All Questions
41
questions
1
vote
0
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33
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Different slopes defined by nesting $m$ polygons
I know that the vertices of a regular $n$-gon determines the total of $n$ different slopes. We nest the total of $m \in \mathbb{N}$ polygons by drawing a $(1/2)n$-gon inscribed inside the original ...
user830143
4
votes
0
answers
85
views
How many partitions of natural numbers into composite numbers are there?
In a graph-theoretic context the following question arose:
Given a natural number $n$. In how many ways can it be written as
the sum of composite numbers?
As an example, the number $64$ can be ...
- 18.2k
1
vote
1
answer
49
views
How to count the total unique numbers in this set?
If we have the following set:
$\Big\{ \frac{1}{1},\frac{2}{1},...,\frac{k}{1},\frac{1}{2},...,\frac{k}{2},...,\frac{1}{k},...,\frac{k}{k} \Big\}$
it is very clear that there are some doublets of ...
1
vote
1
answer
71
views
bounds for number related to colourings
Let $R(r,s)$ be the minimum $n$ so that for all colourings of the edges of the complete graph $K_n$ on $n$ vertices with $2$ colours green and orange, there is a complete subgraph of $K_n$ with $r$ ...
user747916
0
votes
1
answer
423
views
Complete Directed Graph Indegree and Outdegree summations
Let the indegree of a vertex $v$ be $i(v)$ and the outdegree be $o(v)$. Consider a single tournament (a directed graph obtained by assigning a direction for each edge in an undirected complete graph) ...
1
vote
2
answers
195
views
2n points on a circle in two different colors. Prove that pairwise distances of same-color points are the same
There are $2n$ points on a circle. The distance (defined by shortest distance you would take to walk from one point to another along the circle) between adjacent points are the same. $n$ points are ...
- 173
1
vote
0
answers
71
views
Worst-case minimization of size of intersection of an offset set with a given set
Consider the set of first $N$ natural numbers i.e., $S_N$ = $\{1,2,3, \dots,N\}$. Let $S^k$ be a finite subset of $S_N$, where $|S^k|$ = $k$. We define another set $O$ whose elements $\in$ $\{0,1,2,\...
- 104
1
vote
1
answer
34
views
Pentagonal Theorem and Ferrers Diagrams
I have this question about the "Bijective Proof" on The Pentagonal number theorem.
What do they mean by the right most 45degree line? In this example it would be the second and third row?
o ...
- 6,277
0
votes
0
answers
57
views
A variant "3 jugs problem" requires its general proof of non-existence of its solution.
Here is a bank interview question:
You have 3 bottles (or jugs whatever) which have 8, 8 and 3 liters volume. Initially, you have those 2 8-liter bottles filled. Now you are supposed to pour the ...
- 527
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
10
votes
1
answer
660
views
Sparse Ruler Conjecture
Sparse Ruler Conjecture, hard: If a minimal sparse ruler of length $n$ has $m$ marks,
easy: $m-\lceil \sqrt{3*n +9/4} \rfloor \in (0,1)$.
hard: $m+\frac{1}{2} \ge \sqrt{3 \times n +9/4} \ge m-1$...
- 21.4k
3
votes
1
answer
96
views
Counting certain graphs defined over consecutive sets of natural numbers
Let $[n] = \{1,\ldots, n\}$ and call a subset $M \subseteq [n]$ a consecutive subset if $$M = \{m,m+1,\ldots,m+|M|-1\}$$
for some $m \in [n]$, i.e., $M$ contains a smallest and a largest number and ...
- 18.2k
3
votes
1
answer
181
views
A certain partition of 28
Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than ...
- 1,162
-1
votes
2
answers
569
views
How many connected components does the following infinite graph have?
Consider the undirected infinite graph whose nodes are the positive integers $1,2,3,4,...$, and where there is an edge between two nodes $n$, $m$ if and only if $\frac{n^2+n}{2} = m$. How many ...
- 3,285
9
votes
1
answer
246
views
Recovering a partition of 50
The sum of 10 numbers, not necessarily distinct, is 50. When placed appropriately in the circles of this diagram, any two numbers will be joined by a line if, and only if, they have a common divisor ...
- 1,162