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Contest 10.5: Three logic/math questions and an ad (Ended)


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Innocent Bystander
RJ: Matthew Matician

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Thanks for playing, everyone! Here are the results:

1) Postdoc in Problem Solving: Nwabudike Morgan for 2000 influence points!

1) Doctorate in Deduction: Weicong Sng for 1000 influence points!

2) Masters in Mind Games: Scott/Ratan Joyce for 1000 influence points!

3) Bachelors in Brain Busting: Tony Wooster for 500 influence points!

---

Timeline:
James Grey: #3 correct
Scott/Ratan: 2, 3
Tony Wooster: 1, 3
Weicong Sng: 2, 3
Nwabudike Morgan: 1, 2, 3
Weicong Sng: 1, 2, 3
Adam Ybart: 1, 3
Tony Wooster: 1, 2, 3

---

Alas, my disguise is no longer relevant, as I am no longer a bystander (though I do maintain my innocence). To that end, I shall reveal that my real name is Matthew Matician, or just Matt Matician. I expect to take over Econosia using the power of mathematics! (and good hygiene)

Given these goals, I will be unveiling the true names of my companies within the next few days (no one could have suspected that 'Totally Legit, Inc' wasn't my founding company's actual name) and will post an ad announcing so when the time comes. There might even be an IPO, who knows. To celebrate my mathematically-inclined world-domination goals, I have created a follow-up to the last logic game with Ratan "Axiomatic-to-the-Max" Joyce's blessing.

Description:
I'll post 3 logic/math questions. Ratan has graciously offered influence rewards for correct answers. [He may edit the reward values I've put in my post, I just copy/pasted, assuming the rewards would be the same.]

Rewards:
1. First to answer all 3 questions correctly will be rewarded 2000 influence!
2. Second to answer all 3 questions correctly will be rewarded 1000 influence.
3. First to answer 2 questions correctly will be rewarded 1000 influence.
4. Second to answer 2 questions correctly will be rewarded 500 influence.
5. No influence given for participation.
6. If you are the second person to guess the same set of answers as someone else and they don't win a prize for their post, you will have 500 influence points deducted your total winnings per occasion (total winnings will not be negative, though). [Example: if you copy another person's answers and they aren't the first person to get 2 correct or 3 correct, any potential award you get for later posts is diminished.]

Each person can only get 1 reward.

The Twist:
1. You must submit your answer in a post as a reply to this thread. So once posted, your submission is available for everyone else to see/copy. If you had 2 correct answers, and 1 wrong answer, someone else can simply copy your correct answers, and if (s)he answered the other question correctly, (s)he gets the top prize.
2. Editing your post automatically voids your entry. But you may delete your post and repost it. Reason being I don't want to make a function to sort the posts by their edit time.
3. You may ask for clarifications on the problems (by PM or by posting), and I will add them into the clarification part of this post. But I will not comment on the answers until the deadline of the contest.

Deadline:
Wednesday, Apr. 25, 10:00 P.M. (might be bumped later if there's still interest at the deadline and they haven't been solved)

A Twistier Twist:
I will post the first question by itself to give the first readers an hour or two [or more if I have a hard time thinking up good puzzles] to think about it before I post the second and third questions. [To everyone reading this after I post all of them: sorry!]

Question the first:

John and Jane were both trying to snipe batches of B2B water that were being sold on-the-cheap as they were posted (they were the only two players buying B2B water and they had more than enough money to continue this game for as long as they wanted). John snagged the first batch of water and Jane snagged the second batch. After that, the chance that John snagged the next batch was equal to the number he had snagged so far divided by the total batches that he had tried for. (For instance, if at some point he snagged 5 of the first 15, his chance of snagging the 16th one was 5/15 or 1/3.) What is the chance that John got exactly 60 of the first 120 batches? Express your answer as a/b, where a and b are integers.


Question the second:

Frieda likes studying mattresses and the buttons that are sewn onto them. Each Econosian mattress has a rectangular grid of buttons on it: 7 across and 13 down. She happens to own a mattress-slicer that can cut an existing mattress piece (or full mattress) into two smaller pieces along a perfectly horizontal or vertical cross-section of the mattress (it can't cut more than one mattress piece at a time and it can't cut through buttons). Define x to be the fewest number of times she needs to use the mattress slicer so that each mattress piece contains exactly 1 button.

