The tag has no wiki summary.

learn more… | top users | synonyms

184
votes
4answers
12k views

The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
81
votes
8answers
11k views

Are half of all numbers odd?

Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may ...
44
votes
10answers
8k views

Is zero odd or even?

Some books say even numbers start from two but if you consider the number line concept, I think zero should be even because it is in between -1 and +1 (i.e in between 2 odd numbers). What is the real ...
27
votes
7answers
3k views

Do odd imaginary numbers exist?

Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
17
votes
4answers
1k views

$\sqrt 2$ is even?

Is it mathematically acceptable to use Prove if $n^2$ is even, then $n$ is even. to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ ...
14
votes
0answers
528 views

$f(x)=\sum_{t}{x \choose t}{n-x \choose k-t}$ - even or odd?

The following function popped in my research: $$f(x)=\sum_{\array{0\le t\le k \\ t\equiv_p a}}{x \choose t}{n-x \choose k-t}$$ Where: n,k are natural numbers and $k\le n$. t is taken over all ...
11
votes
4answers
2k views

$C(n,p)$: even or odd?

Can we determine if a binomial coefficient $C(n,p)$ is even or odd, without calculating its value? ($p\lt n$, $p$ and $n$ are positive integers)
9
votes
3answers
748 views

Proving a statement regarding a Diophantine equation

FINAL EDIT : Prove that if $p^z|n^2-1$ $$p^{x-z}(p^{z}-1)=\dfrac{ n^2-1}{p^z}-3$$ doesn't hold for any chosen values of $p,x,n$ and $z$. Here $p>3$ is an odd prime , $x=2y+z, \ ...
6
votes
3answers
145 views

Is odd continuous function differentiable at $x=0$?

Suppose that $f(x)$ is continuous and odd: $f(-x) = - f(x)$. Does it have a derivative at $x=0$? Here is what I got so far: First we calculate $f(0)$ using $f(-0) = -f(0)$, from which $f(0) = 0$. ...
5
votes
2answers
1k views

Generalizations of the number theory concepts of “even” and “odd”?

One of the very first number theory concepts introduced to students -- even before primeness, divisibility, etc. -- is the idea that a natural number can either be "even" (that is, evenly divisible by ...
5
votes
1answer
215 views

Why is this function odd?

Suppose a complex valued function $f$ is entire, maps $\mathbb{R}$ to $\mathbb{R}$, and maps the imaginary axis into the imaginary axis. I see that $f(x)=\overline{f(\bar{x})}$ on the whole real ...
5
votes
1answer
46 views

Erasing numbers from circle and writing down sum

There are $50$ copies of the number $1$, and $50$ copies of the number $-1$, written alternately in a circle. In each step, we pick an arbitrary number, write down the sum of the number and its two ...
4
votes
3answers
290 views

Knight on a chessboard moving from a1 to h8

I was given a puzzle to solve which goes as:- Can a knight start at square a1 of a chessboard, and go to square h8, visiting each of the remaining squares once on the way ? I reasoned that this won't ...
3
votes
2answers
135 views

Why is the cube-root of $x$ 'odd'?

I am trying to understand why $\sqrt[3]{x}$ is an odd function; can anyone explain how I could come to this conclusion?
3
votes
2answers
424 views

Frog Jump Problem

Lotus leaves are arranged around a circle. A Frog starts jumping from one leaf in the manner described below. In the first jump it skips one leaf,next jump it skips two,three the next jump ...
3
votes
2answers
42 views

$(2^a -1)(2^b -1)=2^{2^c}+1$ has no nonnegative integer solutions

$(2^a -1)(2^b -1)=2^{2^c}+1$ is not possible for a,b,c nonnegative integers. Any solutions using parity Approach: $(2^a -1)(2^b -1)=2^{2^c}+1\Rightarrow$ $2^{a+b}-2^a-2^b=2^{2^c}\Rightarrow$
3
votes
4answers
79 views

Can decimals/fractions be odd or even? [duplicate]

At school I asked the question and I kept wondering "Can fractions or decimals be odd or even?"
2
votes
2answers
64 views

Prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous

How does one prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous? I'm unsure of where to begin. Pertinent definitions: ...
2
votes
2answers
1k views

Parity and Inverse of Permutations (Odd and Even)

I want an explanation on knowing how to know whether a permutation is odd or even. For example, I have a few permutations of [9] that I need explained for parity, inverse, and number of inversions if ...
2
votes
3answers
308 views

Parity confusion

I am confused. I have to show that $f(x)$ that satisfies the ODE below "has definite parity or can be chosen to have def parity". $$\frac{d^2}{dx^2}f(x) = (h(x)+c)f(x)$$ where $h(x)$ has even parity ...
2
votes
1answer
26 views

Parity Bit Detecting Odd Bit Errors

I'm going over a past paper which has a true or false question with the following statement. A single parity bit computed over 128 data bits can detect an error when bit-flips occur in exactly 93 ...
2
votes
1answer
22 views

Undetected Errors in 2 Dimensional Parity

Given a two dimensional parity with a data block of two rows and two columns what is the probability that a four bit error goes undetected? The naive method would be to look at all ways in which an ...
2
votes
1answer
91 views

Infinite Parity Function

I was looking at this problem, and I have a solution for a finite board with $2^n$ squares, that I want to extend to a countably infinite board. Label the squares from $0$ to $2^n-1$. Consider the ...
2
votes
2answers
151 views

Similarity between integer and logical operations through parity

Lets observe the parity property of integers while adding them or multiplying. It's simple to notice that when we add two numbers, the parity of the result depends on parity of summands: ...
1
vote
2answers
61 views

Linear independence of linear combination of functions

This question is inspired by this question. Is it always true that if $\{f_n(x)\}_n$ are linearly independent then so is $\{f_n(x)+f_n(-x)\}$ given that each $f_n(x)$ do not have definite parity?
1
vote
3answers
36 views

Series representation of $\sin(nu)$ when $n$ is an odd integer?

