It was nice that the volume equation canceled out the square root.
Yes, that's what I had in mind (disc integration). It's interesting because if you tried to find the area of the 2D shape (not the volume of the 3D egg formed as a surface of revolution), you would get an elliptic integral which is not elementary.
p4wn3r wrote:
Prove that, given the exact sequence (*), if one can "reverse the arrows", that is, find an exact sequence of the form
0 -> C -> B -> A -> 0,
then the sequence (*) splits.
This is just the splitting lemma, right? Anyway I trust that what Wikipedia says is true since I'm not going to analyze it any further.
I ended up learning that "abstract nonsense" is an actual term, used by actual mathematicians. And here I thought it was a term p4wn3r made up. Now I'm even more confused.
I have to apologize. The proposition I wrote is not true in general. Initially I thought there was a way to construct a map so that we could apply the traditional splitting lemma, but when I would write it down, I saw it did not work.
I guess that changes the exercise to: find two exact sequences 0 -> A -> B -> C -> 0 and 0 -> C -> B -> A -> 0 such that none of them split. The answer is:
0 -> Z2 -> Z4 -> Z2 -> 0,
where Zn is the abelian group of integers where the binary operation is addition mod n. Abelian groups are also modules if we choose the ring to be Z. Since the sequence is symmetric, reversing the arrows doesn't matter, and it does not split, since Z4 is not isomorphic to Z2*Z2
My bad...
This one is very fun if you solve it geometrically. However, if you want to use it as a calculus exercise, go ahead!
Find the minimum of f(x)=sqrt(x^2+4x+13)+sqrt(x^2-8x+41)
I did this via calculus but would like to see the geometric solution. It may be helpful to note that x2+4x+13 = (x+2)2+9 and x2-8x+41 = (x-4)2+25. Take the first derivative of f(x) to get
f'(x) = 2(x+2)/sqrt((x+2)2+9) + 2(x-4)/sqrt((x-4)2+25).
Now we need to find x such that f'(x) = 0. This is determined by the numerator of the fraction sum, leading to the equation
(x+2) sqrt((x-4)2+25) + (x-4) sqrt((x+2)2+9) = 0.
Move one of the products to the other side of the equation and square both sides to get
(x+2)2((x-4)2+25) = (x-4)2((x+2)2+9),
which simplifies to
25 (x+2)2 = 9 (x-4)2.
The solutions to that equation are x = 0.25 and x = -11, but only the former satisfies f'(x) = 0. So the minimum of f(x) is f(0.25) = 10.
This was quite fun indeed, although finding a geometric solution probably took me longer than it would have taken to just do the differentiation :P
The function f(x) can be thought of as the sum of the magnitudes of the vectors (x+2, 3) and (x-4, 5). If you draw a picture and think about it, you'll see that minimizing f(x) is equivalent to the following problem:
what point C on the central horizontal axis in the figure is such that AC + CB is minimal? This is solved by reflecting the triangle through the axis and constructing the straight line from A to B' (purple line in the diagram). In this case, the angles to A and B from C are equal, which means
(4-x)/5 = (2+x)/3.
Solving this gives x = 1/4, which is the same as what Dacicus found.
edit: just realized my solid/dash styling is inconsistent, but I can't be bothered changing it now. Sorry!
f(x) can just be thought of as the sum of the distance from (x+2,0) to (0,3) and the distance from (x+2,0) to (6,-5), as shown in the diagram below.
Based on the triangle inequality, to minimize the sum of the two distances, (x+2,0) must be collinear with (0,3) and (6,-5), giving two similar triangles. Splitting 6 in the ratio of 3:5 gives 9/4:15/4. So x+2=9/4, and so x=1/4 minimizes f(x).
