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Heroes Community > Tavern of the Rising Sun > Thread: Let's talk about Maths!!!
Thread: Let's talk about Maths!!! This thread is 55 pages long: 1 2 3 4 5 ... 10 20 30 40 ... 51 52 53 54 55 · «PREV / NEXT»
Corribus
Corribus

Hero of Order
The Abyss Staring Back at You
posted April 22, 2011 10:20 PM
Edited by Corribus at 22:23, 22 Apr 2011.

I'm getting old, I think - memory is dying, lol.

Alright, how about another new problem, then - hopefully one we haven't done before!  I don't have the answer to this one.

Two Darts

Two darts are thrown at a large circular target of diameter = 1.  Supposing they strike the target at random places, what is the expected distance separating the darts?

<break>

Alright, maybe I'll let Ihor find his answer to the ant problem and I'll post the answer I found later at his request.
____________
I'm sick of following my dreams. I'm just going to ask them where they're goin', and hook up with them later. -Mitch Hedberg

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dimis
dimis


Responsible
Supreme Hero
Digitally signed by FoG
posted April 22, 2011 10:24 PM
Edited by dimis at 22:27, 22 Apr 2011.

Yeah, I see. I found the receipt on the trash can.

I had on the equivalent first equation
3 T0 = 3* (1/3) *(3* T1) + 3 * 1
so
T0 = T1 + 1/3

go figure ... lol

Anyway, for the third problem you can only get a bound in general dimension. Which problem is this ?

I really got to go though.
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The empty set

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ihor
ihor


Supreme Hero
Accidental Hero
posted April 22, 2011 10:41 PM
Edited by ihor at 23:00, 22 Apr 2011.

Actually I also built a system of equations for that problem.

I denoted t - needed time, t1 - average time to get to meal from vertices, adjactent to where spider at initial point, t2 - average time to get to meal from vertices, adjactent to where ant located.

I've got a system as follows:

t2 = 1/3 + 2/3*t1
t1 = 2/3(1+t2) +1/3(1+t)
t  = 1 + t1

From here I've got my result 3.5 minutes.
Now I think I made a mistake in first equation. It should be

t2 = 1/3 + 2/3(t1+1)

And then the result would be t = 10 minutes. Is that correct now?

Edit: Okay, its midnight here and I'm pretty tired, would like to hear your remarks, will look tomorrow, also will think about the Darts problem... Nice discussion today... Good night.

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Corribus
Corribus

Hero of Order
The Abyss Staring Back at You
posted April 22, 2011 11:07 PM

Yes, that's the answer I had as well, ihor.
____________
I'm sick of following my dreams. I'm just going to ask them where they're goin', and hook up with them later. -Mitch Hedberg

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Corribus
Corribus

Hero of Order
The Abyss Staring Back at You
posted April 22, 2011 11:47 PM

Well, I don't know what got into me because this took a hell of a long time, but dimis I have made the list of problems.  This may not be 100% comprehensive.  I didn't include small homework problems, and also some long problems related to HOMM statistics.  But I think most of them are here.  Maybe you can put a note about this on the opening post or something.

Math Problems Posted

Page 1, Poster: Corribus

John wants to paint his dining room.  He figures it will take him 3 hours to paint the room.  It would take his son, James, 5 hours to paint the room.  If both of them paint it, how long will it take them to finish the room?

Page 3, Poster: Broadstrong

How can you determine whether any given natural number (so no fractions or decimals or such) is divisible by 2, 3, 4, 5, 6, 9, 10 and 11?

Page 5, Poster: Corribus

You have a bag that contains two coins. One coin has heads on both sides and the other has heads on one side and tails on the other. You select one coin from the bag at random and observe one face of the coin. If the face is heads, what is the probability that the other side is heads?

Page 5, Poster: Mamgaeater

you have 4 bags of marbles with 5 1 ounce marbles in each. But another bag contains 5 marbles that weigh 1.1 ounces each.
How can you figure out which bag has the marbles that weigh 1.1 using a single load scale once?


