don't know how to show that mathematically. You just start swimming directly away from the wolf, and then you constantly turn your path to swim directly away from the wolf. You will end up spiraling, but you will slowly get closer to the edge while the wolf will never make any progress around the circle towards you. (or maybe he will make progress, i havent done the math) I still don't get it
Keep swimming straight, but make sure your back is to the wolf the entire time. You will be at the farthest point from the wolf the entire time and you will spiral out of the lake slowly. When you get out, he will still be across the lake from you. I don't know the math behind it though...
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Let R be the radius of the lake.
If you're at 1/4 * R distance from the center, you swim as fast as the wolf runs if you swim in a circle.
That means that at 1/4*R - Ɛ you can swim the inner circle faster than the wolf runs the outer circle, meaning that after you've swum long enough, your distance to the outer circle is 3/4*R ( + Ɛ) , and the wolf's distance to the same spot is π*R.
![[image loading]](http://i.imgur.com/xzLjE.png)
π*R / (3/4*R) = 4π/3 = 4,1888 > 4
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On March 29 2011 20:08 slmw wrote:Let R be the radius of the lake. If you're at 1/4 * R distance from the center, you swim as fast as the wolf runs if you swim in a circle. That means that at 1/4*R - Ɛ you can swim the inner circle faster than the wolf runs the outer circle, meaning that at 1/4*R your distance to the outer circle is 3/4*R ( + Ɛ) , and the wolf's distance to the same spot is π*R. π*R / (3/4*R) = 4π/3 = 4,1888 > 4
Thanks for the explanation, I think I get it now. So you only spiral until you are at at R/4 distance from the center and that the wolf is at the total opposite direction. Then swim straight to the shore, right ?
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On March 29 2011 20:17 endy wrote:Show nested quote +On March 29 2011 20:08 slmw wrote:Let R be the radius of the lake. If you're at 1/4 * R distance from the center, you swim as fast as the wolf runs if you swim in a circle. That means that at 1/4*R - Ɛ you can swim the inner circle faster than the wolf runs the outer circle, meaning that at 1/4*R your distance to the outer circle is 3/4*R ( + Ɛ) , and the wolf's distance to the same spot is π*R. π*R / (3/4*R) = 4π/3 = 4,1888 > 4 Thanks for the explanation, I think I get it now. So you only spiral until you are at at R/4 distance from the center and that the wolf is at the total opposite direction. Then swim straight to the shore, right ?
No need to spiral at all.
First swim straight to the 1/4*R circle, then swim in circle until the wolf is at the opposite side of the lake and then swim straight to the freedom.
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For the horse problem there are ways you can be slightly more efficient in your elimination, though im gonna have to test it out a bit more thouroughly on paper
If you use information about the fastest horse from previous races, you can potentially eliminate more than 2 horses per race in the first 5 races
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Think outside the box and inside the box also for the wolf lake problem.
Know your variables. 1 human, 1 wolf.
Swim a little side to side and the wolf will run the entire distance around the lake again and again and tire itself out.
Then proceed in any direction. Remember the math solution doesn't help you if the lake is so small the wolf always catchs you; you want a solution that encompasses all possible lake sizes which the math only solves for large lakes considering the speed of a human swimming is.
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Just tesitng the horse thing, i've found a solution with only 6 races, though i havnt checked combinatorically whether it will ALWAYS be 6 solutions, only that 6 solutions is possible
I'm gonna test worst and best case in a sec and if i get 6 each time then im pretty sure its 6
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Ooo this thread is awesome, don't have much time so i only did first question.
1) I believe the answer is 8. Get 5 horses race em, keep the top 3 and put 2 untested horse in the mix to race again. The problem with racing the best of 5 seperate groups is that u run the risk of putting 5 of the fastest horses in the first race. Will try the rest tmr!
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EPT
