Questions tagged [number-theory]
A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. Use with [mathematics]
303
questions
2
votes
2answers
175 views
Powerful Octagon
Place different integers on the vertices of an octagon so that the sum of the integers in any two vertices joined by one of its edges is a power of 2. Do so in such a way that the largest integer used ...
2
votes
1answer
272 views
Squares and chords in a circle
The whole numbers 1 to 2n are placed in order around a circle. For which n is it possible to draw n non-intersecting chords (one from each number) such that each of them joins two numbers whose sum ...
-2
votes
3answers
164 views
Sequence with all terms divisible by 8
Let's have the following infinite sequence
3968, 13224, 30624, 59048, ?
What is the next term, replacing the question mark?
Why are all the terms of this infinite sequence divisible by 8?
7
votes
1answer
309 views
Can you distribute the balls equally into 2 boxes?
You have 2 boxes and an even number ($2n$) of balls in the first box. Your goal is to distribute the balls equally into the two boxes, so that each box contains $n$ balls. You must obey the following ...
8
votes
1answer
646 views
The largest Saturday number [closed]
No weekend love yet shown, therefore I will fix that.
A Saturday number is a number in which for all $1 <= i <= l$, where l is the length of the number, the first $l$ digits (from the left) ...
7
votes
1answer
783 views
Inequality derived from a famous problem
Let's have the following inequality: $\frac{2}{3}(\sqrt 5-1)^3\lessgtr\sqrt[3]{2}$.
Which part is greater, the left or the right? No calculator solutions are accepted.
7
votes
2answers
1k views
Coloring positive integers 'black or white'
Each of the positive integers from 1 to n is colored either black or white.
You can repeatedly choose a number m and recolor m together with those numbers, which are not coprime to m. At the beginning ...
0
votes
2answers
133 views
-1
votes
1answer
90 views
Consecutive number division puzzle 2 [closed]
Find 4 consecutive numbers that divide 𝑤, 𝑥, 𝑦, 𝑧 respectively, where 𝑤, 𝑥, 𝑦, and 𝑧 are also 4 consecutive positive numbers greater than 1, or prove it's impossible.
Bonus: What if w > the ...
1
vote
1answer
241 views
Pythagorean triplets generated in a unique way
Let's have the following sequence of Pythagorean triplets $25^2=24^2+7^2,1201^2=1200^2+49^2$,
$58825^2=58824^2+343^2, ?, ?$
What are the next two triplets in this sequence?
How have these triplets ...
5
votes
3answers
184 views
Numbers with minimal sum at the vertices of a cube
The eight vertices of a cube are marked with numbers from 1 to 8 such that
the sum of any three numbers on any face is not less than 10.
What is the minimum sum of the four numbers on a face?
4
votes
1answer
77 views
A 4x6 grid with adjacent integers with gcd > 1
You are given a 4x6 square grid. Each square of the grid
should be filled with different positive integers.
The gcd (greatest common divisor) of any two adjacent (horizontally or vertically)
squares ...
3
votes
1answer
80 views
Divisors ending with digits 0-9 each
What is the smallest positive integer, which has - for each of the digit 0-9 - a divisor ending with this digit?
0
votes
1answer
90 views
Self-indulgent numbers
Let's call a positive integer N self-indulgent of degree K>2 if for every positive integer k<K the following is true:
More than half of the first k multiples N,2N,...,kN of N contain with ...
3
votes
2answers
429 views
Find X and Y so that they are never equal [closed]
In a game, your opponent is given an ordered pair of integers (X, Y), and at each step, they can either double X and add one to Y or double Y and add one to X. Here's an example sequence of steps that ...
9
votes
2answers
447 views
Construction of positive integers by given rules
For a positive integer n there are two operations defined:
append one of the digits 0, 4 or 8 at the right end of n
n can be divided by 2 if n is even
Start number is 4. Is it possible to construct ...
2
votes
1answer
136 views
Four-Number Door Puzzle
So I had an idea for a number-based door puzzle for a TTRPG campaign that could readjust itself every time a wrong guess is made. Here's the basic premise:
Given two numbers, find two more numbers in ...