Page 7, Poster: imis

1] Show that the set of natural numbers N = {0,1,2,3,4,...} has the same cardinality (denoted as =_c from now on) as the set of even natural numbers E = {0,2,4,6,8,...}.
Once you show that note that you proved that a proper infinite subset of an infinite set can have the same cardinality as the "bigger" one.
2] Continue the shock by proving that N =_c Z, where N again the natural numbers {0,1,2,3,4,...} and Z the set of integers {...,-3,-2,-1,0,1,2,3,...}.
3] What about P = (0,+oo) and the Reals R = (-oo,+oo)?
4] What about U = (0,1) and the Reals R?
5] What about P = (0,+oo) and U = (0,1)?


Page 7, Poster: dimis

know nothing about their birthdays (of course I know mine! ) but I claim that at least two people among the participants have birthday on the same calendar day (NOT including the year...). Would you bet for or against that statement? Why?

Page 8, Poster: Ecoris

Find 100 consecutive numbers containing no primes.

Page 10, Poster: Corribus

Two men have the following conversation.

John: Hi Matt.  How are your three kids?  I forgot what their ages are.

Matt:  Hi John.  Don't you remember? The product of their ages is 36.

John: I still don't remember.

Matt: The sum of their ages is the same as my house number.

John: I still can't remember!

Matt: The oldest one has red hair.

John: Ah, right.  I remember now!

How old are Matt's kids?


Page 11, Poster: Ecoris

Is it possible to cover a 6x6 board with 9 pieces of the following shape:

X
XXX

(the Tetris "L")

You may rotate and flip the pieces.


Page 12, Poster: dimis

And a question that comes along with a historical note. Can you prove that square root of two is irrational?

Page 13, Poster: Ecoris

Consider the square below. B is the midpoint of the right edge. A is the lower left corner. You may place the point M anywhere on the top edge.

Where should you place M to minimize the total distance from A to B to M? (i.e. the total length of the two lines inside the square)


Page 13, Poster: dimis

Consider a 3x3x3 cube (like Rubik's cube) made of cheese.  We have a mouse that starts eating the cheese from one of the corners. Each time the mouse eats a small cube of cheese, keeps on eating in a hortizontal or vertical direction (i.e. never diagonal) a small cube that is next to what has just been eaten.
Question: Is there a "way of eating" (i.e. a path) such that the mouse eats the "inner-most" small cube last ?


Page 14, Poster: dimis

Suppose we are given a rule:
Rule: If a card has a vowel on one side, then there is an even number on the other side of the card.

And now suppose that someone presents you the following four cards:
A, K, 4, 7

The question is:
Question: Which cards should I turn to verify the rule above?


Page 15, Poster: dimis

problem: Write down the powerset of the powerset of the empty set. (no typo in expression)

I don't know if you prefer this variation
problem rephrased: Write down the set of all subsets of the set of all subsets of the empty set. (again no typo)


Page 16, Poster: Zamfir

Let's say logx(y) means the base x logarithm with the argument y(I wish HC had an equation editor).

We have a, b and c natural numbers, a,b,c>0.
If loga(b),logb(c) and logc(a) are natural numbers, find a, b and c.


Page 16, Poster: Corribus

A race car is traveling around a square track, with each side measuring exactly 1 mile.  Along the first two legs, the car travels at 60 miles per hour, then along the third leg it travels at 30 miles per hour.  In order to average 60 miles per hour around the entire route, how fast must the driver drive along the final leg of the course?

Page 16, Poster: Zamfir

Solve the ecuation:  2^(x^2+x) + log2 (x)=2^(x+1)  (log2 = base 2 logarithm).

Page 17, Poster: Rubycus

Give a proof for this equalation: a^0 = 1

Page 18, Poster: Corribus

John takes a wooden cube of a certain size and paints it blue.  Then, using a saw, he cuts the cube into an exact number of smaller cubes all measuring exactly 1 cm in dimension.  He then sorts the small cubes by the number of faces that are painted blue.  He finds that there are precisely twice the number of cubes with no blue faces as there are cubes with 1 blue face.  What was the size of the original cube?  

