# Questions tagged [number-theory]

A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. Use with [mathematics]

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### 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 ...
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 ...
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?
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 ...
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) ...
783 views

### Inequality derived from a famous problem

Let's have the following inequality: $\frac{2}{3}(\sqrt 5-1)^3\lessgtr\sqrt{2}$. Which part is greater, the left or the right? No calculator solutions are accepted.
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 ...
133 views

### How is this correct? [duplicate]

How is the following equation is correct?$29$ - $1$ = $30$ Hint-
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 ...
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 ...
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?
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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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’...
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 ...
94 views

### First digit of 2021^2021 [duplicate]

Can you find the first digit of $2021^{2021}$ without a computer? Good luck!
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 ...
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 ...
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 ...
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 ...
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 $x$ ...
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 ...
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 ...
138 views

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### 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 ...
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 ...
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 ...