So, out of boredom and curiosity, today I came up with a series representation for $\sin(nu)$ when $n$ is an even integer: $$\sin(nu) = \sum_{k=1}^\frac n2 ...
1
vote
1answer
109 views

An intutive explanation of natural density (asymptotic density)

I was wondering if someone can provide an intuitive explanation to natural density. I understand the concept very basically (pretty much the definition) but I can't seem to understand what natural ...
1
vote
2answers
398 views

Odd or even permutation with matrices

I know that the number of transpositions would determine the parity of a permutation like: A = (1,2,3,4,5) = (1,5),(1,4),(1,3),(1,2) = even But how would that apply to a matrix? Example: 1 2 ...
1
vote
1answer
29 views

Generator matrix of a binary cyclic code

I need to find the Generator and Parity check matrix of a binary cyclic [9,2] code. If I calculated right, the Generator polynomial is x^7 + x^6 + x^4 + x^3 + x + 1 and the check polynomial is x^2 - x ...
1
vote
1answer
65 views

Palindromic binary words with even or odd amounts of letters

Say you have one bag of apples and one bag of oranges. Each bag contains at least one fruit, and the number of fruit in each bag is an odd number. I have found that it is impossible to line up these ...
1
vote
1answer
76 views

Why does an odd number plus one, not necessarily entail it being even?

Why does an odd number plus one, not necessarily entail it being even? For example, sqrt(5) + 1 is not even.
1
vote
1answer
56 views

Why does $\sin{\alpha}\cdot i\sin{\alpha x}$ disappear from this integral?

In a section on fourier transforms, my textbook contains these steps for an example: $$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$ $$= ...
1
vote
1answer
832 views

Obtaining a generator polynomial from a parity check matrix for a binary cyclic code

In general, what is the strategy for obtaining a generator polynomial (and a check polynomial) given a parity check matrix $H$ for a binary cyclic code? Things I know: Each codeword $c(x)$ ...
1
vote
0answers
41 views

Number of divisiors of $n$ less than $m$

I'm looking for a closed- or alternative-form of the function that counts the number of divisors of an integer $n$ that are less than some integer $m$ (interested in $m < n$, obviously): $ ...
1
vote
0answers
46 views

Quadratic Congruence in $\mathbb Z/2^n \mathbb Z$

Given the congruence $ax^2+bx+c \equiv 0 \pmod {2^n}$, how precisely does one go about finding its roots? I'm comfortable with quadratic congruence mod n with n odd, but 2's lack of a multiplicative ...
1
vote
0answers
145 views

How should I interpret Johnson's “Note on the '15' puzzle”?

Following the suggestion of Gerry Myerson who commented below, I went ahead and read Johnson's "Note on the '15' Puzzle". Bearing in mind that I am not well versed in the English of 19th century ...
0
votes
3answers
66 views

Is it right to say that: if $2a+1=2b$ we have a contradiction?

I am trying to prove by contradiction and I have reached the conclusion that $2a+1=2b$. Now I am tempted to say it's a contradiction and call it a night. Is it a contradiction? because one is even and ...
0
votes
4answers
20 views

Is there a quick parity test for integers expressed with odd radicies?

For integers expressed with an odd base, is there an easy way to tell if the number is odd or even? For an even base, if the ones digit is even, so is the integer. But this doesn't hold true for odd ...
0
votes
2answers
24 views

If $f$ is of given parity, what can be said of its derivative and primitive?

If $f$ is of given parity, what can be said of it's derivative and primitive? Clearly for power functions or simple trigonometric functions it seems that the parity of the derivative and ...
0
votes
1answer
75 views

Parity, Set of functions

Given linearly independent $\{f_n\}_{n=1}^N$ how can we form $n$ linearly independent functions $\{F_n\}_{n=1}^N$ such that each $F_n$ is either an even or odd function? Thanks.
0
votes
0answers
46 views

Finding a parity check matrix of a binary code

I'm supposed to find a parity check matrix of a binary [6,3,3] code. Given a generator matrix G I can find a parity check matrix by row reducing until I get the identity matrix, then take $-A^{\top} ...
0
votes
2answers
181 views

Hamming Code Error Detection

I am learning few things about hamming code and error detection so my question may sound stupid. So i know that lets i ahve (7,4) hamming code and i made transpose of parity check matrix H(t). Now say ...
0
votes
0answers
76 views

parity of powers of prime factors

lets consider the prime factorisation of a number N let the powers of the primes in this factorisation be a,b,c ....and so on. Is there a way to determine whether the number of powers that are even ...
-3
votes
1answer
37 views

Coset Leaders and Syndromes

This is my Parity check matrix For Coset Leader 100010 Syndrome is 101 Could any one help me with the procedure, since I figured ...