What would have been the minimum size of the cube if the total number of unpainted cubes exceeded the number of painted cubes (any number of sides)?


Page 18, Poster: Asheera

You have some 'jewels' that are used to upgrade your 'items' to higher power levels. For example, you can have your item to +3 level or +4, etc. 6 is the maximum bonus so we're aiming for that.

However, here's the catch: when you use 1 jewel, there's a 50% chance the power level of the item will increase by 1 point, a 10% chance that it will not do anything and a 40% chance that it will downgrade by 1 point (can't go below 0 though, so when you have your item at +0 the 40% case is like the 10% one)

10 of these jewels cost 29 dollars. Now, there's another type of jewel, which increases the power level of the item by 1 with 100% chance. My question is: how much should these second jewels cost (dollars) so that 6 of them cost the same as how many of the weaker ones you need to buy to get an item to +6 (on average obviously, since there are chances involved)


Page 20, Poster: dimis

Problem: You are given a complete graph with N vertices*. You start from one vertex, and every time you roll a (N-1) die to decide which of the (N-1) neighboring vertices you are visiting next. What is the expected number of steps to visit all vertices at least once?

Page 20, Poster: DagothGares

Okay, I have six groups:
Group A, group B, group C, group D, group E and Group F ( can be renamed groups 1-6, if you like)

They're volleyball teams. Now, I'll have to make them play so that each of them has played against another team without any repeating (so if graoup a played against group b, I don't want to see that combination again). All teams have to play during one turn and I want to do this four turns. I want to know whether it's possible or not and I would like an explanation.


Page 21, Poster: FriendofGunnar

A magician calls up an audience member to the stage. "Pick 5 cards out of this standard deck, randomly if you want.  Then take them to my assistant."  After the audience member takes the cards to the assistant, the assistant picks one and tells the audience member to put it in his pocket.  Then the assistant arranges the other 4 cards and tells the audience member to bring the 4 cards back to the magician, place them face down on the table in front of the magician, and one-by-one, pick them up and announce them to the magician.  After this the magician announces the card that is in the audience members pocket, guarenteed.
How does he do it?


Page 22, Poster: FriendofGunnar

An orc slavemaster boards a rowing ship with 8 long chains in his hand.  On the ship there's 8 gnomish rowers divided into 4 port (left) rowers and 4 starboard (right)rowers. The slavemaster holds out his hand and instructs each rower to take one end of a chain in each hand.  The rowers are allowed to grab their ends from their side only, they aren't allowed to reach over to the other side and grab an end.  The slavemaster announces that if they form one and only one circle, he will let them all go.  Of course he is playing with them, he has no intention of letting anybody go. Still though, the gnomes get excited because they think their chances at freedom are actually quite good.

So the question is: What are the chances that the gnomes will form one circle?


Page 23, Poster: ihor

Let's say there are 100 wise men in the prison. One day they all have to be executed. The executor creates a column of wise men and gives a hat of red, green or blue color to each man. The column is created in such a way, so each wise man could see a color of hat of all the persons standing in front of him, but he couldn't see the color of his hat and hats of the persons behind him. Then the executor begins to ask each wise man (begins from the end of column) the color of his hat. If the answer is correct then wise man will be free, else he will be executed.
The wise men could agree before execution their actions to maximize the number of survivors.
What is this maximum number, that will survive for sure?


Page 23, Poster: Rarensu

There are three boxes, one containing nails, one containing screws, and one containing both nails & screws. But the labels have been switched up; they are all labeled incorrectly!

Can you determine which box is which by pulling a single item out of a single box? No looking inside or shaking or weighing the boxes. You just observe a single item to see if it is a nail or a screw.


