# 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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### Find the missing number in 3×3 square [closed]

8 12 9 10 7 20 3 10 ? Find the missing number from the 3×3 square.
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### The Bogotá Marathon

a) Runners at the coming Bogotá Half Marathon have been assigned consecutive numbers starting at 1. Lorena, running for the first time, has noticed that the sum of the numbers of all the athletes with ...
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### Guess the Permutation

Let $p$ be a prime. I chose a secret permutation $a_0,a_1,...,a_{p-1}$ of $0,1,...,p-1$ unknown to you. Now, you can ask the following types of questions to me: Type $1$: Tell me two integers $i,j$ ...
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A positive integer is said to be an Ecuadorian number if for some m, the sum of the first m of its n digits, m < n, is equal to the sum of all its other digits. Numbers such as 11, 134, 235, 2024 ...
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### Geometry Puzzle: Tangent Circles with Integer Radii

Take as a semi-related example a series of circles with radii 10, 9, 8, ..., 2, 1. Place the first (largest) circle in the center and subsequent circles around it, keeping tangency between subsequent ...
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### The 50 game between two players, selecting numbers between 1 and 10 inclusive + variations

This is a cross-post from MSE: https://math.stackexchange.com/questions/4900281/the-50-game-between-two-players-selecting-numbers-between-1-and-10-inclusive Let's play a game with two players, with ...
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### Number operation problem - find minimum number of operations

On the blackboard, there are nine numbers from 1 to 9. In each operation, two of the numbers are chosen, erased, and replaced with their sum and difference. If two identical numbers appear on the ...
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### 75 integers are squared or cubed: minimum distinct results? [closed]

75 different integer numbers are written on a blackboard. Each is erased and replaced with either its square or its cube, the operation being random for each. What is the minimum quantity of different ...
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### Which expression is larger?

Without using a calculator/computer, please show steps to determine which expression is larger: $(2^{90})!\quad vs.\quad2^{30!}$
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### mathematical magic trick

Here’s a conversation that took place between two students. A: Hey, want to see a magic trick? B: Sure, how does it go? A: Think of any number. Any nonnegative integer, I should say. B: Okay. A: ...
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### Melissa's Numbers

Melissa has noticed that certain positive integers can be written as the product of other integers, none of which uses any of the digits in the number itself, 512, for example (512 = 4 x 4 x 8). Ten ...
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### An Amazing Configuration

Ed Pegg found in December 2019 this amazing configuration consisting of 22 points in 28 lines of 4. On those points place 22 different positive integers such that the sum of any of the four points in ...
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### Henry Ernest Dudeney puzzle

An officer explained that the force to which he belonged originally consisted of 1000 men, but that it lost heavily in an engagement, and the survivors surrendered and were marched down to a ...
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### First five digits of a googol factorial

Based off of: MacPOW 1134 Without using a calculator/computer, can you determine the first five digits of $10^{100}!$ (a googol factorial)? I came up with this when trying to solve the question this ...
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### sums and differences in consecutive grid [closed]

Fill in each square of the grid with a number from $1$ to $16$, using each number exactly once. Numbers at the left or top give the largest sum of two numbers in that row or column. Numbers at the ...
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### magic square operations

Fill in the grid with the numbers $1$ to $6$ so that each number appears exactly once in each row and column. A horizontal gray line marks any cell when it is the middle cell of the three consecutive ...
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### While 2024 arrives

There are about $9.266 \times 10^{45}$ partitions of 2024, a handful! To each of these partitions corresponds a graph in which the vertices are each of the parts, two of which are joined by an edge if ...
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### Relatively prime numbers

Can you fill in the circles with numbers such that: Each pair of circles connected by one line contains relatively prime numbers Each pair of circles connected by two lines do not contain relatively ...
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### Primeable numbers

Say a positive integer is primeable if it is prime or some permutation of all its digits (leading 0s allowed in permutations) is a prime. Thus the first few primeable numbers are 2, 3, 5, 7, 11, 13, ...
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### Can you find a 3x3 white square somewhere in this relatively prime graph?

