Questions tagged [parity]
This tag is for questions relating to "Parity", a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd.
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Minimum sum between permutations of elements in a matrix
So, for the purposes of creating a heuristic for a Sokoban AI, I have to solve the following problem (if you're curious, the elements of the matrix each represent taxicab distances from boxes to ...
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Parity of the output of a polynomial
$n \epsilon Z+$
For $n=1,2,3...$
$f(n)=(1/2)(n^2-n)$ produces an output of 2 odd integers followed by 2 even integers, $(2O2E)$
$g(n)=(1/6)(-n^3+6n^2+1)$ produces an output of 3 odd integers ...
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Number of triangles a point lies in on a plane
On a plane , there are $2n+1$ points where no three points are co-linear. Show that for any point $P$ which is one of the points, the number of triangles the interior of which $P$ lies in is always ...
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How to find generator matrix and parity check matrix?
The $[n, n − 1, 2]$-parity check code has messages consisting of binary strings of length
$n − 1$. Messages are encoded by adding an extra symbol which is the sum (mod 2) of
the previous symbols.
I ...
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Assigning to every odd number a triangular number. Does the converse hold?
Context:
A triangular number is a number of the form $n(n+1)/2$ where $n\in \mathbb N$.
Question:
After application of two theorems in my textbook (the first one being that "a number $n$ is ...
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52
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Parity in the Schrödinger equation of the hydrogen atom
Hi I'm studying quantum mechanics with Stephen Gasiorowicz's book.
It says for the $n=2$ states of the hydrogen atom, the $l=0$ state has even parity and the $l=1$ state has odd parity.
As far as I ...
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Prove or disprove: the average of any two odd integers is always odd
Prove or disprove the following statement: the average of any two odd integers is always odd.
The proof that this statement is false is given below. Please, verify whether the one is valid or not.
$\...
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Finding the Code words generated by a parity check matrix.
I'm trying to find the code words generated by this matrix and then decode some given words and find the bit with the error for this parity check matrix(H):
$$
\begin{bmatrix}
1 & 0 & 1 & ...
1
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1
answer
95
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Is there a notion of a category where morphisms have parity?
In category theory, is there a name/notion for categories where the morphisms have a well-defined parity? Formally, this should mean that there is a non-constant functor from the category to the one-...
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Ordered sequence of even products of nonnegative integers and possible code
I am trying to find the sequence of the even products of nonnegative integers, m and n, after sorting them. For example, $$1*6=6, 2*3=6, 6*1=6, 3*2= 6$$
What I think is the answer is the sequence is ...
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Three grasshoppers jumping on a line
The following problem appears on pp. 86 Terrence Tao's "Solving Mathematical Problems: a personal perspective".
Three grasshoppers are on a line. Each second, one (and only one) grasshopper ...
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65
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Question in pseudovectors
I learnt that under parity transformation a vector
$\vec{A}$ <---(Parity)------> $-\vec{A}$
and a pseudovector can be written as $\vec{c}=\vec{A} $ $\times$ $\vec{B}$ and since A goes negative A ...
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Variations of coloring problems with parity constaints
I am looking for a variation of a graph colouring problem without the restriction that every two adjacent vertices have different colours (i.e. not a proper colouring), but instead with some parity ...
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Let α1 , …, αr be distinct even permutations, and β an odd permutation. Show then α1β, …, αr β are r distinct odd permutations.
Q: Let $\alpha_1 , …, \alpha_r$ be distinct even permutations, and $\beta$ an odd permutation. Show then $\alpha_1\beta, …, \alpha_r \beta$ are $r$ distinct odd permutations.
Proof: $\alpha_i$ can be ...
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Proving that a couple must sit together on the same motorcycle - a problem with mutually exclusive propositions
$65$ couples go on a spring outing, there are a total of $65$ motorcycles, each motorcycle carries one boy and one girl. Suppose for any two motorcycles, the following two propositions are exactly one ...
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Ratio of odd to even $2n-$tuples
Let $o(n)$ be the number of $2n-$tuples $(a_1,a_2,\ldots,a_n,b_1,b_2,\ldots,b_n)$ such that each $a_i,b_j\in\{0,1\}$ and $a_1b_1+a_2b_2+\ldots +a_nb_n$ is odd. Similarly, let $e(n)$ be the number of $...
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Proving If a is an odd integer then $a^2+3a+5$ is odd.
