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I just discovered a serious conceptual but subtle error in my solution. In one of the steps in my approach I've assumed something that's not the case. Quite a learning experience! I will need to spend some time correcting my approach. I will probably come back to admit other errors. I just...

Thank you for the good work. Just one request. Can you please show me the algorithm used in your code so that I can better relate to your code? Thank you for your response.

You wrote above that: "so that the second roll is now treated as the initial roll". This means that you are actually ignoring the very first roll of the ENTIRE game. But you cannot do that as we must look at the TOTAL sum of the ENTIRE game. If the first roll is, say, five and the second roll...

Let's have two 'parallel' but uncommingled arguments going on, if you don't mind. So, please do not mingle this post (or SUB-THREAD) with the one we have been having so far. Let's call this sub-thread the quantyst's approach. Would you please just tell me where I went wrong with my solution...

I am puzzled as to how you are so certain of your attributions onto me when you write: "you [meaning I] assumed that the third roll is Roll1 and the fourth one is Roll2". I've done no such thing. Let's be concrete about this. Suppose the first roll of the entire game is 5 and the second roll...

Dear AlexandreH, Thank you for your response which I read completely. It is apparent, and I will show it to you if only you read and honestly try to understand my response, that you've missed the sequence of events here. First off, let's begin with peterruse's last step when he writes...

Dear Rados Radoicic, Please stick with the subject matter at hand. I would rather finish one thing at a time rather than mix many unrelated things together in one post. If you want to do a thorough analysis of my exchanges with peterruse, I'd be glad to do it later. So, on to your claim that...

Please stick with the subject at hand. Please do not evade the main issue. I have pinpointed EXACTLY where you went wrong with your solution. Please be a gentleman and take a look at what I said regarding your error.

You are funny. Again you are reverberating with your classic fallacy. You hardly read people's posts even when they show where you went wrong.

The underlined above is actually incorrect! You forget that by this time you actually have two different odd rolls, whose sum is even, and it is the parity of the sum that matters here regardless of what the actual number are.

You are committing a classic fallacy. You say that because my method and answer are different from yours, then my answer must be wrong. By your own logic, why shouldn't your answer be wrong precisely because it is different from mine? So, what's the resolution here? How do we go about...

Answer: Probability of Sum being Even is 4/7 and Probability of Sum Being Odd is 3/7. Here's why: Let p denote the probability of the sum being even when the game stops. Upon the first throw, we have Case (O) an odd roll with probability 1/2, and Case (E) an even roll with probability 1/2...

Here's a solution: Obviously for y<0: f(y)=0. For 0<=y<=1: f(y)=exp(y)=e^y. For 1<y: f(y)= 1 + INTEGRAL{f(u) [where u runs from (y-1) to y]}, which upon differentiation, gives f'(y)=f(y)-f(y-1). Now we can partition the interval [1,infinity) as [1,2)U[2,3)U[3,4)U...U[n,n+1)U.... so that for...

Nice problem. I'd like to generalize it as follows: Let X(1), X(2), ... be independent and identically distributed uniform random variables over [0,1]. Let y be any real number. Set N(y)=min{n: X(1)+X(2)+...+X(n)>y}. Find expected value E[N(y)]. As has been shown, E[N(1)]=e. If we let...

Interview (Thirty minutes) 1) Suppose you are given the opportunity to bid for a treasure chest, which you know with 100% confidence to be priced anywhere between $0-$1000. If you bid equal to or above the price, you win the treasure chest (at the cost of your bid). If you bid below the...

You wrote: "I'm guessing that Problem 3's choice b is a typo and that first x should be x^2." Assuming you are correct, then the situation would be worse, not remedied. If the choice (b) of Problem 3 is [(x^2)+2x-1]^2, then it would have two repeated (degenerate) roots, namely, -1+rad(2)...

Look at Problem 2. Tell me what happens to the limiting case of a triangle when all its sides with the same rate approach a length zero. You get a point, just as you'd get the same thing when the radius of a circle approaches zero. I don't see a difference between a circle, triangle, ellipse...

Let's label the statement you made below as follows: S: Some things just can't be proven or reasoned because of our conceptual limitations. Can you prove statement S? Is statement S one of those that can't be proven? Let's assume that you say the following statement, as labeled: T...

Unfortunately some of the questions and answers in the following list are just badly stated or incorrect. Problem 2 What logic? The logic that (i) the sum of the numbers is 1, or the logic that (ii) the product of the number is 0, or the logic that (iii) the absolute value of the...

Words, like people, do not live in isolation. Each has an identity linked with those of others, quite like us humans linked to our relatives, cultures, and history. By knowing the roots of words, the suffixes and prefixes, you can end up learning the meanings and usages of many by just...

I went, I saw and I was terrified! woohoohoo! Except I was not convinced but was left quite unimpressed. One can apply the same binary grid to any contingency and would reach the same "inescapable" conclusion. Without actual probabilities, the grid has only educative value. I am reminded of...

I am with you alain in your position; namely: "I think I need to understand more about the whole Global Warming thing form the physics standpoint... but not from political standpoint but from the scientific standpoint." The "I" above refers as much to you as it does to me. But given the...

alain courteously asks whether "there is a middle ground?" And I'd say: Yes, of course. On a straight line, my position is (or could hypothetically be) as far away from John's as alain's, except on the opposite side. And BTW, do you think Galileo would've done right by creating science that...

Let n denote a positive integer and let f(n) denote the number of consecutive zeros at the right end of n! expressed in base ten. Explicitly express f(n) in terms of n, and find limit of f(n)/n as n approaches infinity.

