The chief of the forty thieves has to devise a strategy that will give him the highest possible probability of survival.
Let us call strategy ##(\alpha,\beta)## the strategy that consists in placing ##\alpha## white and ##\beta## black balls in box 1, and let us define the events
##A##...
A function is a relation between two sets ##A## and ##B##, which might contain any type of object you like, but with the restriction that each object of ##A## is in relation with at most one object of ##B##, which is zero or one object. The definition set of a function is the subset of ##A##...
Ok, thank you everybody for your participation. I think that this game can end now as it does not interest the public it is for. I hope that you liked the problem as much as I did.
@BiGyElLoWhAt , if you still want to give away the solution, the last word belongs to you.
1- Any choice is allowed, knowing that the only thing that will save him is to pick a white ball. An empty box leads to his execution
2- No handling possible after the distribution
Homework Statement
This is a problem that I really liked and that I want to share with you. Firstly because of the story around it, secondly because of the unexpected solution, and finally because it can be investigated with a computer for those who are the least comfortable with maths...
Hmmm, not the funniest thing to do during summer time :wink:
This is where you should start : https://en.wikipedia.org/wiki/Archimedean_property
Then you will try to prove that between any 2 real number, there is a rational number. By induction, you will deduce that there is an infinite...
If I understand your problem is something like : find ##\delta \in \mathbb{Z}## such that ##\delta^2 = z##, ##z\in \mathbb{Z}##
Assume ##\delta = x + iy ## and ## z = X + i Y = \frac{D^2 \frac{ct}{k_0}}{D^4+4\left(\frac{ct}{k_0}\right)^2} + i \frac{ -2...
Could it be more efficient to generate an auxiliary table of prime numbers less than 256, there are 54 of them, and try to find the prime factors of ##y-b## for all b's with their multiplicity ?
There is a solution that saves you 1 inner loop
for(b=0; b<256; b++){
for (x=1; x<256; x++){
if ( ( (y-b) mod x) == 0)
// then m = (y-b)/x
// save m, x, b and exit the loop
}
}
}
But you will get better help posting this...
If you want to convert a positive integer from base 10 to base b (b integer greater than 1), you can do it by successive euclidean divisions :
## n = b q_0 + r_0 ##
## q_0 = b q_1 + r_1 ##
...
## q_{k-1} = b q_k + r_k ##
...
until the quotient is 0.
In the end, ##n = \sum_{k=0}^m r_k b^k ##...
As an example, take ##E## as the vector space of polynomials of degree less or equal to ##n##.
And take ##f : E \rightarrow E ## as the linear mapping defined by ## f : P(X) \rightarrow P(X+1) ##. What is the matrix of ##f## relative to the basis ##(1,X,X^2,..., X^n)## of ##E## ?
You know that...
You must be right, and I don't have the knowledge to elaborate about what you say. This is how I have been taught about matrices, and I thought it could be a starting point for the OP.
Matrices are representations of linear mappings
Any linear mapping ##f## from ##E\rightarrow F##, where ##E## and ##F## are finite dimensional vector spaces, can be represented by a matrix.
If ##(e_1,...,e_n)## is a basis of ##E##, ##f## is entirely determined by ##f(e_1),...,f(e_n)##.
Each...
I know this, it's called Cesaro's theorem.
For the proof you can write:
## |(\frac{1}{n}\sum_{k=1}^n a_k) - L | \le \frac{1}{n}\sum_{k=1}^N |a_k - L| + \frac{1}{n}\sum_{k=N+1}^n |a_k - L| ##
The first term tends to 0 as n tends to infinity.
The second term is bounded by ##\frac{n-N}{n}...
You can explicitly calculate ##\sigma(n)## and ##\tau(n)##, but you will need the prime decomposition of ##n##, say ##n = \prod_{i=1}^N p_i^{r_i}##.
The positive divisors of ##n## are ##\text{Div}(n)=\{ p_1^{q_1} \times ... \times p_N^{q_N},\ 0\le q_i \le r_i,\ i = 1...N \}##
There is an...