Answer:
Use u = xphi, du = phi*xphi-1dx => dx = du*(x1-phi)/phi = du*(u1/phi-1)/phi. Notice that phi satisfies the equation phi² - phi - 1 = 0. If we move stuff around, we can obtain phi - 1 = 1/phi. Plugging this, we get dx = du*(uphi-2)/phi. Therefore, we reduce to the following integral:
This is just an integral representation of the Beta function: https://dlmf.nist.gov/5.12
Applying formula 5.12.3 of the given reference for a=phi-1, and b=1, we see that the integral is just B(phi-1,1)/phi. The answer follows from simple algebraic manipulation of beta and gamma functions:
So, we get the quite amazing result that the integral is simply 1.
You beat me to it :P
I did it more or less the same way. Let p be the golden ration. Starting from 5.12.3 here https://dlmf.nist.gov/5.12, I made the substitution t = z^p. Then setting a = 1/p, b = p - 1/p = 1, you get
1/p * B(1/p, 1) = I,
where I is the desired integral. Since B(1/p, 1) = p, it follows that I = 1.
This result was a bit surprising to me. Can you prove it?
Given a function that gives evenly distributed random real values between 0 and 1, let's call it R(), the probability distribution of sqrt(R()) is exactly the same as the probability distribution of max(R(), R()).
Given a function that gives evenly distributed random real values between 0 and 1, let's call it R(), the probability distribution of sqrt(R()) is exactly the same as the probability distribution of max(R(), R()).
I assume by R(), you are indicating a random variable (uniform distribution on [0,1]). Then the above statement can be proved by showing that their cumulative distribution functions (CDFs) are the same. Let F(x) be the CDF of sqrt(R()), and G(x) be the CDF of max(R(),R()). Let b be any number in [0,1]. Then:
F(b) = Pr( sqrt(R())≤b ) = Pr( R()≤b2 ) = b2
and
G(b) = Pr( max(R(),R())≤b ) = Pr( R()≤b and R()≤b ) = (b)(b) = b2
It is obvious that both F(x) and G(x) are zero when x<0 and one when x>1. So then, since F(x) = G(x), the probability distributions of sqrt(R()) and max(R(), R()) are the same. QED
Note: The probability function is just the derivative of the CDF: F'(b)=G'(b)=2b when b is in [0,1], and zero everywhere else.
Here is a counting exercise that requires some basic musical knowledge*: How many distinct rhythms are possible in one measure of 4/4 time if you are limited to using whole, half, quarter, and eighth notes or rests and any number of ties among them? The rhythm must be contained entirely within the measure, so ties across bar lines are not allowed.
* There are numerous music theory tutorials online for those who do not know what the terms mean but want to learn.
Here is a counting exercise that requires some basic musical knowledge*: How many distinct rhythms are possible in one measure of 4/4 time if you are limited to using whole, half, quarter, and eighth notes or rests and any number of ties among them? The rhythm must be contained entirely within the measure, so ties across bar lines are not allowed.
* There are numerous music theory tutorials online for those who do not know what the terms mean but want to learn.
Just for clarification: Is a tie between two eighth notes equivalent to a quarter note, or are they distinct? (similarly for other ties of two or more notes)
Just for clarification: Is a tie between two eighth notes equivalent to a quarter note, or are they distinct? (similarly for other ties of two or more notes)
Just for clarification: Is a tie between two eighth notes equivalent to a quarter note, or are they distinct? (similarly for other ties of two or more notes)
They are equivalent.
I also have a question. Are two rhythms distinct if they're related by a shift/delay in time? For example, if I have quarter notes on beats 1 and 3 (rests on 2 and 4), is that distinct from having quarter notes on beats 2 and 4 (rests on 1 and 3)?
Are two rhythms distinct if they're related by a shift/delay in time? For example, if I have quarter notes on beats 1 and 3 (rests on 2 and 4), is that distinct from having quarter notes on beats 2 and 4 (rests on 1 and 3)?
This turned out to be harder than I thought. I wouldn't be surprised if there's an easier way to do it, but here's what I found.
