Logic/math questions from the interview

[Pint0 Xtreme]

So I got back from my five and half hour interview and I felt compelled to share a few logic/math questions they grilled me on (which I managed to answer all correctly in a reasonable amount of time ). If you have the answer, PM me so it isn't spoiled for everyone else. But of course, feel free to ask any questions for further clarification. To be honest, they're quite easy but fun otherwise.

1.) There is a triangle with an ant on each vertex. Hence, three ants each on a vertex. Each ant is about to traverse on the edge of the triangle to another vertex. So each ant can choose one of two paths to traverse. What is the probability that the ants will not collide with each other?

2.) There is a rectangular cake. Some freeloader has cut a piece of the cake, which happens to be rectangular as well. The rectangular cut can be of any arbitrary size, in any arbitrary position and in any arbitrary orientation. With a single straight slice, how would you determine where and how to cut the cake so that you separate the cake into two equal halves?

3.) There are four people who need to cross a bridge at night and there is one flashlight. Only two can cross the bridge at a time and the flashlight is required to cross it. Person A takes 1 minute to cross it, Person B takes 2 minutes to cross it, Person C takes 5 minutes to cross it and Person D takes 10 minutes to cross it. When two people are crossing the bridge simultaneously, the time it takes to cross the bridge is determined by the slowest person. For example, if B and D crosses at the same time, it would take them 10 minutes to cross the bridge since D takes 10 mins while B only takes 2 minutes. The question is how would you get all four people across the bridge in 17 minutes or less?

Enjoy. (If one of the mods feel this should be moved to somewhere else more appropriate, feel free to do so)

[Darth Servo]

1) One third.
2) You have to cut the hole from the missing piece in half with the cake as well I think.
3) Clarification: the flashlight needs to be used to cross the bridge but can two people, one with the light one without cross going different directions?

[Bob the Gunslinger]

3)Send 1 min and 2 min across. 1 brings back the flashlight. Send 5 and 10 across. 2 brings back the light. 1 and 2 cross the bridge. Done.



I'm not sure I understand #2. Do you get to chose where the rectangle piece is removed from or do you have to come up with an answer for any arbitrary rectangle?

[Pint0 Xtreme]

1) One third.

Incorrect. Try again.

2) You have to cut the hole from the missing piece in half with the cake as well I think.

You need to be more specific than that.

3) Clarification: the flashlight needs to be used to cross the bridge but can two people, one with the light one without cross going different directions?

No. The light can only be used to cross in one direction. People on the other side of the bridge cannot cross simultaneously in the other direction.

[Pint0 Xtreme]

3)Send 1 min and 2 min across. 1 brings back the flashlight. Send 5 and 10 across. 2 brings back the light. 1 and 2 cross the bridge. Done.

Dammit guys. I said PM me the answers!

I'm not sure I understand #2. Do you get to chose where the rectangle piece is removed from or do you have to come up with an answer for any arbitrary rectangle?

You have to come up with an answer for any arbitrary rectangular cut.

[Pint0 Xtreme]

To clarify #2, it's basically a rectangular cake with a rectangular hole in it. The hole can be of any size, any position and of any orientation. Hope that helps.

[R. U. Serious]

You extend a straight line from the two center points of the rectangles (the big, cake one, and the "hole"), cutting on that line will cut both, the cake and the hole, in half.

Of course the easier solution mathematically is to cut it horizontally, but that's rather dificult, and who likes to eat only the bottom of a cake...

[Surlethe]

1.) There is a triangle with an ant on each vertex. Hence, three ants each on a vertex. Each ant is about to traverse on the edge of the triangle to another vertex. So each ant can choose one of two paths to traverse. What is the probability that the ants will not collide with each other?

Each ant has two choices. Hence, the total number of possible paths (e.g., (1-2,2-3,3-1)) is eight. Out of those choices, we need to determine how many result in ants not colliding. Clearly, given one ant's choice and the assumption that the ants won't collide, the other ants' choices are completely determined. Thus, since each ant has two choices, we have 2/8 = 0.25.

(Note that this is the rotational subgroup of S_3, which is isomorphic to the group of rigid motions (rotations and flips) of a triangle. Another way of phrasing the problem is to ask, "How many ways are there to cut a triangle out of a piece of paper and replace it without flipping it at all? Obviously, the answer is two: two directions of rotation. Since there are eight possible paths the ants could all take (note that some will not be in S_3 -- e.g., (1-2,2-1,3-2)) we arrive at the answer.

2.) There is a rectangular cake. Some freeloader has cut a piece of the cake, which happens to be rectangular as well. The rectangular cut can be of any arbitrary size, in any arbitrary position and in any arbitrary orientation. With a single straight slice, how would you determine where and how to cut the cake so that you separate the cake into two equal halves?

Cut the cake in half horizontally.