Let X(0) ~ u[0,1], and for k=1,2,3,... let X(k) ~ u[0,X(k-1)]. Find 1. P{X(n)<a} where 0<a<1. 2. P{X(n-1)-X(n)<d} where 0<d<1. 3. P{X(i)-X(j)<d} where 0<d<1 and i<j. 4. P{S(n)<s} where s>0 and S(n)=SUM{X(k) [k:0 to n]}. 5. P{S(n)-S(m)<s} where s>0 and n>m.

Average Distance From The Center Certain events occur with uniform probability at various points within a sphere of radius R. Find the average distance of the locations of these events from the center of the sphere. The probability that an event will occur within a region of volume V inside...

I have previously alluded as to why I choose in this case (and many other future cases) to not describe the game. I would rather leave it to each interested party to observe the game in its naturally presented setting and subsequently to interpret the observation according to his/her...

Your premise, your first sentence, is incorrect. It is not necessarily the responsibility of this site to preempt the appropriateness of a user's decision to click or to not click on a link, as long as the user is given ample forewarning of the nature of the link. While there is a wide range...

As of this writing, wong's post above has received -4 rating (or thumbs down). So, I gather a net of 4 people think that wong's post was WHAT? Not "helpful"? If I understand what the word "helpful" means, then most posts are not helpful to most of us. So, just about every post should receive...

Thank you for the explanation. Nonetheless, it is important to see the problem in its original (and may I say, natural?) environment, and your approach, therefore, remains too restrictive. Here's an alternative solution: Why not we -- i.e., you -- come up with a symbol or label (like...

Within the past 12 hours I posted in the Quantnet Lounge a serious math/probability question that included a link to a video involving Howard Stern. But that post is excised from these forums. WHY?

Try MIT OpenCourseWare: http://ocw.mit.edu/OcwWeb/web/home/home/index.htm

Analysis is the foundation of virtually all that follows. Nothing can be properly understood without a solid grasp of Analysis. Do it right and build your math career on Analysis. Good Luck!

Here's is hardly any math proof: Let x=.99999999999999...... Claim: For any positive number y: if y<1, then y<=x. Proof: If y!=x, then in the decimal expansion of y, a digit of y to the right of the decimal point is (has to be) less than the corresponding digit of x given that all digits of x...

The cardinal rule is (perhaps) to simplify or transform a (difficult) problem in math to something more manageable. Here's a suggestion: Let's assume x(n) can be expressed as a rational a(n)/b(n). I use functional notation instead of subscript notation for the sequences. Upon substitution...

Nice question! Just to mention the trivial case, for n=1, the answer is "yes", with the number of flips being zero. Assume n>1. Suppose corner (n,n) is +1. Let r(i) denote the number of times row i is flipped before all the entries of row i become +1. Likewise, let c(i) denote the number of...

Every How I Became a Quant is different, and depends greatly on the panelists for its quality and information value. As a member of the audience I participated in one sponsored by the MFE Program of Claremont Graduate University (CGU) in the Southern California region. Although the panelists...

Answers: 1. This is a standard calc question. The anti-derivative of sec(x) is log[sec(x) + tan(x)]. So the answer is (1/2)log(3). 2. The answer is 2/5. The problem is equivalent to the following: Suppose there are two drawers, L and R. In drawer L there are n white socks (for some n) and...

Jelly Beans in the Pocket! There are two pockets, a left pocket and a right pocket. Let (L(0),R(0)) denote the initial numbers of beans in each pocket respectively. Let the sequence (L(i),R(i)), where i=1,2,3,..., denote the number of beans in left and right pockets respectively after a single...

Quote: (Another interview question) You have N cars that are all traveling the same direction on an infinitely long one-lane highway. Unfortunately, they are all going different speeds, and cannot pass each other. Eventually the cars will clump up in one or more traffic jams. In terms of...

N Cars On A One-Way Highway Quote: (Another interview question) You have N cars that are all traveling the same direction on an infinitely long one-lane highway. Unfortunately, they are all going different speeds, and cannot pass each other. Eventually the cars will clump up in one or more...

Find The Number ... Given a natural number N, let P(N) denote a function that is the product of all factors of N. For example, for N=6, the factors are 1,2,3,6, so the product of these factors P(6)=36. Or, P(9)=27. Questions: 1. Is P one-to-one? That is, if M and N are two unequal natural...

Thank you very much! Great and convincing answer! Really well done!

Show me how.

Notice a key term is "guarantee". You are always guaranteed to know the color of the last card. And there is no way of knowing in advance the sequence of colors for the first 51 cards. Now imagine that the cards are shuffled a thousand times and just by pure chance, unbeknownest to you...

Is it not a great one amongst all the salaries?

A Guard Dog's Area of Coverage One end of a guard dog's leash is tied to the corner of a rectangular building of dimensions AxB (where A<B), and the other end is tied to the dog. Find the total area that the guard dog can cover if the length of the leash is L. Express the answer as an explicit...

Instead of cereal boxes, below I consider an n-sided biased die whose side k comes up with probability pk. The question is: on average how many time does one need to roll such a die until all the numbers 1,2,...,n come up? The formula for this problem is monstrously unyielding and will not be...

Maybe I should refrain from saying this, maybe not. Anyway, even if the comic is a true depiction of the society today, one fact -- an obvious one -- is that just about everyone, but except the bum, over his/her life has, on average, a positive net worth. No? Yes? A bum, by its nature, is the...

Roll A Die Till You Get All Six An unfair (biased) die is such that the number k has the probability k/21 to come up, where k is in {1,2,3,4,5,6}. On average how many times does one have to roll the die until all six numbers come up?

