I'll assume that up and down must be exactly 7 moves each, left/right must be exactly 1 move each and that left/right moves must alternate with up/down moves.. Solution:
Spoiler :
Let's discuss Mathematics
- Thread starter ParadigmShifter
- Start date
Sorry, forgot to say that you should visit one square exactly once. (Corrected to my previous post now).
No. The chessboard is bipartite: every move you make goes from white to black or from black to white. After making a fixed number of moves, you know if you should end up on white or black. Bottom right and top right have opposite colors, so they cannot both be reached after the same number of moves.
I hope others gave it at least a try. It's frustratingly difficult to prove until you realize to use the colours and the even number. After that you wonder why didn't you think of it before.
I'm confused. The solution I had in mind for the "upper left corner to upper right corner" task was 7D, 1R, 7U, 1R, 7D, 1R, 7U, 1R, 7D, 1R, 7U, 1R, 7D, 1R, 7U. That transverses all 64 squares, each exactly once. What's the solution to the "upper left corner to lower right corner" task?
My bad again. The example should've been that and the problem going from up left to right down.
Sorry.
Sorry.
Mise
isle of lucy
Guys, is there an easy proof/simplification for the following thing:
If you have a 1 in x chance of something happening, then after x attempts, the probability of that thing not having happened yet is p = (1-1/x)^x. Now, as x->infinity, the probability tends to 1/e (I know this because I recognise the numbers when I calculate it). But how do you prove that? Do you have to do some sort of clever series thing to expand the bracket when raised to the power of x?
The rule of thumb is useful in board games: if I select a thing that has a 1 in 20 chance of happening during each turn, then I know that, after 20 turns, there is a ~2/3 chance of it happening at least once. In Settlers of Catan, for instance, it's easy to calculate the probability of rolling a 4 as a 1 in 12 chance (in fact, the number of dots on the hex tells you that it's a 3 in 36 chance). So you can estimate that, with 4 people playing, there'll be a 2/3 chance that a 4 will be rolled at some point during the next 3 rounds (i.e. after the dice have been rolled 12 times). Obviously for small x it doesn't work so well, but for small x it's easy enough to do in your head.
If you have a 1 in x chance of something happening, then after x attempts, the probability of that thing not having happened yet is p = (1-1/x)^x. Now, as x->infinity, the probability tends to 1/e (I know this because I recognise the numbers when I calculate it). But how do you prove that? Do you have to do some sort of clever series thing to expand the bracket when raised to the power of x?
The rule of thumb is useful in board games: if I select a thing that has a 1 in 20 chance of happening during each turn, then I know that, after 20 turns, there is a ~2/3 chance of it happening at least once. In Settlers of Catan, for instance, it's easy to calculate the probability of rolling a 4 as a 1 in 12 chance (in fact, the number of dots on the hex tells you that it's a 3 in 36 chance). So you can estimate that, with 4 people playing, there'll be a 2/3 chance that a 4 will be rolled at some point during the next 3 rounds (i.e. after the dice have been rolled 12 times). Obviously for small x it doesn't work so well, but for small x it's easy enough to do in your head.
The exponential function ex satisfies the equation
ex = lim (n--> infinity) (1 + x/n)n
(https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function)
Let x = -1 to get the result you seek.
ex = lim (n--> infinity) (1 + x/n)n
(https://en.wikipedia.org/wiki/Characterizations_of_the_exponential_function)
Let x = -1 to get the result you seek.
Mise
isle of lucy
Ahh okay, I'm not familiar with that one! Makes sense. Thanks!
EDIT: The "equivalence of 1 and _" sections of the wiki article are especially useful. I couldn't really "intuit" #1 initially, but the equivalence proofs helped me make sense of it.
EDIT: The "equivalence of 1 and _" sections of the wiki article are especially useful. I couldn't really "intuit" #1 initially, but the equivalence proofs helped me make sense of it.
Robert FIN
Emperor
A short question.
Can you divide by cos (or sin) in an equation? I'm just worried if it is zero and then it would be wrong. Do I have to test it afterwards in case it was 0?
Thanks. I hope the question makes sense.
EDIT: I meant "cos x" or "sin x" ofc..
Can you divide by cos (or sin) in an equation? I'm just worried if it is zero and then it would be wrong. Do I have to test it afterwards in case it was 0?
Thanks. I hope the question makes sense.
EDIT: I meant "cos x" or "sin x" ofc..
azzaman333
meh
I'm pretty sure by definition tan(x) = sin(x)/cos(x), so yes, it's definitely possible.
Robert Can't
Éponine
Possible for all x not equal to (pi)(2n-1)/2.
The function is not defined for those values.
Robert FIN
Emperor
Are you prepared for another one?
Let's say f(x) = x tan2x . Then we count f'(x). I think it is called derivation in English.
So the counting goes right when I do this:
tan2x * 2 + x(1+tan2x^2) * 2
But that seems weird. I forgot why did I put the first * 2 and I don't know why the last * 2 is there neither.. I know that it has something to do with the inner function but then is it required in the first * 2 and I cannot see how the second * 2 comes from the inner function. I totally lost the idea..
Let's say f(x) = x tan2x . Then we count f'(x). I think it is called derivation in English.
So the counting goes right when I do this:
tan2x * 2 + x(1+tan2x^2) * 2
But that seems weird. I forgot why did I put the first * 2 and I don't know why the last * 2 is there neither.. I know that it has something to do with the inner function but then is it required in the first * 2 and I cannot see how the second * 2 comes from the inner function. I totally lost the idea..
Lohrenswald
世界的 bottom ranked physicist
derive sin(x)/cos(x) instead
you know the formula for derivating fractions right?
you know the formula for derivating fractions right?
Robert FIN
Emperor
You mean converting tan2x to sin2x/cos2x ? That might work but it feels kinda tricky...
Lohrenswald
世界的 bottom ranked physicist
It's probably easier than trying to fiddle with cotangens
Fraction derivation require less insight to do I think.
If I have time and resolve later I'll post how to do it.
Fraction derivation require less insight to do I think.
If I have time and resolve later I'll post how to do it.
Robert FIN
Emperor
Alright, thanks.
So the thing is that you have to count "D" for the outer function and then you count that times the "D" of the inner function. But the weird thing in my counting was that why do we need to count the "D" of inner function even though we didn't have to count it for the outer function...
So the thing is that you have to count "D" for the outer function and then you count that times the "D" of the inner function. But the weird thing in my counting was that why do we need to count the "D" of inner function even though we didn't have to count it for the outer function...
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