Page 23, Poster: Rarensu

You're on a game-show. There are three doors. Behind one of them is the grand prize. The other two are empty. You select a door at random. The show host then opens one of the doors that you didn't select and shows you that it is empty (he always chooses an empty door, because he has insider knowledge). Now you have a choice - you can change your selection to the other door that's still closed.

Your friends are all telling you that it doesn't matter, the probability of each door having the grand prize is the same. Are they wrong? Are the probabilities different for the two remaining doors? More importantly - is it a good idea for you to switch?


Page 24, Poster: Corribus

I need to find a general solution to the determinant of an n x n matrix (call it M) with elements M(ij) defined as:

M(ij) = x, if i = j
M(ij) = a, if |i-j| = 1 and i+j = 3, 7, 11, ....
M(ij) = b, if |i-j| = 1 and i+j = 5, 9, 13, ....
M(ij) = 0 otherwise.

Express the (general) determinant as a function of x, a, b and n.


Page 25, Poster: Gnollking

3 men go to a restaurant and each one of them orders a pizza.
The men pay 6 euros for their pizzas, (so it's 3x6 = 18 euros)
The waiter takes the money and counts it. There are 5 euros too much. The real cost of the pizzas would have been 13 euros.
The waiter doesn't know how to give the 5 euros to 3 men, so he gives each man 1 euro, and keeps 2 euros for himself.

So, each man paid 5 euros (not 6, because they each got one euro back), and the waiter has 2 euros.

5€ x 3 = 15€
The waiter has 2€
2€ + 15€ = 17€

At the beginning each man paid 6€, which is (3x6 =) 18€
At the end there are only 17€
Where did the 1 euro disappear?


Page 26, Poster: dimis

(0,1) =_c [0,1]

Back with a difficult problem this time.
Show that (0,1) has the same cardinality as [0,1].
Initially it might seem easier to show that (0,1) has the same cardinality as (0,1].


Page 32, Poster: Mytical

Trying to figure out how to get the volume area of a glass that has a base that has a diameter of 6", is 12" tall, and has a top diameter of 8".

Page 34, Poster: dimis

It's easy to show that the sum of the five acute angles of a regular star (like the ones in the American flag or the one in the Soviet flag) is 180ΒΊ (180 degrees). Prove that the sum of the five angles of an irregular star is also 180ΒΊ (180 degrees).

Page 34, Poster: dimis

Using each of the numbers 1, 2, 3, and 4 twice, I succeeded in writing out an eight-digit number, in which there is one digit between the ones, two digits between the twos, three digits between the threes, and four digits between the fours. What was the number ? (A. Savin)

Page 35, Poster: dimis

Q3. Now I'm four times older than my sister was when she was half as young as I was. In 15 years our combined age will be 100. How old are we now ?

Page 35, Poster: dimis

Q4. Thirty people took part in a shooting match. The first participant scored 80 points, the second scored 60 points, the third scored the arithmetic mean of the number of points scored by the first two, and each subsequent competitor scored the arithmetic mean of the number of points scored by all the previous ones. How many points did the last competitor score ? (N. Antonovich)

Page 35, Poster: dimis

Q5. The product of a billion natural numbers is equal to a billion. What's the greatest value the sum of these numbers can have ? (G. Galperin)

Page 36, Poster: dimis

Q6. Three girls went out in white, green, and blue dresses. Their shoes were also one of these three colors. (Each wore a matched pair of shoes!) Only Anna had the same color dress as shoes. Neither Betty's dress nor her shoes were white, and Katie's shoes were green. What was the color of each girl's dress ?

Page 37, Poster: ihor

Q7. There is an island with 100 inhabitants. Some of them are knights, who always tell truth and the others are liars, who always lie. Each of the inhabitants worship exactly one of three Gods (Sun, Moon, Earth). You asked three questions to each inhabitant:
1)Do you worship God of Sun?
2)Do you worship God of Moon?
3)Do you worship God of Earth?

And you got 60 "yes" on the 1) question, 40 "yes" on the 2) and 30 on the 3).
How many liars are there on this island?