Hi Bastian, I think your answer is off the mark. The correct answer (in the case of equal probabilities) is 4*{1+(1/2)+(1/3)+(1/4)}=25/3. See my two solutions (as well as that of Han Z). The original problem did explicitly say the probabilities are equal. I have not done the case in which...

Maybe you noticed, maybe not, that I did 'reject' my own answer by saying "But then, I don't think this answer is correct." My example of data points 1,2,3,4,1000000000000 clearly shows that the first lowest values 1,2,3,4 are all useless. Certainly one should just ignore the lower values as c...

maxrum wrote: Quote: ________________________________________ The mean of 5 randomly picked numbers is: (%5Cmu_1%20=%20%5Cfrac%7B%5Csum_%7Bi=1%7D%5E%7Bi=5%7D%7Bx_i%7D%7D%7B5%7D) The mean of a uniform distribution: (%5Cmu_2%20=%20%5Cfrac%7Bc%7D%7B2%7D) If we are assume that...

An intuitive approach: ON AVERAGE, the five numbers will be equally spaced along the interval (0,c), so if L is the largest of the five numbers, then the best estimate for c is (6/5)*L. But then, I don't think this answer is correct. Why not also take advantage of the smallest of the five...

Toy-Containing Cereal Boxes Generalized The following is inspired by a problem posted by vkaul under the title "hardest question I got in an interview!" in the Education forum. Cereal boxes of a top brand contain toys out of a set of n different types of toys, where n is a known fixed number...

Now that the problem is solved in at least two different ways, I would like to add to the complexity of the problem by changing some aspects of it. Here it goes: Suppose each box contains precisely two toys such that (i) each toy has an equal probability to be in the box regardless of the...

Here's another but more direct solution to the problem. First we'll address a preliminary issue. Suppose a repeatable experiment X results each time in an outcome E with probability p. Then it is easy to show that the expected number of experiments X that need to be done in order to produce...

An n-sided fair die is rolled again and again until all the numbers come up. On average how many times does one need to roll the die until all the numbers come up? Suppose that at a certain point of this process we notice that of the n numbers c have come up and r are still remaining and we...

In an interview environment, just about any question that's sort of unexpected, will look difficult, even if you have confidently done problems of that sort or even more difficult than that sort in the past. This is rather an easy problem (in a sense ...). This problem is akin to rolling a...

Univorm RVs Produced From Other RVs Let [[.]] denote greatest integer value function. Define RV Yn=(X1+X2+...+Xn) - [[X1+X2+...+Xn]] for any n in {1,2,3,...}. Now it is not a trivial task to demonstrate that Y1, Y2, Y3, ... is a sequence of uniform RVs over [0,1] if the X1, X2, X3, ... are...

Parabola's Interesting (& Useful) Property We've all heard of the special property that parabolas have; namely, when a ray in the interior of a parabola and parallel to the parabola's axis of symmetry hits the parabola it then is such reflected that it passes through the focal point of the...

From the get-go, I was under the misconception that the word "clump" meant that when a faster car reaches a slower car, then both come to a stop and no longer continue to move forward. If the cars came to a stop, then of course initial distances do matter. This conception in fact gives rise to...

Try finance-research.net at FINANCE RESEARCH

Does there exist a text of the talk that you can make it available to all by posting it here? Or a video of the talk itself?

The answer stands as given. For the case N=0, X={ }, the empty set, and with probability 1, A=B=X, which agrees with (3/4)^N=(3/4)^0=1. Now let N=1. So, X={x}. Then B={ } with prob 1/2 or B=X with prob 1/2. In case B={ }, then A is subset of B with prob 1/2 as A can only be the empty set...

It is not clear what you mean. Do you mean that the answer to the problem will be the same whether or not we consider the initial distances between the cars? If so, then we need to prove that. Furthermore, do you see what my example for the case N=4 tries to show? It shows that we need to...

Quote: One-lane Highway (Another interview question) You have N cars that are all traveling the same direction on an infinitely long one-lane highway. Unfortunately, they are all going different speeds, and cannot pass each other. Eventually the cars will clump up in one or more traffic jams...

Quote: The Monty Hall Problem Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat (At this...

Quote: Ants 100 ants (zero-length points) walk on a meter stick (a line) at 1 cm/second. When two ants collide, they both reverse direction. If an ant reaches the end of the stick, it falls off. What arrangement of ants maximizes the time before all ants have fallen off? How long can they...

Quote: One more probability puzzle Given a set X with N elements, and sets A and B which are subsets of X, what is the probability of A being a subset of B. Ans: (3/4)^N.

Please tell us more about yourself. Tell us who your organization is, where you are located, what your website is, etc. The more, the merrier. Thanks!

A LA ANDY: Assume you have (\pi) up to a sequence S of one million digits to the right of the decimal point. Write an algorithm to find the largest matching pair of n-digit subsequences of consecutive numbers within the sequence S.

Let's make an initial observation. The knowledge of one million (1M) digits to the right of the decimal point of (\pi) is way more than necessary. The first matching of two 3-digit numbers occurs within the first 3003 digits. Here's why: Let (\pi)=3.x(1)x(2)x(3)x(4)...where x(i) denotes the...

QUOTE: Assume you have (%5Cpi) up to 1M decimal, write an algorithm to find the first matching pair of 3-digit numbers. What is your bigO? Example: if (%5Cpi)= 3.14159525957... then the first pair is 595 A question: Suppose (%5Cpi)= 3.141585857026001410089... What's the answer now...

The Best and Worst Jobs in the U.S. - WSJ.com Remember that kid in elementary school who always had a pencil and calculator nearby, and while the rest of us drew pictures, read comic books or played cards, that kid was happily crunching numbers -- for fun. Fast forward 20 years or so, and it...

