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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.

6
votes
2answers
270 views

Swap — A Puzzle I Created

This puzzle is called Swap. Let's find out why! Suppose you are given a random $\rm N\times N$ matrix (grid) with all the integers from $1$ to $\rm N^2$ each belonging in every grid square (a.k.a. ...
-7
votes
2answers
168 views

What number + 1 equals itself? [closed]

Answer in numerical form. The answer is not an integer. The answer is in string form. The answer is not my love life...
7
votes
2answers
1k views

An Accountant Seeks the Help of a Mathematician

The accountant complaints to the mathematician: “I lent money to five other faculty members and still haven’t been paid back. You are one of them; the other four owe me 12 dollars altogether, but ...
19
votes
4answers
670 views

Consecutive integers around a circle

Find a block of positive consecutive integers that can be placed around a circle in some order so that any two adjacent numbers always have a common divisor greater than 1.
8
votes
5answers
635 views

The Legend of Four

As far as I know, all numbers have a root of 4. What I mean by this is as follows: Starting with any number, for example 384, I take the number of letters in that number. Then I repeat this process ...
13
votes
2answers
716 views

Digit sums of successive integers

For a natural number $x$ both, the digit sum of $x$ and the digit sum of $x+1$ are multiples of $7$. What is the smallest possible $x$?
2
votes
2answers
361 views

Can the sum, difference and product of 2 numbers be perfect squares? [closed]

If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?
3
votes
2answers
78 views

Can you have the harmonic, geometric, arithmetic and quadratic mean of 2 numbers all being integers? [closed]

I created this problem looking at the hm-gm-am-qm inequality (hm = $\frac{2xy}{x + y}$, gm = $\sqrt{xy}$, am = $\frac{x + y}{2}$, qm = $\sqrt{\frac{x^2 + y^2}{2}}$ and hm $\le$ gm $\le$ am $\le$ qm). ...
11
votes
4answers
3k views

Create the numbers 1 - 30 using the digits 2, 0, 1, 9 in this particular order!

Inspired by the last year's "2018 four 4s challenge", I thought it's time to welcome 2019 by a similar challenge. This time you have to use the digits 2, 0, 1, 9 in this particular order to create the ...
4
votes
8answers
527 views

Finni's tricky game

Finni’s game: Person A thinks of a number (1 to 10). This number is called n. Person B says a number (1 to 10). This number is called x. Person A tells the absolute difference of n and x. This ...
3
votes
1answer
151 views

Three-digit multiplication puzzle, part III: Return of the Hex

Followup to: Three-digit multiplication puzzle and Three-digit multiplication puzzle, part II: ever heard of senary? Place different three-digit hexadecimal numbers (000-FFF) on each of the seven ...
3
votes
2answers
142 views

Three-digit multiplication puzzle, part II: ever heard of senary?

Followup to: Three-digit multiplication puzzle I'd never heard of "senary" either, until creating this series of three puzzles. It seems that "senary" is the word for base-six notation. Who knew? ...
4
votes
1answer
188 views

Three-digit multiplication puzzle

Place different three-digit numbers (000-999) on each of the seven nodes of the following diagram: There are six lines in the diagram, on each of which are three of the nodes. On each of those lines, ...
7
votes
1answer
497 views

Fourteen numbers around a circle

Place the numbers 1 to 14 around this circle so that both the sum and (absolute) difference of any two neighboring numbers is a prime.
6
votes
2answers
294 views

The Magic Letter H

Place seven different positive integers on the empty disks of the H figure below so that the product of the three numbers in any straight black line is always the same. Now place seven other numbers ...
13
votes
1answer
735 views

My Social Security Card Number

I have forgotten my social security card number. All I remember is that it is the largest integer with the property that the block of any two of its digits that are adjacent is either a two-digit ...
9
votes
2answers
1k views