Page 37, Poster: dimis

Q8. There are 12 persons in a room. Some of them always tell the truth, the others always lie. One of them said, "None of us is honest"; another said, "There is not more than one honest person here"; a third said, "There are not more than two honest persons here"; and so on, until the twelfth said, "There are not more than eleven honest persons here". How many honest persons are there in the room ? (D. Fomin)

Page 38, Poster: dimis

Q9. I've thought of a three-digit number such that each of the numbers 543, 142, and 562 coincides with it in exactly one decimal location. Guess what this number is. (V. Proizvolov)

Page 38, Poster: dimis

Q10. Two numbers are called mirror numbers if one is obtained from the other by reversing the order of digits - for example, 123 and 321. Find two mirror numbers whose product is 92,565. (A. Vasin, V. Dubrovsky)

Page 38, Poster: dimis

Q11. At a party, each boy danced with three girls, and each girl danced with three boys. Prove that the number of boys at the party was equal to the number of girls. (V. Proizvolov)

Page 38, Poster: dimis

Q12. Winnie-the-Pooh and Piglet went to visit each other. They started at the same time and walked along the same road. But since Winnie-the-Pooh was absorbed in composing a new "hum" and Piglet was trying to count up all the birds overhead, they didn't notice one another when they met. A minute after the meeting Winnie-the-Pooh was at Piglet's, and four minutes after the meeting Piglet was at Winnie-the-Pooh's. How long had each of them walked?

Page 39, Poster: dimis

Q13. Solve the number rebus
USA + USSR = PEACE .
(The same letters stand for the same digits, different letters denote different digits.) (B. Kruglikov)


Page 39, Poster: FriendOfGunnar

Here is an interesting problem that comes from the risk game.  Let's say that somebody is trying to break your continent bonus and they need to conquer two territories to do it.  For example you have south america and you control both Sao Paulo and Dakar.  If you have 12 armies what is the best way to distribute them between the two territories to form the best defense?

Page 39, Poster: Corribus

Q14. Let a pair of numbers (x,y) be called square if their sum (x + y) and product (xy) are both perfect squares.  For instance, 5 and 20 are square because 20+5 = 25 (5^2) and 20*5 = 100 (10^2).  Prove that no pair can be square if one of the members is equal to 3.

Page 40, Poster: Corribus

Q15.  100 Prisoners and Light Bulb.

There are 100 prisoners in solitary cells. There's a central living room with one light bulb; this bulb is initially off. No prisoner can see the light bulb from his or her own cell. Every day, the warden picks a prisoner equally at random, and that prisoner visits the living room. While there, the prisoner can toggle the bulb if he or she wishes. Also, the prisoner has the option of asserting that all 100 prisoners have been to the living room by now. If this assertion is false, all 100 prisoners are shot. However, if it is indeed true, all prisoners are set free and inducted into MENSA, since the world could always use more smart people. Thus, the assertion should only be made if the prisoner is 100% certain of its validity. The prisoners are allowed to get together one night in the courtyard, to discuss a plan, but once visits begin, they may not communicate. What plan should they agree on, so that eventually, someone will make a correct assertion?

Just to make this more interesting for the math whizzes: using your strategy, what is the probability that the prisoners will have been picked within 2 years?


Page 40, Poster: dimis

Problem: Given N coupons that are drawn with replacement, each coupon having the same probability of being selected, we want to find out:
(a) When are we expected to draw all coupons.
(b) What is the probability after k trials to miss at least one coupon.


Page 41, Poster: dimis

Q16. A number of bacteria are placed in a glass. One second later each bacterium divides in two (of equal size as the initial), the next second each of the resulting bacteria divides in two again, etc. After one minute the glass is full. When was the glass half-full ?

Page 41, Poster: Corribus

What is the next number in the sequence:

1, 1, 2, 9, 96, 2500


Page 42, Poster: dimis

Q18. The son of a professor's father is talking to the father of the professor's son, and the professor does not take part in the conversation. Is this possible ?