I've written a proof for which looks correct, but I wanted to know it from someone else as well. Also I feel like my proof could be made a bit shorter, so if anyone has advice for that I would ...
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Parity of number of subsets having odd intersection with fixed number of sets
I am currently working on a combinatorial problem that reduces to the following counting problem:
Let $i$ be a positive fixed integer. Given a set $S$, Is there a family of subsets $\mathcal{B}^i_S$ ...
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29
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Decrement of a Permutation vs Number of Transpositions
Given a permutation with $n$ items and $c$ independent cycles, the decrement is defined as $n-c$. Here is a simple proof showing that this number is also equal to the number of transpositions in the ...
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2
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Switch off all the n bulbs, (n-1) at a time. A parity-based puzzle?
So, here is a puzzle-like problem:
You have n light bulbs. Each one is connected to a special switch that flips the state of all the other n-1 bulbs except the bulb it is connected to.
If you start ...
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Possibility of arranging $1$ and $-1$ in a grid such that the sum of the products is $0$
Consider an $11$ x $11$ grid, where in each square, the number $1$ or $-1$ is written. One multiplies the numbers in each row and column, and then sums up these $22$ products. Is it possible for this ...
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Maximal accuracy of logistic regression on the n-parity problem
Consider the standard logistic regression ('LR') function, $ y = \sigma(w^T \cdot x + b) $, where $\sigma$ is the logistic function ('sigmoid'). When checking the accuracy, we will consider the argmax ...
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30
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Checking the parity of an infinite sum
While going through a paper I stumbled on the following equality :
$\begin{equation} \rho(\alpha)=\frac{1}{\pi}\sum\limits_{k\in\mathbb{Z}}\frac{e^{2ik\alpha}}{\cosh(k\zeta)}=\frac{1}{\pi}\frac{\...
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1
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Scalar Triple Product of Two Polar Vectors and a Pseudovector
So I have recently been trying to use methods of exterior algebra to solve quantum mechanics problems, and I still cannot seem to come to a good conclusion about the cross product of two vectors ...
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1
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85
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When is the 8 puzzle solvable? [duplicate]
I'm struggling to find the solution to this eight puzzle:
1
0
2
3
4
5
6
7
8
When I expressed it as a permutation with the cycle notation, I got (23456780)
The signature of this permutation is (-1)^...
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Is there a simple way to find the parity of the remainder/quotient without performing the division?
The context is the following: I am performing some calculations on long integers in Montgomery form, and I need to know their parity without converting them back to normal form (which is slow). An ...
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The parity of permutation $(b_1, ..., b_{n-k},a_1, ..., a_k)$
Let $k$ be an integer between $1$ and $n$.
Let $\sigma_1$ be the parity of $(a_1, ... ,a_k, b_1, ... ,b_{n-k})$ as a permutation of ${\{1, ... n}\}$ . So $a_1, ... ,a_k, b_1, ... ,b_{n-k}$ are all ...
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Proving that for an even $k$ you can't move $k$ rocks from $k$ platforms to the same platform, given a set of rules
We have $k$ rocks on $k$ different platforms placed in a circle. You can only move two different rocks at a time (not more, not less), and you can move them only one platform to the left or to the ...
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How would you detect soft locks in a tile sliding game?
I have created a tile sliding game in python (using pygame). It's a 3x3, and an online example can be found here.
When my and my friends started testing this game out, comparing it to the online ...
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76
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Parity of $p(n)$ if the parity of the self-conjugate partitions is even
Let $p(n)$ be the number of partitions of the positive integer $n$. What can we say about the parity of $p(n)$ if we know that the parity of the self-conjugate partitions is even?
Actually I manually ...
user1007173
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Probability a factor is odd
Take the above question. In the solution it says there 18 factors of 2 in the number. But how do they work this out? I see no reasoning or intuition between the line of working 10 + 5 + 2 + 1 = 18. ...
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3
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Parity Pattern of Triangle Numbers
I noticed that the triangle numbers given by $\frac{n(n+1)}{2}$ have a parity pattern of odd, odd, even, even, ... and I'm curious as to why.
E.g:
...
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Can we draw a closed path made up of 9 line segments , each of which intersects exactly one of the other segments?
The solution given in Fomin's book is as follows.
If such a closed path were possible, then all the line segments could be partitioned into pairs of intersecting segments. But then the number of ...
user907912
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Induction problem on organising a round robin tournament of n football teams
Prove that for all positive integers n, it is possible to organise a round-robin tournament of n football teams in:
a) n-1 rounds if n is even
b)n rounds if n is odd
A round is a set of games in which ...
