All Questions
29
questions
6
votes
0
answers
215
views
The sequence $0, 0, 1, 1, 3, 10, 52, 459, 1271, 10094, 63133,...$
Let $a_0$ be a permutation on $\{1, 2, ...,N\}$ (i.e. $a_0 \in S_N$) . For $n \geq 0$:
If $a_n(i+1) \geq a_n(i)$, then $a_{n+1}(i) = a_n(i+1) - a_n(i)$.
Otherwise, $a_{n+1}(i) = a_n(i+1) + a_n(i)$.
$...
- 1,029
0
votes
0
answers
34
views
Expected maximum occupation number for randomly distributed objects [duplicate]
Suppose you have $M$ distinguishable objects distributed amongst $N$ distinguishable boxes. Can you calculate the expected maximum occupation number $E_\text{max}(N,M)$? (in other words, the expected ...
- 2,339
3
votes
1
answer
340
views
Arrange numbers 1-12 around a circle so that any three consecutive numbers have a sum $\leq 20$?
My friend sent me the original question. Then he asked if 1-12 can be arranged in a way so that any three consecutive numbers have a sum that is not larger than 20. We guess the answer is no, since we ...
- 77
3
votes
2
answers
194
views
OEIS entry - A316312 has a question: Is it true that if k is a term then 100 * k is a term? [closed]
Refer https://oeis.org/A316312 - the sequence in OEIS.
The sequence says
Numbers n such that sum of the digits of the numbers 1, 2, 3, ... up
to (n - 1) is divisible by n.
A few terms from the ...
- 57
2
votes
0
answers
58
views
Which sets of $n-1$ non-multiples of $n$ can't make a multiple of $n$ using $+,-$?
This is a follow up to my previous question (see linked question).
In short, there it is shown that if $n$ is prime, then any set can make it.
I want to characterize sets $\mathbb A_n$ of multisets $...
- 12.5k
12
votes
2
answers
506
views
Integers less than $7000$ achievable by starting with $x=0$ and applying $x\to\lceil x^2/2\rceil$, $x\to\lfloor x/3\rfloor$, $x\to9x+2$
Problem
Robert is playing a game with numbers. If he has the number $x$, then in the next move, he can do one of the following:
Replace $x$ by $\lceil{\frac{x^2}{2}}\rceil$
Replace $x$ by $\lfloor{\...
- 707
7
votes
1
answer
154
views
For any $n-1$ non-zero elements of $\mathbb Z/n\mathbb Z$, we can make zero using $+,-$ if and only if $n$ is prime
Inspired by Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n?, I wanted to first try to answer a simpler version of the problem, that considers only two ...
- 12.5k
0
votes
1
answer
118
views
Alcuin's problem of inheritance.
A certain father died and left as an inheritance to his three sons $30$ glass flasks, of which $10$ were full of oil, another $10$ were half full, while another $10$ were empty. Divide the oil and ...
1
vote
0
answers
216
views
A $10$ digit number with distinct digits such that the following holds:
A $10$ digit number with distinct digits is given and using all of its digits two new numbers are created. The sum of the two new numbers is $99999$ and their product is the same as the $10$ digit ...
- 12.7k
0
votes
3
answers
188
views
How to find a number $n$ such that $\frac{n}{\phi(n)} > 10$?
How to find a number $n$ such that $$\frac{n}{\phi(n)} > 10,$$ where $\phi(n)$ denotes the Euler's phi function?
I was trying to find the smallest one, so was keeping each prime once.
I tried with ...
- 12.7k
4
votes
1
answer
218
views
Longest consecutive runs of sums of $k$-subsets of first $n$ primes
Table of contents
[$1.$] Definition
[$2.$] Implication. (Motivation.)
[$3.$] Question. & Computed data.
[$4.$] Solutions of simplified variations.
[$5.$] Progress on solving the question.
[$6.$] ...
- 12.5k
4
votes
0
answers
176
views
Smallest number not expressible using first $n$ powers of $2$ (exactly once each), with $+$, $-$, $\times$, $\div$, and parentheses?
Motivation
Solution to this problem is a lower bound for a more general problem.
Problem
Given first $n$ powers of two: $1,2,4,8,16,\dots,2^{n-1}$ that all need to be used exactly once per number ...
- 12.5k
19
votes
0
answers
533
views
Largest consecutive integer using basic operations and optimal digits?
If you are first time reading this, you may want to read the summary section last.
Solution summary and questions
Sequence values
If the allowed operations are $(+,-,\times,\div)$ and parentheses $(...