A few blocks down, Fred is slicing chocolate cake (before frosting; can assume the cake is a perfect cylinder) into pieces. After each slice, he leaves the cake in the same position so that the next slice is applied to the whole cake. Fred is a little picky: he always makes his slices so that, in the end, every piece is exactly the same (in shape and size). Define y to be the most identical pieces he can create using 5 slices.

What is x*y?


Question the third:

Tex (no relation to Tex Corman) is thinking of upgrading one of his three existing factories. He'd like to choose one of them at random, with the condition that each choice is equally likely. He decides to use a computer to pick for him, so he purchases the latest and greatest CPU from East Eosia Company (nice, quality 25!) through the import market.

Luckily for him, this chip has a random number generator on it that spits out a sequence of '0' bits and '1' bits! Unluckily for him, East Eosia Company has put out a notice saying that their random number generator is faulty: it doesn't return 0 and 1 with equal probability. For better or worse, each bit (generated independently) has an x% chance that it will be a '0' bit and a y% chance that it will be a '1' bit (x% + y% = 100%), where x and y are constant but unknown (should've gone for the quality 40 chip).

Using this stream of 0's and 1's that the CPU spits out, how can Tex devise a way so that each of the three options has exactly the same chance of being picked? [Your strategy can make use of as many bits as you need, but assume that you will never know the exact values of x and y throughout.]

Clarifications:

1) The rule that defines the chance that John snags a particular batch only applies from batches 3 and up. In particular, batch 2 is not subject to this rule (but he did try to snag it); so for batch 3, he has a 1/2 chance of snagging it.

2) In Q2b, I do in fact mean: "What's the most number of identical partitions one can create by slicing a cylinder with 5 planes?" Sorry for getting overly fancy with the wording.

3) In Q3, x > 0, y > 0. The random number generator only spits out two numbers: 0 and 1. But you can generate as many of these numbers as you'd like. There are many right answers.
Scott (Admin)
RJ: Ratan Joyce
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Clarification needed:
Did John try to snatch the 2nd batch?
If he did not, after the first 2 batches, John has a 1/1 chance of getting the 3rd one, and therefore 1/1 chance of getting each batch thereafter?

Edit: left out the not. But Matt got the point :-)
Innocent Bystander
RJ: Matthew Matician

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(Clarification #1 added.)
James Grey
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1. Not sure at the moment, struggling but will post answers I've got so far as I'm having a break :P

2.a (13-1) + (7-1) = 18 = x

2.b C(0) = 1
C(1) = C(0) + 1 = 2
C(2) = C(1) + 2 = 4
C(3) = C(2) + 3 = 7
C(4) = C(3) + 4 = 11
C(5) = C(4) + 5 = 16 = y

2. 18 * 16 = 288

3. Print out 4 bit strings repeatedly, if 0011 comes up first upgrade factor A, if 0110 comes up first upgrade factory B, if 1100 comes up first upgrade factory 3.
Scott (Admin)
RJ: Ratan Joyce
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1. 1/2. Assuming John tries to grab every single batch.

2. 1620

3. Assuming both 0's and 1's do exist. Spit out length L strings (L > 0) for each factory, whichever has the most 1's gets upgraded. In the case of a tie, just keep generating more bits.
Tony Wooster
RJ: Johnny Appleseed

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1. 1/119

2. The wording is odd for the second half. Assuming that I can cut the cake across ANY of its symmetries, and that the cake and its pieces are entirely immobile, rephrasing the question as "what is the maximum number of pieces that can result from cutting a cylinder with 5 planes such that every piece is identical", then the total answer is: 1296.

Could you re-phrase the question to make it clearer?

3. We need two good bits to decide. We can get a good "bit" from the computer by reading two bits and encoding it down to one bit as follows:

01 -> 0
10 -> 1
00, 11 -> try again

P(B=01) == P(B=10) == 1/2

So, read as many two bit pairs (and only pairs) as is required to get two non-biased bits. If the value is 1, 2, or 3, upgrade that factory. If 0, start over.
Weicong Sng
RJ: Park Min Young
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3. can someone clarify if the RNG spits out decimals or whole numbers or what? my experience with RNGs spits out a decimal between 0 - 1 to 3 decimal places. (not sure including 0 and 1)
and the question is binary or what? like 00000100 = 4, 00001000 = 8, 00010000= 16?

