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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21 votes
2 answers
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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 ...
Will Octagon Gibson's user avatar
6 votes
1 answer
220 views

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. ...
Retudin's user avatar
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0 votes
1 answer
133 views

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?
gyancey's user avatar
  • 519
7 votes
2 answers
798 views

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.
gyancey's user avatar
  • 519
5 votes
2 answers
691 views

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 ...
Bernardo Recamán Santos's user avatar
13 votes
1 answer
604 views

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,...
Peter's user avatar
  • 583
12 votes
3 answers
758 views

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 ...
Peter's user avatar
  • 583
9 votes
1 answer
785 views

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 ...
Bernardo Recamán Santos's user avatar
5 votes
1 answer
241 views

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 ...
NODO55's user avatar
  • 761
17 votes
4 answers
3k views

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 ...
Tyler Seacrest's user avatar
16 votes
4 answers
1k views

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 ...
Will Octagon Gibson's user avatar
5 votes
1 answer
248 views

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'...
Bernardo Recamán Santos's user avatar
5 votes
2 answers
509 views

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 ...
Bernardo Recamán Santos's user avatar
6 votes
1 answer
264 views

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 ...
Maff's user avatar
  • 601
20 votes
7 answers
1k views

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 ...
loopy walt's user avatar
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1 vote
2 answers
177 views

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 ...
Maff's user avatar
  • 601
4 votes
2 answers
264 views

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 ...
Maff's user avatar
  • 601
3 votes
2 answers
153 views

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 ...
Maff's user avatar
  • 601
12 votes
1 answer
288 views

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 ...
Maff's user avatar
  • 601
10 votes
1 answer
750 views

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 ...
Bernardo Recamán Santos's user avatar
3 votes
1 answer
109 views

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. ...
Maff's user avatar
  • 601
3 votes
1 answer
211 views

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 ...
Maff's user avatar
  • 601
3 votes
3 answers
379 views

One million positive integers [closed]

How many different (multi)sets of one million positive integers are such that their sum equals their product?
Bernardo Recamán Santos's user avatar
11 votes
5 answers
3k views

Lucas buys five items at a store

My nephew Lucas bought five items at his local store, none for more than $100, and all different prices. He claims that their total cost was equal to the five values multiplied together. Can I believe ...
Bernardo Recamán Santos's user avatar
4 votes
2 answers
311 views

Find the value of $\bigstar$: Puzzle 11 - No such thing as equality

This puzzle replaces all numbers 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 solve Puzzle 1 first....
NODO55's user avatar
  • 761
2 votes
1 answer
147 views

Consider the equation a?b?c =d [closed]

Consider the equation a?b?c =d. Here a, b and c are 3 distinct integers from 0 to 9 (both inclusive) and "?” represents any signs out of “+", "-", “x” of “÷”. Note that the 2 ...
Epic's user avatar
  • 37
7 votes
3 answers
500 views

Avoiding arithmetic progressions in square grids

a) Is it possible to place the integers 1 to 25 in a 5 x 5 grid so that no column or row contains an increasing or decreasing 3-term arithmetic progression (A.P.)? b) Can this be done in a 6 x 6 grid ...
Bernardo Recamán Santos's user avatar
8 votes
2 answers
1k views

Number of divisors equals square root

A colleague of mine mentioned that there was something special about the number nine. He noticed that nine had three positive integer divisors {1, 3, 9} and the square root of nine is three. His ...
Will Octagon Gibson's user avatar
18 votes
7 answers
6k views

Make 2 0 2 2 2 0 2 2 = 2022 [closed]

Inspired by this puzzle, I've come up with the following: Can you find a way to make $2 \; 0 \; 2 \; 2 \; 2 \; 0 \; 2 \; 2 = 2022$ by only adding any of the following operations or symbols: $+,\ -,\...
user avatar
3 votes
2 answers
370 views

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 ...
Charlie's user avatar
  • 635
10 votes
4 answers
2k views

Ages of Widow and Her Children

On New Year's Eve, a census taker gathering information calls a woman and asks specific questions about her family and their (integer) ages. She replies, "I don't like to give out specifics, but ...
JLee's user avatar
  • 17.5k
6 votes
5 answers
984 views

A peculiar number

A five digit number is multiplied by 9, the resulting number is reverse of the given number. What is the five digit number? This question was asked in KVPY 2020, SA.
I'm Nobody's user avatar
  • 1,304
0 votes
1 answer
262 views

Sum of digits of numbers

Let S be a function such that S(N) is the sum of digits of N. N belongs to natural numbers, and N < 10²³. N does not contain a zero digit in it. The numbers are in base 10. Find the number of N ...
I'm Nobody's user avatar
  • 1,304
4 votes
1 answer
495 views

