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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.

40 questions
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2answers
7k views

### A general solution to the decanting problem? (aka jug-pouring, water-pouring)

Take a look at these two questions: - A Set of Water Jug Challenges - Pouring problem Now I'm asking for a generalised solution to that problem. I define the problem as follows: You are required ...
6answers
451 views

### Find the value of $\bigstar$: Puzzle 1 - Evaluation

This puzzle replaces all numbers with other symbols. Your job, as the title suggests, is to find what number fits in the place of $\bigstar$. All symbols abide to the following rules: Each symbol ...
3answers
265 views

### Find the value of $\bigstar$: Puzzle 2 - Switch-a-roo

This puzzle replaces all numbers with other symbols. Your job, as the title suggests, is to find what number fits in the place of $\bigstar$. All symbols abide to the following rules: Each symbol ...
4answers
240 views

### Find the value of $\bigstar$: Puzzle 3 - Substitution

This puzzle replaces all numbers with other symbols. Your job, as the title suggests, is to find what number fits in the place of $\bigstar$. All symbols abide by the following rules: Each symbol ...
2answers
175 views

### Find the value of $\bigstar$: Puzzle 4 - In Between

This puzzle replaces all numbers with other symbols. Your job, as the title suggests, is to find what number fits in the place of $\bigstar$. All symbols abide to the following rules: Each symbol ...
2answers
233 views

### Find the value of $\bigstar$: Puzzle 5 - Every little Symbol

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:...
3answers
380 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:...
18answers
15k views

### Nine gangsters and a gold bar

One night nine gangsters stole a gold bar. When the time came for dividing the bar, they faced a problem: two of the criminals put guns to each other's faces. Now it's up to fate whether one of them ...
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 ...
1answer
1k views

### Deducing Two Numbers based on their Difference and Ratio

Yesterday I met the perfect logicians Divvy and Subtra. I told them: I have chosen two positive integers $x$ and $y$ with $2\le x\lt y\le 100$ and with $x$ a divisor of $y$. I will now whisper ...
2answers
237 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 ...
13answers
5k views

### Teacher, teacher on the wall, Who's the dumbest of them all?

A maths teacher writes a very large number on the blackboard and asks her pupils (of whom there are $n$ in the room) about its factors. The first pupil says, "The number is divisible by 2." The ...
1answer
404 views

### What is a Composite Word™?

Here are some Composite Words™: ...
1answer
190 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, ...
1answer
180 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 ...
6answers
4k views

### What are the numbers?

Three perfect mathematicians with extremely strong memories are taking an exam. The examiner tells each of them a certain piece of information about $x$ and $y$, which two positive integers between ...
24answers
20k 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 ...
6answers
6k views

### 3x3 “Magic Square” of Prime Numbers

During the thinking and analysis of some mathematical problems, I came up with this puzzle: Just like any magic square, one has to fill in $9$ different numbers $P_1, P_2, \dots P_9$ to a $3 \times 3$...
1answer
3k views

### The largest Monday number

A Monday number is a positive integer $N$ with the following three properties: The decimal representation of $N$ does not contain the digit 0 The decimal representation of $N$ does not contain any ...
11answers
10k 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 ...
3answers
486 views

### A Gathering of Number-Theorists

A certain number of the 5000 members of the World Arithmetical Society (each of which has a different membership number between 1 and 5000) got together to discuss a problem. Much to their surprise, ...
1answer
145 views

### Another loop of integers with consecutive terms adding to a square

The integers 1 to 50 are placed around a circle in such a way that the sum of any two of them which are adjacent is a perfect square. Of these integers, all but the prime integers were removed. ...
1answer
761 views

### 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 sum of any two of them which are adjacent is a perfect square. Of these integers, the even numbers are then removed. Restore them....
5answers
2k views

### Divisible by seventeen

Determine the smallest integer $n \geq 0$ for which the decimal digit sum of n is a multiple of 17 the decimal digit sum of $n+1$ is a multiple of 17. No computers! The puzzle has a nice direct ...
1answer
492 views

### Labelling a graph with a partition of 100

Label the vertices of this graph with positive integers (repetitions allowed) whose sum is 100 in such a way that any pair of vertices are joined by an edge if (and only if) they have labels with a ...
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 ...
1answer
474 views

### Relevant primes

Let's call a prime number $p$ "relevant" if there exists an integer $n>1$ such that the integer part of the sum $$\sum_{k=1}^{p^n} \sqrt[n]{\frac{1}{k^{n-1}}}$$ is $2016$. How many "relevant" ...
8answers
547 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 ...
1answer
246 views

### Curios observation about a special grid - Why?

[0,1,6,4,3] [4,5,6,0,9] [9,9,0,1,1] [1,0,4,5,6] [7,6,4,9,0] This 5x5 table has unique properties. Each number in a cell means : The cell = last digit of (sum ...
2answers
116 views

### A pile of chips involving powers of 2

Ann and Bob play alternately on a pile of chips. On each play, any number of chips, which is a power of 2 (including 1=$2^0$), can be removed from the pile. Obviously the number of chips to be removed ...
4answers
3k views

### Smallest PRIME containing the first 11 primes as sub-strings

In Smallest number containing the first 11 primes as sub-strings, @Alconja successfully found the smallest number which contains the first eleven primes (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31) as ...
2answers
746 views

### IX-NAY on the IX-SAY

Will this sequence ever have a 6 in it? 9, 1, 1, 1, 10, 3, 1, 1, 10, 5, 1, 1, 10, 1, 5, 2, 1, 1, 10, 1, 1, 1, 5, 4, 1, 1, 10, 3, 1, 1, 5, 1, 1, 1, 5, 2, 1, 1, 10, 5, 1, 1, 5, 3, 1, 1, 5, 4, 1, ... ...
3answers
1k views

### 3x3 “Magic Square” of Prime Numbers — Part II

Glad to know the previous puzzle, which was the first puzzle I posted in Puzzling, was warmly welcomed (Thank you!), and an optimal solution was found. Inspired by the comments there, here is the ...
2answers
5k views

### Martin Gardner - Persistence

A number's persistence is : The number of steps required to reduce it to a single digit by multiplying all its digits to obtain a second number Then multiplying all the digits of that number ...
4answers
873 views

### Why did this prime-sequence puzzle not work?

While attacking a recent puzzle (whose solution ended up being entirely different from what I was trying), I was inspired to create a number-sequence puzzle with a sequence $(p_n)$ of primes where the ...
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 ...
2answers
356 views

### Arrange numbers 1 to 9 into the upsilon grid

Arrange numbers 1 to 9 into the octagon, so the operation is correct. C is a constant. Do the math operation in sequence, ($×$) and ($/$) is NOT HIGHER than ($+$) and ($-$).
2answers
152 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? ...
1answer
219 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 ...
2answers
227 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...