If you ask me, qn3 isn't exactly mathematics, more like creative thinking. How do you exactly put a RIGHT or WRONG to our answers? That is unless you're telling me that the answer involves mathematics as well.. too.
Innocent Bystander
RJ: Matthew Matician

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(Added clarifications #2 and #3.)
Weicong Sng
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1. (59!)^2 / 119! (you can't possibly want me to put it in explicit form right..?) the answer applies no matter what sequence of events, as long as both john and jill gets 60 each out of the 120.
2. x=90 y=18 x*y=1620
3. Instead of using the RNG to generate 1 single number to determine the factory to be upgraded (which is unfair since 0 and 1 occurence are not 50:50 and thus factories will get unequal chances), simply generate one number for each of the 3 factories, and largest wins.
Eg. factory A gets 10111110, factory B gets 11110000, factory C gets 01111111.
Convert to binary, Factory B clearly wins.
As long as the lottery system does not discriminate against any factory, and that is by making it so that the 0-1 discrepancy doesn't matter at all.
Nwabudike Morgan
RJ: CEO Nwabudike Morgan
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1. The probability that John and Jane each get sixty batches of water is 1/119.

2a. Since the mattress cutting machine cannot cut more than one mattress piece at a time, 90 cuts are required.
2b. I believe the maximum number of pieces is 18. I haven't come up with a way to attain 24.
2. The product of 90 and 18 is 1,620.

3. Define a trial as follows. Take three separate sequences of two bits for each factory: (a_i), (b_i), and (c_i). Repeat this process until all three two-bit sequences contain one 0 and one 1, AND one sequence is distinct from the other two. Choose the factory that corresponds to the distinct sequence.
Weicong Sng
RJ: Park Min Young
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Please disregard earlier post, i got Q1 wrong.

1.(JUST REALISED IT IS THE PROBABILITY OF 1 SPECIFIC CASE) (59!)^2 / 119! (you can't possibly want me to put it in explicit form right..?) the answer applies no matter what sequence of events, as long as both john and jill gets 60 each out of the 120.

(59!)^2 / 119! x 118 nCr 59
(59!)^2 / 119! x 118!/[(59!)^2]
= 1/119


2. x=90 y=18 x*y=1620
3. Instead of using the RNG to generate 1 single number to determine the factory to be upgraded (which is unfair since 0 and 1 occurence are not 50:50 and thus factories will get unequal chances), simply generate one number for each of the 3 factories, and largest wins.
Eg. factory A gets 10111110, factory B gets 11110000, factory C gets 01111111.
Convert to binary, Factory B clearly wins.
As long as the lottery system does not discriminate against any factory, and that is by making it so that the 0-1 discrepancy doesn't matter at all.
Adam Ybart
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1) 1/119
2) x = 72
y = 26
xy = 1872
3) Generate 4 bits, if it's 0101 upgrade factory A, if it's 0110 upgrade factory B, if 1001 upgrade factory C. Otherwise regenerate until it's one of these.
Basically looking at all possible values of 4 bits any time there's a two consecutive bits the probabilities are unequal and so we must regenerate.
Tony Wooster
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It's too late now, but revising my answer for 2: 1620
zxektok megatron
RJ: zxektok

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i dislike this new format of first to the thread. why are contests now like this?
almost an activity test?

:p
Innocent Bystander
RJ: Matthew Matician

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I understand where you're coming from, I enjoy puzzles in an untimed format. As for why I chose this format? Just monkey see, monkey do: I'm following Scott/Ratan's format for the last one.
zxektok megatron
RJ: zxektok

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well make them last a few days so i can actually get a shot at playing - some people work

probably once a week, also please try an avoid the silly hypotheticals

for example : if the mattress cutter can only cut one "piece" at a time then i can still fold it over and cut a single mattress piece. in which case i can get less than 90...
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