Six Different Rectangles

a) Six different rectangles, none a square, have all integer sides chosen from a, b, c, and d. If I take any two of these rectangles with no common side (there are three ways of doing this), the ...
Bernardo Recamán Santos's user avatar
16 votes
4 answers
2k views

The Game of Barranca

Barranca is played with sixteen cards, numbered 1, 2, ... , 16. Two players alternately choose a card, until each has eight. The winner is the one who has a (sub)set of numbers whose product is 220, ...
Bernardo Recamán Santos's user avatar
6 votes
2 answers
383 views

Splitting the integers 1 to 36

Split the integers 1 to 36 into two sets, A and B, such that any number in set A has a common divisor greater than 1 with no more than two other numbers in A, but for every number in B there are at ...
Bernardo Recamán Santos's user avatar
5 votes
4 answers
1k views

Six positive integers

Find six different numbers (positive integers) such that each of them has a common divisor with precisely three of the other numbers. How small can the largest of the six numbers be? What if $2n$, $n&...
Bernardo Recamán Santos's user avatar
3 votes
1 answer
229 views

A number of ten different digits, divisible by 8, 9, 10, and 11

Each of the digits 0 through 9 is used exactly once to create a ten-digit integer. Find the greatest ten-digit number which uses each digit once and is divisible by 8, 9, 10, and 11.
Akshaya Gunnam's user avatar
-6 votes
1 answer
163 views

The peculiar inequality [closed]

Let's have the following relation $\sqrt[3]{\frac{(x+1)^3+x^3}{2}}\lessgtr\frac{2x^2+2x+1}{2x+1}$ where $x$ a positive integer greater than zero. Which inequality is valid?
Vassilis Parassidis's user avatar
10 votes
2 answers
556 views

Permutations of first 10 natural numbers such that all the prefix sums are distinct

I posted this question on Math SE as well. Did not receive any help. This is a question that I was asked in a Quant Interview. I would like you all to have a crack at this. I could not find a problem ...
bigbang's user avatar
  • 201
5 votes
1 answer
380 views

Insert Plus Signs and Add

If you take any integer (in base 10) and insert plus signs, "+", in between its digits (as few or as many as you like), and carry out the indicated sum, you will end up with a smaller number ...
Bernardo Recamán Santos's user avatar
1 vote
1 answer
103 views

Equal row-products and column-products in a given array [closed]

I don't know if this is the right place to ask this question, but I'm stuck on this and can't figure out how to even proceed. Any hints anyone? Is it possible in a 5 × 5 array of integers for all row ...
happysoul09's user avatar
12 votes
1 answer
1k views

A hidden number everyone is talking about

The following describes an 8-digit positive integer. Identify this number, and explain the title of this puzzle. The number is in the form of 2021____. It has 24 ...
Bubbler's user avatar
  • 11.5k
3 votes
1 answer
353 views

How many consecutive integers to ensure one has digit sum divisible by 19?

How many consecutive positive integers are at least required, such that there is always a number in such a sequence whose sum of digits is divisible by 19?
ThomasL's user avatar
  • 11.9k
12 votes
2 answers
1k views

Introducing S-sequences: which is the shortest to contain all integers 1 to 20?

Consider a sequence (finite or infinite) of different positive integers, such as the following, in which the first term is 1, and thereafter the nth term is either the previous term plus n, minus n, ...
Bernardo Recamán Santos's user avatar
4 votes
2 answers
635 views

The Divisibility Graph... Again!

The divisibility graph of a set of positive integers is the graph whose vertices are the integers, two of which are joined by an edge if one divides the other. What is the smallest positive integer ...
Bernardo Recamán Santos's user avatar
4 votes
1 answer
198 views

Guessing Two (or Three) Different Integers

I am thinking of two different positive integers between 1 and 100 (both inclusive). At most how many questions do you need to ask to find my two numbers if I will answer your questions truthfully and ...
Bernardo Recamán Santos's user avatar
8 votes
2 answers
751 views

Divisibility Graph

The divisibility graph of a set of integers is the graph whose vertices are the integers, two of which are joined by an edge if one divides the other. What is the largest integer N such that the ...
Bernardo Recamán Santos's user avatar
7 votes
2 answers
282 views

Positive integers as sum or difference of consecutive square numbers

Is it possible to represent each positive integer n in the form $n=\pm1^2\pm2^2\pm3^2...\pm m^2$ ? Examples: $1=+1^2$ $2=-1^2-2^2-3^2+4^2$ $3=-1^2+2^2$ $4=-1^2-2^2+3^2$
ThomasL's user avatar
  • 11.9k
10 votes
2 answers
746 views

Splitting the Integers

For which n is it possible to split all the integers 1, 2, 3, ..., n into two non-empty disjoint sets such that the product of the sum of the elements in one set and that of those in the other is a ...
Bernardo Recamán Santos's user avatar

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