Page 42, Poster: ihor

Q19. You have a sequence {a} with following properties:
a_1 = 1
a_2 = 2
a_3 = 3

For every positive integer n:
a_n+3 = a_n + a_n+1 - a_n+2

What is the value of a_2010 ?


Page 42, Poster: dimis

Q20. Comparing Numbers

Which number is greater: 2^{300} or 3^{200} ?


Page 42, Poster: Corribus

Q21.  A friend of yours visits you and asks if you would like to play a card game.  The card game works as follows.  He deals four cards from a standard deck of cards onto a table.  If the four cards are the same suit, he pays you one dollar.  If the four cards are one of each suit, you pay him one dollar.  If the four cards are any other arrangement of suits, you both keep your money.  Should you play?

Page 42, Poster: FriendofGunnar

Very long problem, see text, last post, page 42.

Pag3 43, Poster: FriendOfGunnar

Q23. You have a ruler and compass and are given a circle.  Draw a square that is the same area as the circle.

Page 45, Poster: Binabik

Stacking firewood puzzle - very long, see text.

Page 46, Poster: ihor

You have square with edge length equal to 1. There are 4 bugs in all vertices of the square (bug 1, bug 2, bug 3, bug 4). Neighbour bugs are of different sex and it is said 1 loves 2, 2 loves 3, 3 loves 4 and 4 loves 1.
At the same time and with the same constant speed they all start moving in the direction to the bug each love. Obviously they'll meet in the center of the square. What is the distance each of them will pass before the meeting?


Page 47, Poster: Corribus

Darts and target puzzle - long with diagram, see text.

Page 47, Poster: dimis

Balloons puzzle - long, see text.

Page 48, Poster: ihor

1. Peter has several sticks. Some of them are 5cm long, other - 6cm long. Total length of all sticks is 6 meters. Prove that it is possible to form a regular decagon from all his sticks.
Note 1meter = 100cm

2. Given function f(x) = lg[x] + lg{x}. For particular real number a, f(a)=2. Prove that f(a^2)>4.
Note lg(x) - is common logarithm (logarithm to base 10). [x] - is floor function of the number, {x} - is a fractional part of the number {x} = x - [x].

3. There is a table 11x11 with positive numbers in cells. It is known that product of all numbers in every row is equal to 1, the product of all numbers in every column is also equal to 1 and the product of all numbers in every 3x3 square is equal to 2. Find the product of first and second cells in third row.

4. Given a triangle ABC and points C1 and B1 selected on sides AB and AC correspondingly. BB1 and CC1 intersect at O. Point D is selected, so AB1DC1 is parallelogram. Prove that if D is inside the triangle ABC, then area of quadrilateral AB1OC1 is less than area of triangle BOC.

5. Sequence (x_n), n=0..infinity satisfies following conditions:
x_0 = 1
x_1 = 6
x_n+1 = x_n + sin(x_n), if x_n > x_n-1
x_n+1 = x_n + cos(x_n), if x_n <= x_n-1
Prove that x_n < 100 for all nutural numbers n.


Page 48, Poster: FriendOfGunnar

A) There's a pair of rings drawn on a wall that divide it into three sections.  A center "bullseye" worth nine points, an outer ring worth 4 points, and a vast "out of bounds" area worth zero points.  You have an infinite number of darts.  What's the highest score that is impossible to get?

Page 48, Poster: Corribus

In two decks of cards, what is the least amount of cards you must take to be guaranteed at least one four-of-a-kind?

Page 49, Poster: ihor

You are playing a lottery, so you can choose one of two boxes. You also know for sure, that the amount of money in one of the boxes is exactly twice more than in another. After you choose one of the boxes, you are offered to change your choice and take second box or to not to do that and select first one.
Question: should you change your mind or not and why?


Page 50, Poster: Corribus

In front of you are three buckets, labeled A, B and C. In bucket A are 3 balls which are identical except for the fact that they are labeled by consecutive numbers: 1, 2 and 3.  The other two buckets are empty.  The goal of the game is to get all of the balls into bucket C.