If we call the duration of an eighth note 1, then a bar has duration 8. Because of ties, it's possible to construct notes of any duration from 1 to 8. Suppose first there are no rests, then a rhythm will be an ordered sequence of notes whose durations sum to 8. For example, four quarter notes would be 2 + 2 + 2 + 2 = 8. These are the compositions of an integer https://en.wikipedia.org/wiki/Composition_(combinatorics). The number of compositions of n is given by 2^(n-1), which is the number of rhythms that don't include any rests.
To deal with rests, note that rests can't be tied (as there is no sound). If we look through all the compositions of n, we can replace every instance of a 1 (which corresponded to an eighth note) with an rest of duration 1 to get a new rhythm. We only need to do this for 1s though: replacing a 2 with a rest of duration 2 would give the same rhythm as the same composition with the 2 replaced with 1+1 and both 1s replaced with rests, which we have already considered. If there are p 1s in a certain composition, all choices of notes/rests for the 1s gives a total of 2^p rhythms. The total number of rhythms of duration n is hence given by
sum from p=0 to p=n of 2^p * a(n,p)
where a(n,p) is the number of compositions of n containing p 1s. Now we need to find a(n,p). It's easy to see that a(n,n) = 1 and a(n,n-1) = 0. For p < n-1, a composition of n containing p 1s contains precisely p 1s and a composition of n-p that does NOT contain any 1s. Since order matters, there are (p+1) choices for the locations of the 1s. The number of compositions of n-p not containing any 1s turns out to be equal to F(n-p-1) where F(n) is the n'th Fibonacci number (F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2)). Admittedly, I got this part using a recurrence relation I found by google searching. Hence a(n,p) = (p+1)*F(n-p-1) for p < n-1.
Finally then, the number of rhythms of duration n is given by
2^n + sum from p=0 to p=n-2 of 2^p * (p+1) * F(n-p-1)
When n =8 I get 1257. This does include the rhythm containing only rests (i.e. silence). If you don't count that I get 1256.
I believe that there are errors in the derivation of this formula. If you take n = 5 and p = 1, for example, it yields a(5, 1) = 2 * F(3) = 4. But that Wikipedia link shows that a(5, 1) should be 5.
NxCy wrote:
Since order matters, there are (p+1) choices for the locations of the 1s.
Is this true if n-p has more than one composition that does NOT contain any 1s?
NxCy wrote:
This does include the rhythm containing only rests (i.e. silence). If you don't count that I get 1256.
Please do include the rhythm that contains only rests.
I believe that there are errors in the derivation of this formula. If you take n = 5 and p = 1, for example, it yields a(5, 1) = 2 * F(3) = 4. But that Wikipedia link shows that a(5, 1) should be 5.
Yeah, you're right. I forgot to include cases where the 1s are mixed between the numbers in the composition without the 1s (in your example my argument misses 2+1+2). I'll have to think about it some more.
Here is a counting exercise that requires some basic musical knowledge*: How many distinct rhythms are possible in one measure of 4/4 time if you are limited to using whole, half, quarter, and eighth notes or rests and any number of ties among them? The rhythm must be contained entirely within the measure, so ties across bar lines are not allowed.
This is equivalent to finding sequences of length 8 consisting of symbols R (8th rest), N (new 8th note) and T (tied 8th note) where T is only permitted if the previous symbol is N or T (*).
For length 0, an empty sequence is permitted, and for length 1, there are two sequences: R and N (a sequence cannot begin with T).
For k≥2, a sequence of length k satisfying (*) is normally formed by taking a sequence of length k-1 satisfying (*) and adding R, N or T to the end (**). However, since T is not permitted after R, we would need to subtract the sequences in (**) that end in RT; these are precisely sequences of length k-2 satisfying (*) and adding RT to the end. So if we let an denote the number of sequences of length n satisfying (*), the recursive formula is:
a0=1
a1=2
an = 3an-1 - an-2 (for n≥2)
Using this formula we get a0=1, a1=2, a2=5, a3=13, a4=34, a5=89, a6=233, a7=610, a8=1597. So there are 1597 such rhythms. Note that this is the sequence of odd-indexed Fibonacci numbers.
