Questions tagged [number-theory]

A mathematical puzzle whose solution is heavily based on the arithmetic properties of the integers. General number theory questions are off-topic but can be asked on Mathematics Stack Exchange.

Filter by
Sorted by
Tagged with
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?
2
votes
4answers
352 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 ...
1
vote
1answer
127 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
242 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
102 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
165 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. ...
17
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 ...
0
votes
1answer
444 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
218 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
726 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
269 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
112 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.&...
9
votes
1answer
367 views

The Lucky Number

Lucky numbers are 4 digit numbers that have the following property: they are equal to the sum of the fourth power of their digits. Therefore, they can be expressed as follows: $$1000a+100b+10c+d = a^4+...
12
votes
2answers
432 views

The beginning of a factorial

Is there a positive integer n such that the decimal representation of n! starts with 123456789?
5
votes
2answers
340 views

Seventeen positive integers

Find 17 positive integers such that no four of them have, pairwise, a common divisor greater than 1, but, likewise, no four of them are, pairwise, relatively prime. Do so in such a way that the ...
1
vote
1answer
138 views

On certain triplets of consecutive integers [closed]

While completelely factorizing integers, my student Luciana noticed that the canonical prime factorization of the three consecutive numbers 81=3^4, 82=2x41, and 83=83, use numbers which are all ...
7
votes
3answers
212 views

Perfect power nim

Let $m,n$ be positive integers. Ann and Ben has $m$ stones, and each of them takes exactly the perfect power of $n$ stones ($n^k$, where $k$ is a nonnegative integer) in order, starting from Ann. Who ...
3
votes
1answer
130 views

Fill in numbers on the cube … again!

You are given a cube. You are told to fill in each face randomly with some of the numbers $4, 5, 6, ..., 11$, with no repetition. What is the probability that for each two faces that are connected by ...
3
votes
1answer
126 views

Inheriting my father's “great” investment

My father, you see, was frankly quite gullible. Many years ago he fell for one of these "investment" scams. The scam promised to exactly double your investment every month, and excited by ...
4
votes
1answer
163 views

Fill in numbers on the cube!

You are given a cube. You are told to fill in each vertex with the numbers $4,5,6,...,11$, with no repetition. What is the probability that for each two vertices that are connected by a common edge, ...
5
votes
3answers
861 views

4k reputation special: “I hate square numbers!”

There is a large prison, with exactly 4000 prisoners. The warden noticed that there were too many prisoners, so they lined up all the prisoners, and repeated the following procedure until less than ...
10
votes
5answers
763 views

Length of an integer is part of its digits [closed]

How many positive integers less than 1,000,000,000 contain their length as part of their digit string? Example: 123466 has a length of 6 and 6 is one of its digits. Hence this number needs to be ...
7
votes
3answers
364 views

The 15 Pebbles Game

This is a game for 2 players - Each player uses a different coloured marker or pencil, there are 15 pebbles in total. Players take turns to colour 1, 2 or 3 pebbles (player chooses how many). When all ...
3
votes
2answers
230 views

Cannonballs packing

A pile of cannonballs stacked like a pyramid has a rectangular base. Each layer has a length and a width in terms of cannonballs that are each one less than those of the layer that is directly below. ...
13
votes
8answers
2k views

n(n+1) as a multiple of 100

Here's a puzzle I came up with while walking today: For how many natural numbers $n$ is the number $n(n+1)$ a multiple of $100$? This is true for infinitely many $n$, so "how many" means something ...
1
vote
2answers
125 views

Find a factor #1

The series of Find a factor puzzle is started by Culver Kwan, and asks the solver to identify a factor of a certain large number within a certain range using some mathematical identities. This should ...
9
votes
0answers
358 views

Dead By Daylight

This puzzle is a reference to a game, Dead By Daylight. Though you don't need a knowledge about the game. Consequently, there is no "video-games" tag. Story The Entity wants to have the ...
4
votes
1answer
172 views

What's the graph relation? #3

What's the relation that joins the nodes? Previous What's the graph relation? #1 What's the graph relation? #2
6
votes
2answers
385 views

Primes in a Line

Place the first 20 primes (2 to 71) in a line so that the sum or difference (or both) of any two primes that find themselves next to each other is always a perfect square. For which other values of N ...
6
votes
1answer
156 views

Follow the path of relation through the grid #3

There is a relation between rectilinear-adjacent squares such that there is a unique rectilinear path from the top-left corner of the grid down to the bottom-right corner of the grid. Each square can ...
7
votes
1answer
179 views