Ten-digit number that satisfy divisibilty rules for 2,3,4,5,6,7,8,9,10&11

Question: Arrange the digits 1,2,3,4,5,6,7,8,9,0 to make a ten-digit Number that satisfies all of the divisibility rules for 2,3,4,5,6,8,9,10,&11. BONUS: make the number also divisible by 7
-5
votes
1answer
70 views

What is the pattern that describes these numbers? [closed]

Supposed you are given the following numbers: 6 28 496 8128 33550336 8589869056 What's the relation between them?
6
votes
8answers
515 views

Make $1,\dots,15$ using $3, 9, 9, 9$

This was inspired by many puzzles that use three/four numbers to create other numbers. I chose these numbers in particular because of this post. Can you find a way to make all the natural numbers ...
61
votes
24answers
18k views

Make 0 0 0 0 = 8

Can you find a way to make: $0\ 0 \ 0 \ 0 = 8$ by adding any operations or symbols? You can use only these symbols: $+,\ -,\ *,\ !,\ /,\ \hat\, ,\ ()$. It is limited to this list, and ...
7
votes
3answers
210 views

Is this number unique?

Inspired by Interview Question or Pathbreaking puzzle and A121808. Start with $1$, and count the number of times $1$ occurs, and report this in the format 'number of ones:1', i.e. the next term is $...
10
votes
6answers
2k views

Maximize the number of factorials in your solution to 6 5 4 3 = 1

Using the same rules as Make 6 5 4 3 = 1, but maximize the number of factorials. You may not take the factorial of 1 or 2. This is harder than it looks. A great answer has six factorials, an ...
11
votes
11answers
9k views

Make 6 5 4 3 = 81

Can you find a way to make: 6 5 4 3 = 81 by concatenation and/or adding any of (and only) these mathematical operators: + - × ! ÷ ^ standard parentheses () You cannot add other numbers to the ...
4
votes
9answers
1k views

Make 6 5 4 3 = 1

Can you find a way to make: 6 5 4 3 = 1 by concatenation and/or adding any of (and only) these mathematical operators: + - × ! ÷ ^ standard parentheses () You cannot add other numbers to the ...
22
votes
10answers
14k views

Make 5 5 5 5 = 19 [closed]

Can you find a way to make: $5\ 5 \ 5 \ 5 = 19$ by adding any operations or symbols? You can use only these symbols: $+,\ -,\ *,\ !,\ /,\ \hat\, ,\ ()$. It is limited to this list, and ...
0
votes
2answers
202 views

Just another simple math problem

$4+5=9$ $7+9=13$ $11-5=9$ $17+29=\,?$ Find the value of "?"
5
votes
0answers
188 views

Math Puzzle - What am I?

I am X. I was roaming around some place and found another X. We got attracted. Got into some operation and generate Y. Me and my X got together now and went to roam the places. We found the group ...
8
votes
3answers
480 views

Do they have to be integers? [closed]

$A^2$ + $B^2$, $AB$, and $A + B$ are all integers. Do both $A$ and $B$ have to be integers? If not, what is an example where they are not?
3
votes
1answer
183 views

A football tournament

During a tournament, seven football teams, three European, three South American, and one from Africa, scored a total of 89 goals. The number of goals scored by the African squad was relatively prime ...
22
votes
1answer
723 views

A Tour Around a Triangle

Place the 18 even integers between 2 and 36 in the empty nodes of this triangular graph in such a way that if a path is drawn by coloring in red all the edges joining any two nodes whose numbers add ...
17
votes
1answer
479 views

A partition of 1000 into nine parts

The sum of nine whole numbers is 1000. If those numbers are placed on the vertices of this graph, two of them will be joined by an edge if and only if they have a common divisor greater than 1 (i.e. ...
11
votes
3answers
1k views

Four Marathon Runners

Four marathon runners, each identified with a positive whole number, sit around a table. Each of them notices that their own number has a common divisor with the number of the runner sitting on his ...
0
votes
2answers
231 views

Factor the number 23 into four numbers $a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6}$ [closed]