P.S. The generating function is (1-x)/(1-3x+x2). Using partial fractions and then writing out the series gives the xn coefficient:
since the second term goes to 0 and is always positive. So you can just get the number of such rhythms of length n by evaluating the first term and rounding up to the next integer.
P.P.S. The sequence is OEIS A001519 (offset by 1 index), and the page mentions that this sequence can be thought of as "Number of musical compositions of Rhythm-music over a time period of n-1 units."
Edit: Fixed an error in the image.
This week's Riddler Classic has me puzzled. Not because it's especially difficult-- I have an answer I'm pretty confident in-- but the premise seems off to me.
The premise:
Make a block of ice melt more slowly by cutting off a piece of it.
Intuition would point you in the direction that the best way to have your ice melt slowly is to have more ice. Intuition is often wrong, however.
So can this be proven? One obvious issue is that we don't have a model for our melting ice. I could mumble a few words about the heat equation, but really the entire block could be at 0 degrees C while all the interesting physics happens at the boundary of the block as it melts. Then on top of that are the boundary conditions. I'd guess Dirichlet, but I'm vaguely aware that there are subtleties in heat transfer. Does our choice of boundary conditions even matter?
Basically, given some model (which we'd have to decide on first), suppose we have two blocks of ice, one a proper subset of the other. Is it possible to prove that as they melt, the smaller block of ice remains a proper subset of the first at all times? The more I think about it, the more I imagine the answer is "yes", but I'm also excited by the idea of a counterexample.
I haven't really put pen to paper in an attempt to solve it, largely because I don't have much confidence on the theoretical side of PDEs. One way I've looked at it is to consider the ratio of the heat flux to the block's volume. A second line of attack (which seems too good to be true, like it's almost a proof but might be either three lines or 50 pages away from a proper proof) is to just imagine that the smaller block is embedded in the first and point to its boundary where it lies within the larger block and say, "See? No heat entering here, so it melts strictly more slowly in the presence of the piece we lopped off."
Any thoughts on this?
Joined: 10/27/2004
Posts: 1978
Location: Making an escape
I'm not qualified to solve this problem, but I think you might be overthinking it. It may have more to do with the ratio between volume and surface area: by slicing off a chunk from a cube (supposing our ice block is a cube), you'll remove some ice, but you'll also remove some surface area, making it more difficult for heat to get in. You'd need to calculate a slice that would maximize both volume and surface area.
I'll grant the puzzle as stated is a bit understated, failing to describe what the conditions are. Vague language like this leaves a lot open to interperetation.
A hundred years from now, they will gaze upon my work and marvel at my skills but never know my name. And that will be good enough for me.
I'm not qualified to solve this problem, but I think you might be overthinking it. It may have more to do with the ratio between volume and surface area: by slicing off a chunk from a cube (supposing our ice block is a cube), you'll remove some ice, but you'll also remove some surface area, making it more difficult for heat to get in. You'd need to calculate a slice that would maximize both volume and surface area.
I'll grant the puzzle as stated is a bit understated, failing to describe what the conditions are. Vague language like this leaves a lot open to interperetation.
No, I'm pretty sure I get the motivation of the problem. After all, for a given volume, the shape that (probably) melts the slowest is a sphere. (Rather a lot hangs on that "probably". I'd say something along the lines of the rate of melting being proportional to the curvature or something like that, but it's up to cleverer people than I to actually model that.) The problem is that this isn't "a given volume", we're lopping off some of it. My question is, is this ever worth it?
I've given it a bit more thought and I think I'm homing pretty close to an answer. Let the cube of ice be given by volume U (that is, a set of points U, not the numerical volume). Let the smaller volume of ice be V, so V is a proper subset of U. We've lopped off U\V.