Are the hooks movable, transportable? Or are they stuck in their original locations?

We need to prove whether or not the sequence converges. It should help if we can develop a recursive relation that generates the sequence. In situations like this a bivariate sequence is needed. Here is one: Define f(m,n)=RAD{(m-n+1) + f(m,n-1)}. Then f(m,n)=RAD{(m-n+1) + RAD{(m-n+2) +...

Compute RAD{1+RAD{2+RAD{3+RAD{4+RAD{5+....}}}...}.

Fooled by Ponzi (and Madoff) Fooled by Ponzi (and Madoff) <small>How Bernard Madoff Made Off with My Money</small> by Stephen Greenspan There are few areas of functioning where skepticism is...

Do you know what "the question" in the original post (above) refers to?

Your reasoning is incorrect! There is absolutely NO LOGICAL explanation why M has to be a 3-digit number. If your reasoning is valid, then the x and y in the following system of equations must be both 1-digit numbers: x-y=4 3x-4y=2 It is incorrect to say, as you did above, that "If M or N...

Clearly you abuse the method of defining things. For starters, you cannot take an unknown M and define it to be anything you like. The word "define" is used when one symbol substitutes for another. For example, we can take an quantity like (a+b) and call it x, as in: define x=a+b. I make no...

I will assume that the second X must be a Y. As a result of this assumption, the random variable f is now: f(X,Y,Z) = (X+YZ)/sqrt(1+Z^2). Now let a be a real number. We have: P{f(X,Y,Z) < a}=P{((X+YZ)/sqrt(1+Z^2)) < a}. Conditioning on Z, we have: P{f(X,Y,Z) < a}=E[P{((X+YZ)/sqrt(1+Z^2))...

Where is the Y in your function? Congrats on getting the job!! Which company?

Coinucopia! There are two fair coins and one double-headed coin in a jar. A collaborator randomly selects two coins from the jar and performs four rounds of tosses on them and then provides the following report without referencing the results to the coins: {T,H}, {H,H}, {H,T}, {T,H}. Now, of...

Write a bijective bivariate function f: A --> B, where A=[-1,+1]X[-1,+1] and B={(x,y): (x^2)+(y^2)<=1}. A bijective function is onto and one-to-one. That is, a bijective function is a one-to-one correspondence.

No. You committed serious errors from the getgo! You claimed that the equality holds for any real x such that for any natural number n, sin((2^n)*x) is NOT zero and cos((2^n)*x) is NOT zero. There are infinitely many such x, according to you, as solutions. We now know there is NO solution...

emil, Your approach, although worthwhile, has been wrong all along! It is still wrong! But you continue to defend the indefensible! You wrote: "I didn't know the exact solution from the beginning,..." So, why did you write "q.e.d." in your first post? Do you know what it means? Do you...

Unfortunately, you try hard to defend that which is not defensible. The upside down A means "for any", which is a universal quantifier. It does NOT mean "for any n and a certain k". The part "a certain k" means "there exists a certain k", which points to an existential quantifier. And...

For k=1, n=1, you get the interval ((pi), 3(pi)/2) - {3(pi)/4} according to you. For any x in this interval, sin(x)<0, which, as you know, cannot be a solution. Earlier I asked: "Can you come up with a single real x for which sin((2^n)*x)>0 for all natural n?" Instead you have given me lots...

What does it mean to say sin((2^n)*x)>0 for all natural n? So, for which real values of x does the original equality hold? Can you come up with a single real x for which sin((2^n)*x)>0 for all natural n? (BTW, most of the "if and only if" relations (i.e., <=>) in your post above must be...

Suppose "L(x)=R(X)" denotes an equation in x. Let's simplify it to "L=R". Is "L=R" equivalent to "(L^2)=(R^2)"? To see this better, does the RADICALIZED equation above (involving sin(x), etc.) hold for x=-(pi)/3?

Radically Radicalized! Let h=(1/2), RAD(z)=z^(1/2) and f(n)=2^n for each natural number n. For which real values of x does the following equality hold? sin(x)=RAD{h-h*cot(f(1)*x)*(RAD{h-h*cot(f(2)*x)*RAD{h-h*cot(f(3)*x)*RAD{h-h*cot(f(4)*x)*RAD{...

The question is well stated. If you think the system has no solution, then say so and prove it.

Incorrect. The correct answer is min H=(3/5)L. When I did this problem, it took me two pages with explanation. There is MUCH more than meets the eye!

This problem is so elementary that it requires no further analysis or commentary ... if it weren't for your mutilations committed upon so simple a problem and, just as equally, your comment "i dont think your equation works because you assume ..." This problem, in its essence, has nothing to do...

Does A Loop! One end of a (non-stretchable) string of length L is tied to a raised nail in the wall at a point P, and the other end of the string is attached to a point mass M. Directly below the point P, we erect another nail at a point C on the wall. Let H denote the distance between P and C...

Solve XOR Systems of Equations! A refresher first! The (XOR) operation acts on 0 and 1 as follows: 0(XOR)0=0, 0(XOR)1=1, 1(XOR)0=1, 1(XOR)1=0. We also let (XOR) operate on numbers expressed in binary notation on their digits one at a time. This is called bitwise operation. Here are a few...

Please Comment On This Solution This problem is akin to placing n rooks on an nxn chessboard in such a way that no two of them fall within each others' line of attack. There are obviously n! ways of doing this. For each such arrangement of the rooks on the nxn chessboard, we keep track of the...

quantyst's Number contains ALL truth! In base two, construct the following real number 'canonically': q = .0 1, 00 01 10 11, 000 001 010 011 100 101 110 111, 0000 0001 0010 0011 0100 0101 0110 0111 1000 ....etc. The commas serve to keep the digits organized and the progression better...