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Showing whether a intial arrangement of numbers (odd/even) is sortable according to the constraint on sorting each time.
Let $P$ denote some arrangement of the numbers $1,2, \ldots, n$. A move on $P$ consists of exchanging the position of element 1 with the position of another element. For example, if $P=[3,1,4,2]$, ...
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Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials
So, can we transform an even function into an odd function and vice versa?
Let's consider this method:
Transformation even->odd:
Suppose $f_{even}(x)$ is a function which satisfies the following ...
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Are there an odd number of numbers in the unit interval?
Or, four questions that I think are equivalent, but I don't know if will be thought to make sense:
Are there an odd number of numbers in $[0, 1]$?
Are there an even number of numbers in $[0, 1)$?
Are ...
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Sign of permutation related to number of inversions
I define the sign of a permutation as the number of transpositions that are in any re-writing of $\sigma$ in a product of transpositions.
The number of inversions of a permutation $\sigma$ of $S_n$ is ...
3
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1
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91
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Probability that logarithm expression of $500$ values between $(0,1]$ is even
Let $a_1, a_2, a_3, \ldots, a_{500}$ be uniformly and independently chosen from the interval $(0,1]$ at random. Find the probability that $$\lceil \log_2 (a_1) \rceil + \lceil \log_4(a_2)\rceil + \...
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What is the description of pseudo-quantities on a general manifold?
When talking about physics (on $\mathbb{R}^3$), one tends to call pseudovectors $+$-parity objects, since they transform into themselves under the parity transformation $P: (x,y,z) \mapsto (-x,-y,-z)$....
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Formal expression for parity of three integers
During an X-Ray Crystal Diffraction course, we ended up with this sum :
$$(-1)^{h+k} + (-1)^{k+l} + (-1)^{h+l} + 1$$
The condition of extinction for the diffraction is that the sum must be equal to 0 :...
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On the number of ways to get a sum with given parity
Fix some integers $a_1,\dots,a_n\geq 0$. Denote by $E$ and $O$ the number of ways to choose $(x_1,\dots,x_n)$ so that $0\leq x_i\leq 2a_i$ and such that $\sum_{i=1}^n x_i$ is even and odd, ...
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Why do sieves have the parity problem?
Sieve methods have the "parity problem".
Terry Tao gives a "rough" statement of the problem: If A is a set whose elements are all products of an odd number of primes (or are all ...
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Game involving operations on 1, 2, ..., 100 (Leningrad Mathematical Olympiad, 1984)
I have been reading Grade Five Competition from the Leningrad Mathematical Olympiad by Garaschuk and Liu, which is a wonderful source of interesting problems. One of the problems from the year 1984 is ...
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Question on parity from Mathematics Circles book
Forty-five points are chosen along line $AB$, all lying outside of segment $AB$. Prove that the sum of the distances from these points to point $A$ is not equal to the sum of the distances of these ...
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Does the parity of the identity of a group hold any known significance? [closed]
I understand that for permutation groups, the parity of identity is even.
However when considering, for example $\langle\mathbb{Q} \setminus \{0\}, \cdot\rangle$ where $\cdot$ is regular ...
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A question on parity from book "Mathematics Circles" [closed]
Three hockey pucks, A,B, and C, lie on a playing field. A hockey player hits one of them in such a way that it passes between the other too.He does this 25 times. Can he return the three pucks to ...
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$a,b,r,s$ are coprime such that $a^2+b^2=r^2$ and $a^2-b^2=s^2$. Which of the numbers are odd?
$a,b,r,s$ are coprime such that $a^2+b^2=r^2$ and $a^2-b^2=s^2$. I’m trying to show that $a, r, s$ is odd and $b$ is even.
If we add the equations, we get $2a^2=r^2+s^2$. Hence $r^2 + s^2$ is even. ...
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Find all positive integers $x,y$ such that $(x+y)(xy+1)$ is a power of $2$ [duplicate]
Find all positive integers $x,y$ such that $(x+y)(xy+1)$ is a power of $2$
It is clear that as we are given a factorisation of the power of two, both of those terms have to be powers of two. $x+y$ is ...
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Permutation group sending even to even and odd to odd
This question is not about odd and even permutations. Rather, I would like to understand the subgroup of the symmetry group $S_n$ that preserves the parity of all numbers $1, 2, \ldots, n$. So, if e.g....
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