Now I offer to reform block U by appending U\V back on to V to produce V∪(U\V), which is just U again. Allow these two blocks-- V and V∪(U\V)-- to melt for a short time. Any part of V away from where it was cut will melt no slower than U. (I'm unable to show this rigorously, but it seems intuitively true that insulating part of a block of ice should not make a different part melt faster.) Points on the interior of the surface adjoining V and U\V are insulated and will therefore not melt at all, whereas that same ice will melt somewhat in V. The only melting that we really need to consider is the ice at the perimeter of the surface of where V meets U\V. Here too, I imagine that the addition of U\V will only cause this ice to melt more slowly (though once again, I fall short of outright proving this). Thus, after this short time, encompassing block U melts to U' while smaller block V melts to V'. Based on my outline above, no part of U' has melted to within the interior of V'.
The last step is to play this game again, now starting with U' and V'. By all the same arguments above, any parts of V' that melt are in the interior of U' and thus at all times, smaller block V lies entirely within larger block U, no matter how much time has elapsed, proving that a smaller block will always melt faster than a larger one that it can fit completely inside.
Actually modeling this problem will be rather difficult. Especially because shape has a lot to do with how convective flows move around. But a simple model with a fixed convective coefficient is probably good enough for a first order approximation.
Let's consider the two extreme cases, a cube and the sphere which inscribes the cube (both of ice). The cube will have a momentary heat transfer which is related directly to the surface area via the coefficient, so the ratio of heat transfer is dependent on the ratio of the surface areas. In this case the ratio of surface area between the sphere and cube is 24/4pi = 6/pi. Whereas the ratio of volume between the two is 8/(4/3 pi) = 6/pi. Wait what? (Of course it makes sense, because the surface area is the differential of the volume, and that's linear.)
So assuming the ice melts perfectly evenly, both the cube and the sphere will finish melting at the same time. How unsatisfying. But the ice won't melt perfectly evenly on the cube, because the ice is not perfectly at 0 degrees all of the way through, so corners heat up faster, and because, outside of our idealized single convection coefficient world, we have interesting edge effects in convection that cause edges to be affected more than corners. These complications have the effect of rounding off the corners, reducing the surface area of the cube, slowing it down. So our square will melt slower overall. I agree that the question seems ill-posed.
Build a man a fire, warm him for a day,
Set a man on fire, warm him for the rest of his life.
Interesting puzzle from a book I first read many years ago, but only really appreciated when revisiting it recently. It goes like this:
A secret society would like to poll their 60 members on some yes/no question. They are only interested in the parity of the number of yes votes. However, they will only allow a given pollster to interview 50 members, and each pollster may only report back a number between 0 and 50 inclusive. It is believed that up to one pollster may lie or make a mistake in reporting.
* Find the smallest number of pollsters required (easy).
* Find a strategy where each pollster can report a number and the correct result can be guaranteed.
* Find the smallest N such that a pollster can report back only a number between 0 and N inclusive and the correct result can be guaranteed.
It was nice that the volume equation canceled out the square root.
what point C on the central horizontal axis in the figure is such that AC + CB is minimal? This is solved by reflecting the triangle through the axis and constructing the straight line from A to B' (purple line in the diagram). In this case, the angles to A and B from C are equal, which means
(4-x)/5 = (2+x)/3.
Solving this gives x = 1/4, which is the same as what Dacicus found.
edit: just realized my solid/dash styling is inconsistent, but I can't be bothered changing it now. Sorry!
Based on the triangle inequality, to minimize the sum of the two distances, (x+2,0) must be collinear with (0,3) and (6,-5), giving two similar triangles. Splitting 6 in the ratio of 3:5 gives 9/4:15/4. So x+2=9/4, and so x=1/4 minimizes f(x).
since the second term goes to 0 and is always positive. So you can just get the number of such rhythms of length n by evaluating the first term and rounding up to the next integer.
P.P.S. The sequence is OEIS A001519 (offset by 1 index), and the page mentions that this sequence can be thought of as "Number of musical compositions of Rhythm-music over a time period of n-1 units."
Edit: Fixed an error in the image.