Cat & Mouse Chase in 1D On the one-dimensional integer line a cat is originally located at 0 and a mouse, originally located at X(0)=a. In its pursuit to catch the mouse, the cat takes a one-unit step to the left or to the right at each move. At each move, the mouse, unaware of the cat, moves...

Coins and Probability Toss a fair coin 16 times. Which of the following events has a greater probability of occurring? (E1) H H H H H H H H H H H H H H H H or (E2) H T T H T H T T H H H T T H T H

m-gon, n-gon, doggone it! Inscribe a circle within a regular n-gon. Inscribe a regular m-gon within the circle. Find the ratio of the area of the regular m-gon to that of the regular n-gon.

Intersection of Circles Let X, Y, Z, A, B, C be iid uniform random variables over [0,1]. Two circles, one with center (X,Y) and radius Z, another with center (A,B) and radius C, are randomly drawn in the xy-coordinate plane. Let W denote the area of the region of intersection of the two...

Dizzying! Starting at (0,0) in the xy-coordinate INTEGER grid, you take n steps, each step being a one-unit left move, right move, up move, or down move. The following are examples of n-step trajectories, where n=12: (T1) ((0,0), (1,0), (2,0), (3,0), (3,-1), (4,-1), (4,-2), (5,-2), (5,-1)...

Cat Catches Mouse! In the xy-coordinate INTEGER grid, a mouse is originally located at position (x(0), y(0)) and a cat at position (0,0). Assume x(0) >= 0 and y(0) >= 0. In its pursuit to catch the mouse, the cat, at every move, takes a one-unit step to the left, right, up or down. At every...

It's all Brown in here! Let W(t) denote the standard Brownian motion. Define T=inf{t>0: W(t)=1 or W(t)=-1}. Compute E[sin(T)].

How High? On a horizontal surface S a frog (to be considered as a point mass) using sufficient but the least initial speed jumps over a log of radius r and lands on surface S on the other side of the log. How high does the frog reach as it jumps over the log?

America the Banana Republic: Politics & Power: vanityfair.com

UnUniform Suppose Z is a uniform random variable over [0,1]. I.e., Z~u[0,1]. Define X(Z) and Y(Z) to be two uniform random variables over [0,Z] and [Z,1], respectively. I.e., X(Z)~u[0,Z] and Y(Z)~u[Z,1]. Let r be a real number. Find the probability P{ Y(Z) - X(Z) < r }.

MaxAcc A negligible mass m is positioned atop the horizontal surface of another mass M, which is resting on a horizontal surface S. The coefficient of static friction between masses m and M is 1, and the coefficient of kinetic friction between M and the surface S is k. We are given that...

Folded Rectangle Take a rectangular piece of paper ABCD. Point E is somewhere on side AB, point F is somewhere on side BC, G is somewhere on side CD, and points H and K are somewhere on side DA. Now we fold the paper along the line segment HF and flatten the folded paper. Suppose that as we...

LOGUATION Find ALL positive integers x and y such that the following equation (involving Log functions BASE 2) holds: (Log(x))/(Log(y)) = Log(x/y). Prove your answer exhausts all solutions!

Candy Factoryal This is classic. It's also basic but important in combinatorial math. A restatement: There are k colors/types of jelly beans and each jar contains exactly n jelly beans. How many different jars can be produced (or are there)? Solution: Let x(i) denote the number of jelly...

Follow Up on Easy Probability! Jill and Jane are two collaborating magicians. Jill pulls out an ordinary deck of playing cards and gives it to a random member of the audience to both examine and thoroughly shuffle it and put it all inside an empty cardboard box. She then tells the audience that...

Easy Probability? A magician pulls out an ordinary deck of playing cards and gives it to a random member of the audience to both examine and thoroughly shuffle it and put it all inside an empty cardboard box. He then tells the audience that he will randomly pull out a card from the box without...

Buffon’s Needle Breaks in Two! A set S of infinitely many parallel lines, each adjacent pair one unit distance apart, is given in the plane. A needle of unit length is dropped onto the plane. But as it lands it breaks into two pieces of random lengths (with uniform probability) and each piece...

Buffon’s Needle Punctures Intuition Consider a set A of infinitely many parallel lines that are consecutively one unit apart. Consider another set B of infinitely many parallel lines that are also consecutively one unit apart, but are all perpendicular to all the lines in set A. That is, each...

f(z)=cra(z)y Let f:R ---> R be defined as follows: f(z) = 4(z^3) + 12(pi) z cos(z) - 12(pi) sin(z) + 3((pi)^2) z. Find the EXACT extreme values of f(z) and the EXACT points at which it attains these extreme values.

Sun Giveth, Earth Receiveth Assume the Sun with radius R and the Earth with radius r are perfect spheres, and further assume that their centers are a distance D apart. If the Sun releases per second a total energy E uniformly in all directions, how much of it is captured by the Earth?

Don't Google It! Find the volume of an n-dimensional hypersphere of radius R.

E[m/M] Let X(1), X(2), ... be a sequence of iid uniform random variables over [0,1]. Let M=M(n)=max{X(1), X(2), ..., X(n)} and m=m(n)= min{X(1), X(2), ..., X(n)}. Compute E[m/M].