The game has three rules:

1. You can only move one ball at a time.
2. You can only move a ball from a bucket if it is the highest numbered ball in that bucket.
3. You can only move a ball to a bucket if its number is higher than every other ball in that bucket. [Note: this implies you can always move a ball to an empty bucket.]

Question: What is the minimum number of moves you can use to move all the balls from bucket A to bucket C?
Bonus problem: What is the minimum number of moves you can use to move n balls, numbered consecutively 1 to n?


Page 50, Poster: Corribus

Consider a thoroughly shuffled standard deck of 52 cards (no jokers).  You turn one card over and observe it.

Here are the rules of the game:

1. You pick one card at a time.
2. If the card you pick is higher than or equal to the previous card, you get to pick another card.
3. If the card you pick is lower than the previous card, you lose the game.
4. If you successfully turn over 4 cards consecutively (including the first one!) you win the game.
5. Aces are high (though, I don't think it should matter).  All suits are considered equivalent.

What is the probability of winning the game?
Does it hurt or hinder your chances if you draw from an infinitely large pile of cards?
For a bigger challenge, what is the probability of successfully turning over n cards in an infinitely large deck?


Page 51, Poster: Corribus

A spider and an ant are on opposite corners of a regular cube. The ant cannot move and the spider is hungry.  If the spider is able to move only along an edge of the cube, and he choses his direction randomly, and it takes 1 minute to move from one corner to an adjacent corner, what is the expected (i.e., mean) amount of time it takes the spider to reach his meal?

What if the ant is moving as well, at the same speed, also in a random direction?  (Actually, I don't know the answer to this part - I made it up.)


Page 52, Poster: Corribus

Two darts are thrown at a large circular target of diameter = 1.  Supposing they strike the target at random places, what is the expected distance separating the darts?
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ihor
ihor


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Accidental Hero
posted April 24, 2011 10:58 AM

This work deserves a QP .

Regarding Darts, I investigated a bit. Using polar coordinates we may take independent variables.
R1, R2 - uniformly distributed on [0; 0.5]
a1, a2 - uniformly distributed on [0; 2Pi]
And by law of cosines we may build a random variable

D=sqrt(R1*R1 + R2*R2 - 2*R1*R2*cos(a1-a2))

And now the answer will be mean value of random variable D. I don't know now how to find it, but only by generating a sample and approximating the result .
If I'm not mistaken, it will be approximately 0.3624.

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baklava
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posted April 24, 2011 01:18 PM

Oh crap Corey's gone QP hunting.

They just had to make Doomforge admirable, didn't they. Well I hope you mod basterds are happy with what you've done.

God help us all.
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Warmonger
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posted April 24, 2011 01:42 PM

Quote:
Page 11, Poster: Ecoris

Is it possible to cover a 6x6 board with 9 pieces of the following shape:

X
XXX

(the Tetris "L")

You may rotate and flip the pieces.

Let's paint the board like a chessboard:

xoxoxo
oxoxox
xoxoxo
oxoxox
xoxoxo
oxoxox


Now, no matter how hard you try, every piece will cover exactly two field of each kind:

o
xox

or

x
oxo

As there is equal number of both field on a board, it seems possible so far.
But notice that each of these pieces has a diagonal of two fields of same size. Now separate chessboard into 2x2 squares. To cover each square, exactly one brick is needed. You can't try to cover such suare with two different bricks since boundaries would remain empty in such case.
You can't, at same time, cover nine squares as each piece placed in one intersects with another square. Since the number of squares is uneven, there is no match.

Uhh, it turned out more complicated than I though and there's still not so clear proof... I didn't do any maths since first year at university

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Corribus
Corribus

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posted April 24, 2011 04:51 PM

Quote:
Oh crap Corey's gone QP hunting.

They just had to make Doomforge admirable, didn't they. Well I hope you mod basterds are happy with what you've done.

God help us all.

