All Questions
133
questions
1
vote
1
answer
51
views
A problem on a type of {m,n} tree
So here is the tree. For given $\left \{ m,n \right \}$. {m,n} will transform to give these elements which I will represent using a summation operator;
$$\sum^{n}_{k=1}\left \{ m-k,k \right \}$$
...
- 555
1
vote
2
answers
126
views
Square Chessboard Problem [duplicate]
Show that there is a $6$ x $4$ board whose squares are all black or white, where no rectangle has the four vertex squares of the same color. Also show that on each $7$
x $4$ board whose squares are ...
- 1,591
4
votes
1
answer
192
views
Simple recursion or closed form for $\lfloor 2^n \sqrt{2}\rfloor$
Is it possible to find an expression of the form
$$\lfloor 2^n \sqrt{2}\rfloor = \sum_{k=1}^r (\alpha_k + \beta_ki)\cdot\Big(a_k+b_k i\Big)^n, $$
where $i^2 = -1$ and $\alpha_k, \beta_k, a_k, b_k$ ...
- 3,645
4
votes
2
answers
403
views
Maximum run in binary digit expansions
For numbers between $2^{k-1}$ and $2^{k}-1$, how many have a maximum run of $n$ identical digits in base $2$? For instance, $1000110101111001$ in base $2$ has a maximum run of 4.
See picture below ...
- 3,645
2
votes
1
answer
183
views
given 20 diff jewelry, wear 5 per day. proof that after 267 days at least one pair of jewelry would be wear in 15 diff days.
Given 20 different jewels.
which 5 of them can be wear per single day.
proof that after 267 days, there would be at least one pair of jewelry that wears together (in other combination) in 15 ...
- 119
2
votes
3
answers
1k
views
Looking for Distinct Solutions to $x_1 + x_2 + x_3 + x_4 = 100$ Given Certain Conditions
How many distinct solutions to the equation does the following have?
$$x_1 + x_2 + x_3 + x_4 = 100$$ such that $x_1 \in \{0,1,2,... 10 \}, x_2, x_3, x_4 \in \{0,1,2,3,...\}$
My attempt: Ordinarily ...
- 1,203
0
votes
1
answer
98
views
sum of a part over all compositions of fixed length
The number of solutions for $x_1+...+x_k = n$ for positive integers $x_1,...,x_k,n$ is given by $\binom{n-1}{k-1}$, which is the number of $k$-compositions of $n$.
Now we take the sum over all of ...
- 73
2
votes
2
answers
723
views
Proof for : How many numbers cannot be expressed as a sum of two numbers which are chosen from a given set. [duplicate]
We are given two numbers $a,b$ and a range of numbers from $[b,a+b-1]$ where $b\geq1$ and $a\gt2 $. We can add any two numbers $p,q$ (not necessarily distinct) in the range to form more numbers and ...
- 77
-2
votes
2
answers
503
views
Preserving historical information of the Collatz function?
In some sense this two equations are the same, namely $f_2$ preserves the historical information of $f_1^n$, where the exponent is function composition, but I am not sure how to show this rigorously. $...
- 1,074
1
vote
1
answer
43
views
Inverse of a particular bijection
Let $X := \{ (i,j) \in \mathbb{N}\times\mathbb{N} \; | \; 1\leq i < j\}$. I know that the function $T: X\longrightarrow \mathbb{N}$ defined by $$T(i,j) = \frac 12 j(j-3) + i + 1$$ is a bijection. I ...
- 687
-2
votes
1
answer
41
views
Why does taking floored root of a natural number x give me all natural numbers up to x which are perfect squares,cubes,fifths etc.
Eg. The number of perfect squares from $1$ to $10^{10}$ is $\left(10^{10}\right)^{1/2}= 10^{5}$
Context:I got this question from this one: Count the number of integers in the range $1$ to $10^{10}$ ...
2
votes
0
answers
25
views
Number of $k$-sets in $[n]$ so that any two of them share at most 2 elements?
Let $[n]$ be $\{1,2,\dots,n\}$. Let
\begin{align}T_\ell:=\Big|\Big\{&\{K_1,\dots,K_m\} \text{ is "maximal"}:\\ &\text{each $K_i$ is a $k$-subset of $[n]$, and $|K_i\cap K_j|\le \ell$ for all $...
- 2,075
1
vote
1
answer
47
views
can we find multiple of a set of finite numbers that are in the "middle" of numbers mod a prime
$\newcommand\N{\mathbb N} \newcommand\ceil[1]{\lceil#1\rceil}$Let $a_1,\dots, a_k\in \N$ be an arbitrary finite set of positive integers. Can we find a prime number $p$ such that $p>k$ (preferably $...
- 1,667
3
votes
1
answer
332
views
How many different pairs of integers $(x,y)$ modulo $p$ such that $ax^2+by^2+c \equiv 0\pmod{p}$?
Let $p$ be an odd prime number and $a,b,c$ be integers coprime with $p$. How many different pairs of integers $(x,y)$ modulo $p$ such that $ax^2+by^2+c \equiv 0\pmod{p}$ ?
Until now I haven't had any ...
- 63
0
votes
1
answer
90
views
Weights and Scale Problem
a.) What is the minimum number of integer weights to balance a scale with bags of rocks from weights 1 to n.
b.) What is the minimum number of weights to balance a scale of weights on one side and on ...
- 1