Cute Coloring of a Cube An ordinary three-dimensional die consists of six sides (i.e., surfaces), with each side (i.e., surface) having a unique number from the set {1,2,3,4,5,6}. But instead of each side (i.e., surface) being numbered, imagine each side (i.e., surface) had a unique color from...

jonylee wrote: "Anybody has idea how to solve this one: A basket has N balls which are numbered. N is an integer between 1 and 100. You randomly pick a ball and find that it is the 5th ball. What is your best guess of what N is?" Problem Restated: Let K be a positive integer. A number...

Notwithstanding your correction of your answer (changing "x" to "*"), there is still an important detail to attend to: You need to express cos((\pi)/5) in algebraic terms. That is, you've got to express cos((\pi)/5) in terms of radicals and other stuff without trig functions. Now, that's a...

Hi ramnik, What's that x doing in your answer? The correct answer has no mystery in it! It is one definite concrete quantity, without any unknown in it. This problem is more difficult than it seems at first sight.

PENTAGON -----> pentagon Join the diagonals of a regular pentagon to form a smaller pentagon. Each vertex of the smaller pentagon is the intersection of two diagonals of the original pentagon. Find the ratio of the area of the smaller pentagon to that of the original pentagon.

Now, notwithstanding my earlier post, I will give a solution below only because your problem has been around the net long enough that I have seen it before with a specific value for n. There are many distinct solutions to your problem. Below is one, of my own creation. Assume n>3. [Originally...

Your problem is not well stated. It is too vague and engenders a lot of unanswered questions. You need to delineate the components of "protocol for communication" in a systematic way so that the problem solver knows who speaks to whom, how he/she speaks, who hears it, etc. etc...

This problem is what it is BECAUSE of the need to prove that one's answer (however it is discovered or produced) is indeed the MINIMUM given N, k, j. Any other consideration, any strategy, any insight or discovery, as useful as it may be in its own right, does NOT solve the problem without a...

It seems you have misunderstood the problem. You are to determine the j fastest horses, not the j fastest horse.

Please read my solution several times, if need be, until you grasp the situation better.

Prateek Bhatia wrote: " So for N=31 we get x=3 as per the inequality stated above. So we have 3 single cut, and we will have a subchain of (x+1)=4 rings and we will have a subchain of (2x+2)=8 rings and we will have a subchain of (4x+4)=16 rings so we have 1+1+1+4+8+16=31 That is 6 pieces...

WHAT is a chain of rings? WHAT is a chain like? Why don't you get a chain (with two free ends) and horizontally lay it down in front of you and ask yourself what happens if you cut a middle ring. The answer is: you get three separate pieces: the cut ring, the portion to the left of the cut...

Gold Chain Mystery Revealed! Problem: You've got someone working for you for N days. You own a gold chain of N rings to be paid as compensation to the worker. The gold chain consists of N sequentially connected equal rings with two free ends. You must remit the worker one gold ring at the end...

Your answer is incorrect! If you have (k-1) cuts in the chain (to produce the k subchains in addition to the (k-1) single cut rings), why then don't you use these (k-1) single cut rings to give out exactly one single cut ring for every day of the first (k-1) days?

Your answer is incorrect, but just about THERE! Hint: Do you need to include the first term C(n, 0)*F(n, k-1) given that no room is empty?

What you wrote is off the mark. Did you read the problem carefully? What does it mean to say " We race every horse but horse n, and we rank them."? You can have a race of (at most) k horses at a time. What you are proposing is a total disregard for the minimum number of races to determine...

Find The j Speediest Horses! There are n horses. You are to determine the minimum number of races it takes to identify the j fastest horses amongst the n horses. Each horse's speed is constant and different than that of any other horse. Each horse race is comprised of a maximum of k horses...

Unchain the Gold Chain You've got someone working for you for N days. You own a gold chain of N rings to be paid as compensation to the worker. The gold chain consists of N sequentially connected equal rings with two free ends. You must remit the worker one gold ring at the end of every day for...

Circumspectify Circumspecticate Circumspectate Circumspeterize ... It is best to exercise a bit of restraint before pushing the 'submit' button! It's important to get into the habit of checking answers with extreme cases, when n and k (or the variables/parameters) take very small values or...

Let us see if your formula gives the correct answer when the number of rooms, k, is 1. We know when k=1, there is only one way we can place all n mathematicians in just one room. So, we should expect an answer of 1 from a correct formula. Agreed? But what answer does your formula give...

Reply to jayg's "My first post" You have solved a part of the problem, and have done a very good job of solving the Diophantine equation. But the issue is not only where the function f is maximized, but what that maximum value is subject to f<R. How are we supposed to know in advance that...

O.K. I will give a limited answer to a special case of the problem. The special case involves the one in which x and y are allowed to be any integers. The insight we take from this case might be useful in finding a solution for the case where x and y are non-negative integers. Here it goes...

Untrivialized ... Why is it trivial? Because? Because? It's trivial because the code did it. In that sense it is trivial! And so are many other problems. Now, do the same problem analytically, without the advantage of a computing machine (other than the organic one). Now, it becomes fun...

MAXimize! Suppose a and b are two positive rational numbers and let R be any positive real number. For example: (1) a=3/8, b=5/6, R=(37)^(2/5), (2) a=5/12, b=1/10, R=(4067)^(1/3). Define the function f(x,y)=ax+by. Maximize f(x,y) subject to f(x,y)<R where x and y are non-negative...

Deciphered ... awry! Here's a personally silly response, but any one is welcome to take it seriously. Let x denote the eighth number in the sequence. So, we have: 1, 4, 5, 6, 7, 9, 11, x. Now draw a vertical line through the center and subdivide the terms into two groups, one on...

Any number will do! Here's why I say this: For any number (even an irrational number) as the eighth number following the first seven terms of the sequence, I can find a formula that generates the first seven numbers as given, and that generates the eighth number as the new arbitrary number...

