Chill Out Area : math problem

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mklim_irlpl
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Joined: Sun Jul 21, 2013 5:56 pm

Mon Mar 23, 2015 3:32 pm

Doesn't it cost anything from 0.01 to 0.1? :D

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gutuami
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Tue Apr 21, 2015 7:47 pm

You have Albert, Bernard and Cheryl. Cheryl says "my birthday is one of these ten dates"

May 15 May 16 May 19
June 17 June 18
July 14 July 16
August 14 August 15 August 17

She gives Albert the month of her birthday. And Bernard the number.

Then the following conversation occurs:

1) Albert: I don’t know when the birthday is, but I know Bernard doesn’t know too.
2) Bernard: At first I don’t know when the birthday is, but now I know.
3) Albert: Then I know the birthday too.

From that information, work out Cheryl's birthday.

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gutuami
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Joined: Wed Apr 15, 2009 4:06 pm

Tue Apr 21, 2015 9:13 pm

gutuami wrote:You have Albert, Bernard and Cheryl. Cheryl says "my birthday is one of these ten dates"

May 15 May 16 May 19
June 17 June 18
July 14 July 16
August 14 August 15 August 17

She gives Albert the month of her birthday. And Bernard the number.

Then the following conversation occurs:

1) Albert: I don’t know when the birthday is, but I know Bernard doesn’t know too.
2) Bernard: At first I don’t know when the birthday is, but now I know.
3) Albert: Then I know the birthday too.

From that information, work out Cheryl's birthday.
solution: https://www.youtube.com/watch?v=emiMj8cCL5E

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marksmeets302
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Wed Apr 22, 2015 2:17 pm

Awesome!

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marksmeets302
Posts: 505
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Thu Apr 23, 2015 8:17 am

problem.gif
You have to walk from point A to point B 10 times, blindfolded. Every time before you start walking, a fair coin is tossed, and if it comes out heads, a wall in the middle is present (situation 2 in the picture), while if it comes out tails it is not (situation 1 in the picture). As you are blindfolded, you do not know if the wall is present or not.
Even though you are blindfolded, you can move in perfectly straight lines towards any point you choose. Can you give a prescription how to walk from A to B so that the total distance you cover is as small as possible?
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gutuami
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Thu May 28, 2015 12:21 am

if you take the round route that would be √2*2*10=~28.2m
if you take the 50% chance route then:
min=10*2=20m
max=(1+1+√2)*10=~34.1m
avg=27.05m

the standard deviation depends on sample size. For ten times coin toss it is √2.5=~1.58

so taking chances will give large deviation from 27.05m

on 10 walks only I would take the round routes.
on 1000 walks I would take the coin toss routes

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gutuami
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Thu May 28, 2015 12:25 am

Imagine a scientist deals four cards out in front of you. Unlike normal playing cards, these have single numbers on one side and single colors on the other. You see from left to right a three, an eight, a red card, and a brown card. Now he says, “I have a deck full of these strange cards, and there is one rule at play. If a card has an even number on one side, then it must be red on the opposite side. Now, which card or cards must you flip to prove I’m telling the truth?”
Remember — three, eight, red, brown — which do you flip?

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marksmeets302
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Joined: Thu Dec 10, 2009 4:37 pm

Thu May 28, 2015 9:06 am

Regarding the pop-up wall, there's another solution.
PHOTO_20150528_095503_resized.jpg
If you want to walk from A to B, not knowing if the wall pops up or not, walk straight to point X, taking path p0. If the wall is down, take path p3 to arrive at B. If the wall is up, walk around it, by taking path p1 followed by p2.

So it's a matter of minimizing half the length of p0 + p3 plus half the length of p0 + p1 + p2.

According to pythagoras the distance D(x) is

D(x) = 0.5 * 2 * sqrt(1+x^2) + 0.5*(sqrt(1+x^2) + 1 - x + sqrt(2) )

The minimum of D(x) can be found by differentiating and looking where D'(x) equals zero.

D'(x) = 3x / (2*sqrt(x^2 + 1)) - 1/2

D'(x) = 0 when x = 1 / (2*sqrt(2))
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marksmeets302
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Thu May 28, 2015 9:30 am

The rule is of the form P implies Q. So if we check the 8 card (P), it must be red (Q). If it isn't you are not telling the truth. Given a brown card (not Q), we have P -> Q and (not Q), which leads to (not P). So if the brown card is even, you are not telling the truth.

Flipping the other cards gives no information.

If based upon the 8 and the brown card you are still telling the truth.. then I don't know how to definitively prove you are telling the truth. There may still be an even black card in the rest of the deck. I must be missing something here :-)

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gutuami
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Thu May 28, 2015 1:07 pm

The only answer is to turn over both the eight card and the brown card. If you replace the numbers and colors on the cards with a social situation, the test becomes much easier. Pretend the psychologist returns, and this time he says, “You are at a bar, and the law says you must be over twenty-one years old to drink alcohol. On each of these four cards a beverage is written on one side and the age of the person drinking it on the other. Which of these four cards must you turn over to see if the owner is obeying the law?” He then deals four cards which read:
23—beer—Coke—17
Now it seems much easier. Coke tells you nothing, and 23 tells you nothing. If the seventeen-year-old is drinking alcohol, the owner is breaking the law, but if the seventeen-year-old isn’t, you must check the age of the beer drinker. Now the two cards stick out—beer and 17.

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