Follow the path of relation through the grid #2

There is a relation between rectilinear-adjacent squares such that there is a unique rectilinear path from the top-left corner of the grid down to the bottom-right corner of the grid. Each square can ...
10
votes
2answers
274 views

Add a divisor! A game

Let $k$ be a positive integer. Amy and Ben are playing a game, with the number $1$ written on the whiteboard initially. Amy and Ben do the following in order, starting with Amy: Suppose the number on ...
11
votes
1answer
300 views

Floor floor floor inside another floor

Inspired from this question $$ \aleph(x,n)=\lfloor x\lfloor x\lfloor x...\rfloor\rfloor\rfloor\ $$ where $\aleph$ is the inner floor function with $n$ times for $x$. For example; $$ \aleph(x,3)=\...
7
votes
3answers
189 views

The ceiling function and powers of 2

How many integers $1\le x\le2048$ such that $$\Big\lceil \frac x{2^n}\Big\rceil$$ is not a multiple of five for all nonnegative integers $n$? This problem is a 2020 contest problem which has finished....
0
votes
0answers
38 views

Brute force a keypad with minimal keystrokes [duplicate]

Senario Say you have a keypad whose password is some two digit code which you do not know, say 34. Entering digits in succession on this keypad eg. ...
45
votes
4answers
6k views

x⌊x⌊x⌊x⌋⌋⌋ = 2020

Solve for $x$: $$ x \left\lfloor x \left\lfloor x \left\lfloor x \right\rfloor \right\rfloor \right\rfloor = 2020. $$ The floor function $\left\lfloor t \right\rfloor$ has the usual “greatest integer ...
5
votes
2answers
391 views

Consecutive numbers which use all digits a different number of times

Are there arbitrarily long sets of consecutive numbers such that when writing the set down, every single digit (0 to 9) is used a different number of times?
4
votes
3answers
188 views

The death prism

One day, you are caught by a evil wizard. He presents you with a prism, and says, "You can ask me to turn this prism to any $n$-angled right prism. Then you shall fill in $1$ to $3n$ with no ...
6
votes
1answer
208 views

What's the graph relation? #2

What's the relation that joins the nodes? Open the image in a new tab if you'd like to see the diagram with better resolution. Previous What's the graph relation? #1 Hint 1
4
votes
1answer
98 views

What's the graph relation? #1

What is the relation that connects the nodes of this digraph?
6
votes
4answers
1k views

What is larger than largest, your intuitions are tarnished. (What am I?)

You might call me a number, but that would be a blunder. In class you may have been told otherwise, but I tell you now that those were all lies. I'm in the deck of the cards, as big as them come. ...
8
votes
0answers
315 views

I'm a number with a special product, so name me when you think you've got it

In the blocks that come before, their special product tells us more. To guess my scheme you'll need calculation, but only little tests of recreation. Each block contains atoms strong as Thor, ...
10
votes
2answers
992 views

First digit of 2020!

This is a follow up to First digit of 3^2020 Can you find the first digit of 2020! (factorial) without a computer?
33
votes
3answers
5k views

First digit of 3^2020

Inspired by The last digit for 3^(2019) Can you find the first digit of $3^{2020}$ without a computer?
6
votes
2answers
164 views

The Balls of Death

You have nightmares about a pool ball with the number 1 on it, and an empty box. Why? Immortality Imagine, if you will, we are 5000 years into the future. Medicine has evolved and we now are immortal -...
2
votes
2answers
762 views

The last digit for 3^(2019)

Which would be the last digit for $3^{2019}$ ? You can And afterwards
21
votes
4answers
2k views

What's in my pocket?

Well, I can tell you Johnny has memory cards in his pocket. Back Story My brother, Johnny, is a tech nerd. He loves gadgets of all kinds. As a matter of fact, you can be sure at any one given time, he ...
9
votes
3answers
448 views

Number Guessing (Part 1)

I thought up two positive integers with product less than $500$. I told their product to Penny, and their sum to Sandy, and told both of them the constraints and they are both perfect logicians. They ...
5
votes
1answer
10k views

How to programmatically solve math puzzle

I have this puzzle and I want to solve by code. I wrote simple code trying to brute force but it fail in one condition https://dotnetfiddle.net/cJyu3w Anyone know of a link or source how to solve ...
20
votes
1answer
1k views

Four mathematicians and their ages

Four mathematicians, none yet a centenarian, meet for coffee. The graph-theorist among them noticed that the common divisor graph of their ages (that is, the graph whose vertices are their ages, two ...

1
2 3 4 5 6