Bug-gone-it! quantyst wrote: "2. A bug, beginning from the bottom of a 10 meter hole, crawls up at the rate of one meter per hour, but at the strike of every hour the bug falls back one-tenth of its total displacement from the bottom. How long does it take the bug to get itself to the edge of...

A Bug's Life: It's a Crawl! Two puzzles: the first one you've probably heard a variant of, the second one is a variation on the first. 1. A bug, beginning from the bottom of a 10 meter hole, crawls up at the rate of one meter per hour, but every time the bug climbs two meters, it falls back...

Conicus Challenges On... Right before the embarkation, you begin to worry with a new nagging question: 3. Assuming that Conicus is situated in the xyz-coordinate space such that the summit is at point (0,0,H), the circular base is contained in the xy-plane with center at (0,0,0), and that...

Reach The Summit! Because it's there, you've decided to climb Mt. Conicus, which is a perfect right circular cone of (vertical) height of H meters and (horizontal) base radius of R meters (where H > R). Since you can rise at most 1 vertical meter for each L meters of movement along the surface...

Can Eat Cake If You Can Cut It By Two How do you subdivide a rectangular cake into four equal pieces by two cuts when someone has already sucked out an irregular piece from it? All is 2-D. Is it even possible to do so? To solve this problem, it would be a good start to prove that it is...

You need to be a bit more circumspect. This problem comes on the heels of the one preceding it. As before, it goes without saying, you cannot deform them. Can you squeeze steel balls? This is a tough physics problem having to do with angular moments of inertia. A rather (hopefully) simple...

That Was Easy, How About This? Earlier I posted the two spherical balls puzzle, and true to my expectation, it was quickly and easily solved. Now, here's a new challenge, and as of this writing, I have no solution to it: Two outwardly identical spherical balls are same in every respect...

Identical Twins But Different! Consider two outwardly identical spherical balls made of the same material, same weight, same color, same surface structure, same in every respect but one: the first ball is a solid whose mass is uniformly distributed throughout the ball, the second has a...

Dice Mania Consider the following dice game between two players who take turns to roll a number of dice. First off, there is an inexhaustible supply of identical dice. The two players agree in advance on an integer s. The first player rolls a single die. Let Y(1) denote (the number on) the...

Expectation at Infinity Define an infinite sequence of random variables X(0), X(1), X(2), ... as follows: X(0)=1, and for every i>0: X(i) is uniformly distributed over the interval [0, (X(i-1))^a] where a>-1. Find limit of E[X(i)] as i approaches infinity.

Greater Angle Let m(A, B, C) denote the measure of angle ABC. In the Euclidean plane are given a circle and a square that are disjoint but have equal areas. Three points C1, C2, C3 are randomly chosen in the circle, and three points S1, S2, S3 are randomly chosen in the square. Find the...

Greater Distance Let d(A,B) denote the Euclidean distance between two points A and B. In the Euclidean plane there are given a circle and a square that are disjoint but have equal areas. Two points C1 and C2 are randomly chosen in the circle, and two points S1 and S2 are randomly chosen...

Chuck Norris has higher ranking than Andy.

Let's Shoot Some Targets! In the xy-coordinate plane, on the horizontal line y=1 there are infinitely many targets of length L(i), each with endpoints (i, 1) and (i+L(i), 1), where i runs through all integers, positive, negative, and zero. From the position (0,0), a bullet is fired at an...

Chuck Norris can differentiate the path of a Brownian motion EVERYWHERE!

The Disappearance of itative ... NICE! It goes without saying that Chuck Norris alone is the destroyer of itatives big and small! (P.S. Chuck Norris made me do it!)

Plato: For the greater good. Karl Marx: It was a historical inevitability. Machiavelli: So that its subjects will view it with admiration, as a chicken which has the daring and courage to boldly cross the road, but...

Do You Double It? When? From any integer position on the number line, you take a step of integer length k in either direction with probability p(k) where 1<=k<=m, and where m is a positive integer. Let L denote a positive integer greater than m. Starting at position zero, if you reach or...

Thanks!

Thanks!

Partitioned Mathematicians Solution: This solution is an incomplete one. I will not provide a closed-form formula in terms of n and k. Instead I will come up with a recursive relation. The closed-form formula will be presented in another post. Let f(n,k) denote the number of...

Answer: 1-(1-x%)^(d2/d1) 1. Chance of dying in a distance of d1 is x%. 2. Chance of not dying in a distance of d1 is 1-x%. 3. Chance of not dying in per unit of distance is (1-x%)^(1/d1).This is so because to not die along the entire distance of d1 is to not die (independently) along every...

Hi Isaac, You are welcome. Thank you for your interest in maths.

Hi ramnik, Crisp and clean! Well done. Thanks.

May The Calc Be With You! There is a road in the first quadrant of the xy-coordinate plane whose graph is given by xy=1. Beginning from a point S(s,1/s) on this road, where s is extremely small but positive, your objective is to drive your car to the origin E(0,0). You have the option of...

It helps a lot if you take a look at other posters' attempts to solve this problem and the discussion surrounding their attempts. Before submitting a response, take the time to check your soution for special cases, usually extreme values, or endpoint values, or small-numbered and very...

There is a theorem known as Fermat's Theorem (which is a special case of Euler's Theorem), which goes like this: If p is prime and p is not a factor of positive integer A, then A^(p-1) is congruent to 1 modulus p. That is, {[A^(p-1)] - 1} % p == 0. Assume p is different from 2 and 5, as...