4
votes
2answers
255 views
All possible locations of a robot going from $(x,y)$ to $(x+y, y)$ or $(x,x+y)$ [closed]
Suppose I had a little robot on the coordinate grid that moves according to the following rule. If it's at the point $(x,y)$, it can move to either $(x+y,y)$ or $(x,x+y)$. If the robot starts at the ...
6
votes
2answers
340 views
Twin primes and divisibility
Let $p$ and $q$ be a pair of twin primes. Find the smallest possible value of $a+b$ where $a$ and $b$ are positive integers such that $p\;|\;(a+qb)$ and $q\;|\;(a+pb)$.
This puzzle is my own ...
9
votes
2answers
612 views
A Triangle of Squares
Let $T(n) = 1 + 2 + 3 + \text{...} + n$ be the $n$th triangular number.
For which $n>1$, if any, is it possible to split the first $\frac{n(n+1)}{2}$ positive integers into $n$ sets, all of ...
9
votes
4answers
941 views
How to find the 2021st integer co-prime to 15
I recently saw a puzzle where you were to find the 2021st positive integer co-prime to 15 (it was phrased in terms of a game but this is the mathematical core). I wrote code to find the answer but can’...
-3
votes
1answer
133 views
Add or subtract 212 in octal to get a palindrome
The puzzle is as follows:
Suppose you have a three-digit number in the octal system. If you add
or subtract 212 (also in the octal system) from that initial number,
you get a three-digit palindromic ...
0
votes
0answers
94 views
First digit of 2021^2021 [duplicate]
Can you find the first digit of $2021^{2021}$ without a computer? Good luck!
17
votes
3answers
2k views
What makes this polynomial a square number?
For which integer values of $x$ is $x^4+x^3+x^2+x+1$ a square number? Please include a proof that the polynomial cannot be a square number if $x$ is not one of your answer(s).
Source: a math olympiad ...
4
votes
1answer
184 views
More primes and squares, in a summation triangle
Place a different prime number or perfect square in each of the twenty-one
disks that make up the triangle below, so that the number in any disk
that lies on two others is precisely the sum of the ...
5
votes
3answers
354 views
A Circle of numbers
Just saw a Circle of numbers on my Whats App message (source not listed) which is as following
Arrange numbers 1 to 32 in a circle such that any two adjacent
(neighboring) numbers add up to a perfect ...
11
votes
4answers
436 views
Not so boring numbers
This is inspired by this and this and more similar ones.
Let us consider formation of a given positive integer N by the following rules
You may use only one digit, which one is your choice and you ...
7
votes
2answers
469 views
When do decimal-coded binary numbers XOR to zero?
Background definition: XOR on numbers
Given two non-negative integers $x$ and $y$, let $x\oplus y$ denote the bitwise exclusive or (XOR) of the numbers $x$ and $y$. This is the result of writing $x$ ...
9
votes
3answers
383 views
A theorem about angles in the form of arctan(1/n)
There is a famous classical geometry puzzle about the angles formed by integer coordinates:
What is the sum of angle A and B in the following image? Do not use any advanced mathematics such as ...
8
votes
2answers
565 views
Multiple of 3 in any numeric base
Can you find a positive binary number that is a multiple of 3 when it is read in any base from 2 to 10? A binary number contains only digits 0 and 1. For example the binary number "11" is 3 ...
0
votes
1answer
138 views
The mysterious fractions
Let's have the following fractions.
$ \frac{752}{375} + \frac{754}{376}+ \frac{756}{377} + \frac{758}{378}+ \frac{760}{379} \approx 10\times(\frac{5}{375}+1)^{1/5}$
$\frac{752}{375}+ \frac{754}{376}+ \...
-16
votes
1answer
249 views
Equation $X^4-DY^4=Z^4$ (Part 1) [closed]
Let's have the positive integers X,Y,Z. The number D is a terminating decimal always. The numbers X,Y,Z do not have a common factor. Based on the above information, can you solve the following ...
-17
votes
1answer
350 views
Solving the equation X^4 - DY^4 = Z^4 [closed]
Let's have the equations $12^4 - DY^4 = 7^4$ and $24^4 - DY^4 = 19^4$. For what values of D and Y do these equations have a solution?
Secondly, what little trick is required to obtain solutions of the ...
