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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5 votes
1 answer
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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 ...
Vincent's user avatar
  • 53
4 votes
1 answer
433 views

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 ...
Alexander's user avatar
  • 595
19 votes
2 answers
1k views

Which expression is larger?

Without using a calculator/computer, please show steps to determine which expression is larger: $(2^{90})!\quad vs.\quad2^{30!}$
Olive Stemforn's user avatar
4 votes
1 answer
255 views

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: ...
user1134699's user avatar
29 votes
4 answers
3k views

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 ...
Bernardo Recamán Santos's user avatar
19 votes
1 answer
1k views

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 ...
Bernardo Recamán Santos's user avatar
-1 votes
1 answer
236 views

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 ...
Ali MohammadiNia's user avatar
-2 votes
1 answer
233 views

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 ...
CrSb0001's user avatar
  • 2,241
12 votes
2 answers
524 views

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 ...
godlification's user avatar
9 votes
2 answers
998 views

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 ...
godlification's user avatar
6 votes
2 answers
338 views

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 ...
Bernardo Recamán Santos's user avatar
12 votes
1 answer
968 views

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 ...
godlification's user avatar
15 votes
3 answers
1k views

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, ...
Bernardo Recamán Santos's user avatar
22 votes
2 answers
2k views

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
7 votes
1 answer
248 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
  • 8,601
0 votes
1 answer
154 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
815 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
1k 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
15 votes
1 answer
637 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
  • 613
12 votes
2 answers
790 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
  • 613
10 votes
1 answer
1k 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
286 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
6 votes
1 answer
254 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
956 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
7 votes
1 answer
270 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
  • 621
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
  • 21.2k
1 vote
2 answers
213 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
  • 621
4 votes
2 answers
276 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
  • 621
3 votes
2 answers
193 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
  • 621
12 votes
1 answer
291 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
  • 621
11 votes
1 answer
1k 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
117 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
  • 621
3 votes
1 answer
296 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
  • 621
3 votes
3 answers
484 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
332 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
167 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
  • 47
7 votes
3 answers
965 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
375 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
  • 18.2k
6 votes
5 answers
1k 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
  • 1,334
0 votes
1 answer
266 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
  • 1,334
4 votes
1 answer
867 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
766 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

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