Lol.  Desperate situations require desperate measures I guess. Either that or I did this before Doomforge was made admirable.

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mamgaeater
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posted April 24, 2011 06:06 PM

Is there a way to define a two dimensional regular polygon as a function of it's radius in terms of it's terminal angle?

like a circle has a constant radius of r so it's function would be a horizontal line.


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baklava
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posted April 24, 2011 06:50 PM

Quote:
Either that or I did this before Doomforge was made admirable.

Whoa that's some heavy quantum relativistic Schrodinger shyte.
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Ecoris
Ecoris


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posted April 28, 2011 10:13 AM

Quote:
Is there a way to define a two dimensional regular polygon as a function of it's radius in terms of it's terminal angle?

like a circle has a constant radius of r so it's function would be a horizontal line.




Yes, of course. Such a function would be continuous, but not differentiable. You can determine the function explicitly using the sine rule.

@ Warmonger
I don't understand your argument after you have divided the 6x6 board into nine 2x2 squares.

@ ihor
The position of a given dart is uniformly distributed on the circular target. Your presentation (uniformly distributed polar coordinates) does not result in this distribution; the probability to hit some subset of the target should only depend on the area of that subset. But in your presentation the probability increases the closer we are to the centre.
The angles R1 and R2 are uniformly distributed on [0,2*pi], but the radii a1 and a2 are not uniformly distributed on [0,0.5].
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ihor
ihor


Supreme Hero
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posted April 28, 2011 10:24 AM

Yes, thats a valid point Ecoris.
With this calculations become even more deterrent.

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maretti
maretti


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posted April 24, 2012 05:58 PM

What is infinity minus 1?
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dimis
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posted April 24, 2012 07:00 PM

infinity
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maretti
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posted April 25, 2012 12:52 AM

Thanks. Nice to know where to ask if I have that kind of questions.
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dimis
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posted April 25, 2012 01:18 AM

You are always welcome
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dimis
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posted September 27, 2012 12:16 AM

The Limits of Religion (subs included)

Silas
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dimis
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posted January 08, 2013 05:45 PM
Edited by dimis at 17:56, 08 Jan 2013.

On the 6x6 Board Covered by Tetris-L's

It turns out that we could prove something stronger on the problem posed by Ecoris in page 11 which was about covering a 6x6 board with tetris-L's.

So far we have proved that we can not cover ((2(2n+1))x(2(2n+1))) boards. The alternate proof that someone pointed me to is a simple extension of the one that we have, but actually proves that the only way that we can cover a rectangular board with tetris-L's can be done if the board has 2*4*n squares (and hence a 6x6 board can not be covered).

Let a, b be the dimensions of the board. The problem is not trivial if a*b is a multiple of 4 and we are given precisely a*b/4 tetris-L's. (Otherwise, the board can not be covered by tetris-L's trivially.) Since a*b is a multiple of 4, then either a or b is even. Say a is even. Now, we apply Ecoris solution by coloring a/2 rows (or columns) white and the rest a/2 rows (or columns) black interchangeably. Again we use the same observation that every tetris-L in the covering must cover three squares of one color and one square of the other.

If m tetris-L's cover three black squares and n tetris-L's cover three white squares, then 3m + n = a*b/2 = 3n + m. Hence m = n. It follows that a*b/2 = 3m+n = 4m and hence m is equal to a*b/8. (Same argument is true for n.) This further implies that the covering must use 4m tetris-L's => an even number of tetris-L's. Moreover, 8 divides a*b. Except for the 1x8n rectangle, every rectangle of area 8n can be partitioned into exhaustive, disjoint rectangles of dimensions 2x4 and/or 3x8, but both the 2x4 and 3x8 rectangle can be covered with tetris-L's in an obvious way. Hence, the necessary and sufficient conditions are that a and b be greater than 1 and a*b = 8n.

Ecoris power!
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rogerz84
rogerz84

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posted January 11, 2013 09:17 PM

I don't remember how I got my answer but I'm almost positive its wrong anyways....
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