7
votes
4answers
691 views
Sum of digits of sum of digits of sum of digits
The following question was asked in a competitive exam for which I am preparing and I was unable to solve it (in fact I am completely clueless about it).
So, I am asking for help here.
Given a number ...
5
votes
2answers
464 views
Paying bills in Alphagonia
In the Kingdom of Alphagonia, where the national currency is the Alpha, banknotes are available in all whole-number denominations of alphas: 1, 2, 3,...
a) What is the least number of such notes a ...
asked Nov 7 '20 at 13:53
Bernardo Recamán Santos
13.6k2 gold badges23 silver badges123 bronze badges
2
votes
1answer
196 views
Puzzle regarding emptying of cup!
Initially, I have $3$ cups with infinite capacity and some prefilled amount of water(positive integers). I can do only one operation repeatedly by choosing any $2$ out of $3$. The operation is that if ...
-1
votes
1answer
93 views
Fractions with different denominators
Lets have the following fractions.
$121393/28657$
$121393/17711$
$121393/10946$
$121393/?$
What number is on the denominator of the last fraction where the question mark is?
Also, which algebraic ...
5
votes
2answers
137 views
Crossnumber Puzzle
Across:
1. Sum of consecutive integer powers of 21 Across
7. Prime number, not all of whose digits are prime
8. Number that is coprime with 13 Down
9. Sum of three consecutive primes
10. Noble gas ...
9
votes
4answers
2k views
Integers whose arithmetic mean equals their geometric mean
For which positive integers n is it possible to find n integers whose arithmetic mean equals their geometric mean?
0
votes
4answers
378 views
Curious relations between numbers
Lets have the numbers $454+2\sqrt{457}, 16+8\sqrt{85}, 460+4\sqrt{457}, 83+\sqrt{85}, 14\sqrt{457}+42 , 87+3\sqrt{85}$.
How are these numbers related?
How are such numbers generated?
HINT 1:
What ...
0
votes
1answer
144 views
Unusual connections of numbers
Let's have the equation $(DX)^2-Y^2= ± Z^5$ and $x,y$ two positive integers greater than zero. From some facts we can obtain solutions of the above equation by giving integer values at $x,y$. Examples:...
7
votes
1answer
258 views
Slim at any size?
Recall from ŧhis question that we call a positive integer slimdownable or slim for short if it is part of a sequence of integers where each is followed by itself divided by its length, i.e. its number ...
1
vote
1answer
136 views
Break into Goldbach's safe
You need to unlock a safe by typing in the correct password. All you have is the following note:
...
5
votes
2answers
172 views
Highest n where an equal number in all cells is (im)possible
Inspired by Board with all 2020s :
Zeroes are written in all cells of a n×n board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it.
...
18
votes
5answers
1k views
Board with all 2020s
Zeroes are written in all cells of a $5 \times 5$ board. We can take an arbitrary cell and increase by 1 the number in this cell and all cells having a common side with it. Is it possible to obtain ...
-1
votes
1answer
485 views
How are these numbers related?
Let's have the following numbers.
34932, 52428, 10023, 1881, 512, 64764, 63012, 57825, 59367, 65508, 30840, 55449, 18009, 65537, 20148, 39321, 62361, 27756.
(1) What are the relations between these ...
4
votes
1answer
226 views
Two integer prisms
Two rectangular prisms have the same height, but one is 38 times bigger than the other. They all have integer edge lengths and the diagonals on their faces also have integer lengths. What is the ...
27
votes
1answer
761 views
Shifting a digit from right to left
A positive integer n (without leading zeros) has the property that shifting the rightmost digit of n to the left end doubles the number.
Examples: 1->1, 1234->4123, 2020->202
What is the ...
9
votes
2answers
295 views
How to make 2 Euros with smaller coins
You are given n > 0 of each of the standard denomination Euro coins: 1 ct, 2 ct, 5 ct, 10 ct, 20 ct, 50 ct, 1 Euro, 2 Euro.
What is the smallest n such that it is impossible to select n coins that ...
-2
votes
1answer
126 views
How many pigeons are in the flock? [closed]
A crow reaches a flock of pigeons. the crow asks the pigeons' leader: "How many of you are there?"
The pigeon replies: "We and we and a half of we and a fourth of we and you equal 100.&...