- 12.5k
3
votes
0
answers
165
views
Generalization of Four Fours puzzle - optimal set of quadruplets?
Four fours is a math puzzle whose goal is to build numbers out of mathematical expressions using four fours, and a restricted set of mathematical operations and symbols.
Problem
I'm interested in ...
- 12.5k
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
1
vote
1
answer
118
views
Largest smallest value in sudoku like puzzle
In this post, a sudoku like math puzzle is proposed.
The grid must be filled while respecting a unique constraint : the sum of all $3\times3$ sub-squares must equal $2019$.
It is not that difficult ...
- 9,624
0
votes
2
answers
813
views
Common sum in magic square
A magic square of size $N,N ≥ 2$, is an $N ×N$ matrix with integer entries such that the sums of the entries of each row, each column and the two diagonals are all equal. If the entries of the magic ...
- 239
0
votes
1
answer
126
views
Resizing a matrix
I have a problem of resizing a rectangular matrix into a square matrix with certain restrictions:
Suppose you have a rectangle of size $m\times n$, and wlog $m> n$. Now i want to resize this ...
- 2,632
5
votes
0
answers
120
views
Large 2-digit numbers that factor into 2-digit numbers.
$799777779779979$ and $111111811818111$ are numbers of $2$ non-zero digits that can be factored into $2$-digit numbers. They are $97$-smooth numbers.
Are there any larger examples?
- 21.4k
3
votes
0
answers
252
views
Integer-valued Blanche Dissection of a Square
Blanche's Dissection divides a square into 7 noncongruent rectangles of the same area. Seven is the minimum possible.
That solution uses non-integer values. What is the smallest non-trivial ...
- 21.4k
4
votes
1
answer
73
views
On the GCD of two palindromes.
I had an observation. Which I will discuss below. My question will be Is my observation correct? If so, how can one prove it?
Observation:
Consider the string of palindromes below:
$100...01$ and $...
- 2,190
12
votes
1
answer
260
views
On the theorem "$3$ is everywhere"
In this Numberphile video it is stated that "almost all natural numbers have the digit $3$ in their decimal representation", and a proof of this fact is proposed.
A sketch of the proof follows:
...
- 36.9k
13
votes
0
answers
140
views
Every natural number in binary can be cut and added so that it is a power of $2$? [duplicate]
I was watching a google techtalk with Donald Knuth and he mentions for every binary number $\overline{a_1a_2a_3\dots a_n}$ there exists $c_1<c_2<\dots <c_r=n$ so that:
$\overline{a_1a_2\...
- 106k
2
votes
1
answer
521
views
Optimization and Postage Stamp Problem
(1)
Given the set U = {1, 2, 3, ..., 98, 99, 100} of Natural numbers, find the smallest subset S contained in U that:
For every element v belonging to U, there are a, b elements of S, not ...
- 21
1
vote
3
answers
175
views
A set of cards from which I can identify and number $n$ with $1\le n\le N$
I am working with a collection of cards. On each card is written a set of numbers $1\le n\le N$ in ascending order.
When I arrange the cards in lexicographic order on the table, no two adjacent cards ...
- 19
3
votes
1
answer
89
views
Minimal diameter of set of fractions
Let $p_n$ be a pairwise partition of $\{1,2,...,2n\}, n\in \bf N$ where $(a,b)\in p \implies a<b$, and $P_n$ the set of all such pairwise partition. $d(n) := \min_{p_n\in P_n}\big[\max\big(\big|\...
- 9,927
3
votes
1
answer
476
views
Triangle from a given rectangle
We are given a set of (marked) points in a 2D coordinate system and function $f(x,y)$ which counts number of points marked in the rectangle $(0 , 0), (x , y)$ - where $(0 , 0)$ if down-left corner, $(...
1
vote
2
answers
3k
views
How many ways to write one million as a product of three positive integers?
In how many ways can the number 1;000;000 (one million) be written as the product
of three positive integers $a, b, c,$ where $a \le b \le c$?
(A) 139
(B) 196
(C) 219
(D) 784
(E) None of the ...
- 57
6
votes
1
answer
270
views
Constructing Magic Squares over $\mathbb{Z}$ from Magic Squares over $\mathbb{Z}_m$
A magic square over $\mathbb{Z}$ is an n x n matrix whose entries are $\{1, \ldots, n^2\}$, with the sum of every row and column identical (in particular, my magic squares are all normal, but the sum ...
- 7,051