Hi IlyaKEightSix, No mathematician is left out. Each and every one of them will be in some room, but all at the same time. Please take a moment and read the problem one more time. As I said earlier, this problem involves -- to the best of my understanding -- the Inclusion/Exclusion...

Hi IlyaKEightSix, It seems you've misunderstood the problem. Your first paragraph above reveals your misunderstanding. If there is only one room, then there is only one way in which all n mathematicians can be placed in the set of rooms.

Prateek Bhatia wrote: ____________________________________ Assuming that each mathematician and room are distinct, then there are nPk ways to arrange the first k mathematicians. Then there are (n-k) mathematicians left. We can arrange them in k rooms in (n-k)Pk ways. Then there are (n-2k)...

Hi IlyaKEightSix, First let me answer your last question above: Quite to the contrary! You certainly are or can be a quant material! You probably have a better chance of getting a legit quant job than I. To be a successful quant, you need to have at your disposal a sufficient amount of...

Solution: It's easier to do this problem by generalizing it a bit. We are given an n-sided fair die, where n is, of course, a natural number. Now let k be an integer in the set {0, 1, 2, ..., n}. Let us generalize by saying that the game ends on the first roll if the rolled number is...

Good question! And fairly difficult to answer given the vagueness of the word "relevant" in your question. Certainly, this forum provides the participants an opportunity to bang heads with like-minded people, i.e., other quants or wannabes. In this way, it is helpful. Quite a few of the...

Did you notice Prateek's later post indicating his answer is wrong? In fact, it is. Here's the correct answer: (1+1/n)^n. So as n ---> infinity, the answer becomes e. Cheers!

Hi IlyaKEightSix, I think you are over-counting! We know that when k=1, the answer to the question is just 1. You have one room to be filled with n mathematicians. That can be done in only one way. But your answer, namely "nPk*(n-k)^k", gives the result n(n-1), which is obviously incorrect...

Partition These Mathematicians! There are n mathematicians numbered 1 through n, who want to be placed in k rooms numbered 1 through k, such that in each room, there will be at least one mathematician. In how many ways can you do this?

Hi Prateek Bhatia, I am having difficulty understanding a phrase you used. What do you mean by the following phrase, as you used it just recently: " Hence expected value of making it in two turns = 2*(n+1)/2n"? What does " ... expected value ... in two turns ... " mean? Thanks.

Die...Oh...Rama! Keep rolling an n-sided fair die until you get a number that is less than or equal to a previously rolled number, at which point in time the game ends. On average, how many times do you roll before the game ends? What happens as n gets unboundedly large?

This Flea Hops and Hops! Consider a square ABCD along with its two diagonals intersecting in point E. A flea hops from any one of the points A, B, C, D, E to an adjacent point. For example, from point A, The flea hops to D or B or E, but not to C. Assume that the flea is originally...

Hi olepep, Your counting method is incorrect as it blurs the distinction between the points. It is more accurate to allow for permutation of the points by considering the points as distinct. Here's what I mean: let P denote the first point and Q the second. Let P-1 denote the placement of...

The Ways Of Coins On A Checkerboard In how many ways can the following be done? To place k identical coins on an nxn checkerboard -- consisting of n^2 identical unit squares -- subject to the following conditions: 1. Every coin must be placed within a unit square. 2. Each unit square can...

Have an Integral Break! Integrate (f(x)) from (\\\pi/2) to infinity where (f(x)=\frac{sin x + cos x}{e^x + cos x})

Di vi si bi li ty Tests! Does there exist a divisibility test for every prime number? In this forum, a while back there was a question on divisibility tests. The divisibility test for number 11 is rather very simple. Let sigma{(10^k)*a(k) [as k runs from 0 to (n-1)]} denote an n-digit...

Answer: 1-(1-x%)^(d2/d1) Ask me how I did it, and I will explain.

Solution to DUO ON A SQUARE - Prob 1 Prob 1. Two points are randomly selected on the sides of a unit square. Find the probability that the distance between them is less than unity. Consider a unit square in the first quadrant of xy-coordinate plane, two of whose sides coincide with the x and...

The Hands Have An Angle On The Face Letting t represent the time in hours since the most recent 12 midnight, express A(t) as the minimal positive angle in radians between the hour-hand and minute-hand of a regular analog clock. In particular, find A(.25), A(1.08), A(6), A(33), A(100), A(-.25)...

Hi laguy, You wrote: 1. It's currently 5:15 - at what time (to a few decimal places of a minute) will the the minute hand meet the hour hand? Let HH denote the position of the hour-hand between 5 and 6 o'clock. So, HH=5+x, where 0<x<1. Let MH denote the position of the...

I thought it was ..., well, fascinating to see the critical and unhelpful responses to Dream Chazer's post in the careers section, dated 06-10-2008, 10:52 AM. It's fair to say most every responder did understand what he (all singular third person pronouns refer to both sexes equally) said in...

Re: Not So Obvious! dstefan is thinking the right way. The task is to find Q = arcsin[sin(360)]. The title of the post is a hint. So, of course 360, as the argument of the sin, is to be understood as a real number, i.e., it is in radians. By arcsin I mean the "sine inverse" function, which...

Not So Obvious! Find the EXACT value of arcsin[sin(360)].

A DUO ON A SQUARE Prob 1. Two points are randomly selected on the sides of a unit square. Find the probability that the distance between them is less than unity. Prob 2. Let d be a number between 1 and rad 2. Two points are randomly selected on the sides of a unit square. Find the...

Restricted Taxicab Paths Consider an nxn regular square chessboard (consisting of n^2 unit squares). You are to move from the lower-left endpoint to the upper-right endpoint of the board. Every step taken is defined to be a move along the side of a unit square and it must be either a rightward...