This puzzle comes from: http://skepticsplay.blogspot.com/search/label/puzzles Wow, it's been some time since I've posted a puzzle! Here's a simple pure math puzzle off the top of my head. Back in ...
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### Intermingled primes

This puzzle is part of the Monthly Topic Challenge #14: Think inside the (very small) box!. 6 different, 3 digit primes are stacked here in two layers. You only see the sum of overlapping digits. ...
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### The Triangular Cannonball Problem [closed]

How many ways are there to stack an equilateral triangle of cannonballs into a tetrahedron of cannonballs? In other words, how many positive integers are both triangular and tetrahedral?
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### Vector Sum of Pythagorean Triples

Given any finite set of linearly independent Pythagorean Triples, show that the vector sum of this set is never a Pythagorean triple.
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### Number of 1's needed to write all primes up to P

i) Find, if it exists, a prime P such that the number of 1's used to write all the primes from 2 to P is precisely P. ii) Are there infinitely many such P? If not, find them all. These questions ...
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### Self-referential sequence that is sometimes powers of two

I've created an integer sequence where, after the first two elements, every element is calculated using the previous two. If the first two numbers are $1$ and $3$, the sequence goes as follows: 1, 3,...
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### What do 84, 96 and 108 have in common?

There's a certain property that's shared between (as far as I know) infinite positive integers including 84, 96 and 108. Below are the first thousand numbers with this property; I added that many in ...
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### Villeta's Soup of Primes

i) Hidden in this 8 x 8 board are the first 31 primes starting with 2 and up to to 127. They occupy adjacent, non-overlapping cells (up to 3), and are read horizontally (from left to right) or ...
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### Find the value of $\bigstar$: Puzzle 12 - Not enough variables

This puzzle replaces all numbers (and operations) with other symbols. Your job, as the title suggests, is to find what value fits in the place of $\bigstar$. To get the basic idea, I recommend you ...
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### How abundant can a number get?

Famously, a perfect number is equal to the sum of its proper divisors. For example, 28 is equal to 1 + 2 + 4 + 7 + 14. If the sum is more than the original, the number is called abundant, and if the ...
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### An Almost-squarish set of numbers

A set of numbers is called Almost-squarish if it satisfies the following two properties: The set contains only positive integers. The product of any two distinct numbers in the set is one less than a ...
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### Self-numbers and repunits

Self-numbers or Colombian numbers (A003052 in the OEIS) are natural numbers which are not the sum of a smaller number and the sum of its digits. Repunits (in base 10) are numbers consisting only of 1'...
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### Super Star Numbers

A Super Star Number is a positive integer N, such that the 21 vertices of the super star below can be labelled with different positive integers so that the product of the three numbers in any of its ...
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### Same sequence interwoven with itself creating groups

If we take all the digits from 1 to 9 and lay them out in order. 123456789 Now repeat the sequence and add it to the end. 123456789123456789 Now let's copy and reverse everything to create a second ...
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### Save now! All the digits at half the price

... or double the price depending on where you're coming from Consider the set $PD10$ of pan-digital ten-digit numbers, i.e. positive whole numbers whose decimal representation has each of the digits ...
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### How Many Magic Hexagons that use repeated digits?

There exists only 1 normal magic hexagon that uses non repeating consecutive digits for 1 to 19. If We allow digits to repeat we can create something like this hexagon that is made up using ...
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### Magic Hexagon 0 + 1 to 9 twice

Consider the following image. Within the grid the are a total of 19 cells. We have one cell for zero leaving 18 cells. Shading nine cells we create 2 sets of the digits 1 to 9. With one set being on ...
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### Smallest Magic Hexagon Using Repeat Digits

Consider this image below. Its a magic hexagon using repeated digits to create a magic sum of 10. All rows columns and diagonals, meaning the cells in any straight line through the hexagon in any ...
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### Square Sum Problem Summing 3 consecutive digits along the line

In this image from a numberphile video we see a sequence of numbers from 1 to 15 without repeats where any pair of neighbouring digits sum together to make a perfect square number. 15 is the lowest ...
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### Numbers whose product of digits is a multiple of sum of digits

Find three consecutive numbers, greater than 10 and none with a digit 0 in it, each of which is such that the product of its digits is a multiple of the sum of its digits. What about four or more such ...
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### Digital Digits Magic Square 3x3 that can be rotated 180 degrees

In the below image we have a magic square of a size 3x3. The magic number for all its rows, columns and both diagonals is 165. Rotate the grid 180 degrees and all sums still have the magic number 165. ...
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### Smallest 3x3 Magic Square of different square sums

Consider the follow magic square highlighted in yellow. The sum of its rows and columns are in green and the sum of the diagonals in red. All of its sums are a square number with the sum of the whole ...
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### One million positive integers [closed]

How many different (multi)sets of one million positive integers are such that their sum equals their product?
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### allocation of infinity

Suppose you have a hotel which has one floor with infinite number of rooms in a row and all of them are occupied. A new customer wants to check in, how will you accommodate her? What if infinite ...
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