3.) There are four people who need to cross a bridge at night and there is one flashlight. Only two can cross the bridge at a time and the flashlight is required to cross it. Person A takes 1 minute to cross it, Person B takes 2 minutes to cross it, Person C takes 5 minutes to cross it and Person D takes 10 minutes to cross it. When two people are crossing the bridge simultaneously, the time it takes to cross the bridge is determined by the slowest person. For example, if B and D crosses at the same time, it would take them 10 minutes to cross the bridge since D takes 10 mins while B only takes 2 minutes. The question is how would you get all four people across the bridge in 17 minutes or less?

AD: 10 min.
(throw flashlight back across bridge)
CB: 5 min.
That's 15 min.

[Wyrm]

3.) There are four people who need to cross a bridge at night and there is one flashlight. Only two can cross the bridge at a time and the flashlight is required to cross it. Person A takes 1 minute to cross it, Person B takes 2 minutes to cross it, Person C takes 5 minutes to cross it and Person D takes 10 minutes to cross it. When two people are crossing the bridge simultaneously, the time it takes to cross the bridge is determined by the slowest person. For example, if B and D crosses at the same time, it would take them 10 minutes to cross the bridge since D takes 10 mins while B only takes 2 minutes. The question is how would you get all four people across the bridge in 17 minutes or less?

AD: 10 min.
(throw flashlight back across bridge)
CB: 5 min.
That's 15 min.

Better way:

CD: 10 min.
(throw flashlight back across bridge)
AB: 2 min.
Total: 12 min.

[Darth Servo]

1) One third.

Incorrect. Try again.

Damn it, I added the possibilities instead of multiplying them.

[Max]

May I ask what the interview was for? I may have missed it...

[Gil Hamilton]

AD: 10 min.
(throw flashlight back across bridge)
CB: 5 min.
That's 15 min.

I'd assume since the mean time for crossing the bridge is 4.5 minutes that the bridge is sufficiently long that it precludes throwing the flashlight. Besides, it's pitch black out, how would you even catch it?

The solution is:
AB crosses (t = 2 min)
A returns (3 min)
CD cross (13 min)
B returns (15 min)
AB cross again (17 minute)

Dingbats.

[Pint0 Xtreme]

CLARIFICATION: You cannot cut the cake horizontally because that would leave the bottom half with no topping!

[Pint0 Xtreme]

May I ask what the interview was for? I may have missed it...

A programming position at USC.

[KrauserKrauser]

For #2, since the cake was cut but not removed just cut it in half normally.

[Pint0 Xtreme]

For #2, since the cake was cut but not removed just cut it in half normally.

Incorrect. The cut was removed. There is a rectangular hole in it.

[SilverWingedSeraph]

For #2, since the cake was cut but not removed just cut it in half normally.

I believe, while not implicitly stated, it's implied that the piece which was cut was also removed. Cutting it in half horizontally is the way to go.

[KrauserKrauser]

For #2, since the cake was cut but not removed just cut it in half normally.

Incorrect. The cut was removed. There is a rectangular hole in it.

Adding that to the question would be key as proper reading comprehension allows for my answer to be correct.

[Pint0 Xtreme]

I believe, while not implicitly stated, it's implied that the piece which was cut was also removed. Cutting it in half horizontally is the way to go.

Again, that is not correct because the bottom piece would be left with no topping. This question was not designed for you to look for a loophole but for you to apply basic geometric reasoning to solve the problem.

[Pint0 Xtreme]

Hint/Clarification: It just occurred to me that I should clarify that when splitting the cake in half, the two halves do not have to be exactly identical in shape. They just have to be equal in terms of the amount of cake.

[Darth Fanboy]

Hint/Clarification: It just occurred to me that I should clarify that when splitting the cake in half, the two halves do not have to be exactly identical in shape. They just have to be equal in terms of the amount of cake.

OH!

Well in that case wouldn't you just create a line using the exact cetners of both rectangles, the two rectangles being the cake and the piece removed from the cake?

[Pint0 Xtreme]

OH!

Well in that case wouldn't you just create a line using the exact cetners of both rectangles, the two rectangles being the cake and the piece removed from the cake?

Precisely.

[Durandal]

2.) There is a rectangular cake. Some freeloader has cut a piece of the cake, which happens to be rectangular as well. The rectangular cut can be of any arbitrary size, in any arbitrary position and in any arbitrary orientation. With a single straight slice, how would you determine where and how to cut the cake so that you separate the cake into two equal halves?

Cut the cake through a plane running parallel to the surface the cake is sitting on positioned at half the cake's height.

[darthbob88]

2.) There is a rectangular cake. Some freeloader has cut a piece of the cake, which happens to be rectangular as well. The rectangular cut can be of any arbitrary size, in any arbitrary position and in any arbitrary orientation. With a single straight slice, how would you determine where and how to cut the cake so that you separate the cake into two equal halves?

Cut the cake through a plane running parallel to the surface the cake is sitting on positioned at half the cake's height.

Already been disqualified, and the real answer found.

[Braedley]

Since no one else is using the PM, I guess I won't either (dammit, I was going to get #3 )

Anyways, #1 is easy and basically solved but not posted so:

There are only two possible events where the ants will not collide. 1-if they all take the left route, 2-if they all take the right route. There are a total of 2^3=8 possibilities. 2/8=0.25=25%