We know that $23$ is a prime number nonetheless, I'm asking to find 4 numbers $a,b,c,d > 0$ such that $23$ factors. $$ 23 = A \times B \times C \times D \text{ with } A,B,C,D = a + b \sqrt{2} + c \...
1
vote
8answers
7k views

Make numbers 1-30 using 2, 0, 1, 9

This is very similar to the 2, 0, 1, 8 problem. Just try to make all numbers 1-30 using the digits 2, 0, 1, 9. Rules: Use all four digits exactly once Allowed operations: +, -, x, ÷, ! (factorial), ...
1
vote
6answers
403 views

Product of Factorials

In the annual meeting of the International Conference of Puzzle Scenarios, each of $100$ people in a room is given a different number from the set $\{1!,2!,3!,...,99!,100!\}$. One person leaves the ...
7
votes
2answers
196 views

A partition of 1000 into six parts with least and greatest product possible

Find six positive natural numbers, not necessarily distinct, whose sum is 1000 and which, if placed appropriately on the vertices of the following graph, two of them will be joined by an edge if and ...
11
votes
2answers
959 views

What number follows up next? Part 2

I'm trying to figure out what number follows next in this sequence. Can you help me? 5, 21, 341, 5461, 1398101, 22369621
8
votes
1answer
254 views

To Plunder Treasure Islands

Captain Etarip, wants to plunder all the treasure islands that he can. There is exactly one island for every $n\in\mathbb N$. The $n^{\text{th}}$ treasure island contains three cities, each with $n$ ...
1
vote
2answers
136 views

Finding unique number properties

To create my puzzles, I often use the numerical properties of the integers. However, as of recently, I feel like I am running out of properties to use. So, why not make it a sort of game to find ...
3
votes
1answer
158 views

What number follows up next?

I'm currently working on a (difficult) number progression and need your help. How would you continue? 2, 5, 12, 25, 54, 113, 240, 481 ? Thanks in forward!
13
votes
1answer
783 views

The damaged QR Code

Consider the following pixel puzzle which somehow looks like a damaged QR Code with clues on the left of every row and on the top of every column. These numbers represent the total amount of "black ...
2
votes
1answer
244 views

The A-B chocolate puzzle!

Imagine you have 2 types of chocolates (A and B). You randomly pick up two chocolates at once from your bag in a specific pattern. If the same type of chocolates come out, you give them both to your ...
1
vote
1answer
208 views

Find the value of $\bigstar$: Puzzle 9 - Options

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 down, I recommend you solve Puzzle 1 ...
3
votes
0answers
156 views

A special number set [closed]

Find the largest set of stricly positive integer terms in which none divides another and respecting the following rule: given any three of them, one divides the sum of the other two. Source: ...
17
votes
3answers
4k views

Which two students spoke wrongly? [duplicate]

A teacher wrote a large number on the board and asked the students to tell about the divisors of the number one by one. The 1st student said, "The number is divisible by 2." The 2nd student said, "...
14
votes
2answers
584 views

Largest odd factors summing to a square

I just found this awesome puzzle from the Tournament of the Towns (though I'm sure it's appeared other places too). The connection between odd factors and square is surprising, and the proof has a ...
1
vote
1answer
172 views

Find the value of $\bigstar$: Puzzle 8 - Inequality

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 down, I recommend you solve Puzzle 1 ...
7
votes
3answers
258 views

Integers around a circle with consecutive pairs differ to a square

Inspired by this puzzle : Integers around a circle with consecutive pairs adding to a square The integers 1 to 50 are placed around a circle in such a way that the difference of any two of them which ...
4
votes
2answers
227 views

Find the value of $\bigstar$: Puzzle 7 - Boss Battle

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 down, I recommend you solve Puzzle 1 ...
3
votes
3answers
368 views

Find the value of $\bigstar$: Puzzle 6 - Enclosed Operations

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 $\bigstar$. All symbols abide to the following rules:...