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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.

17
votes
2answers
6k 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 ...
10
votes
6answers
439 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 ...
6
votes
3answers
245 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 ...
8
votes
4answers
234 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 ...
6
votes
2answers
170 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 ...
4
votes
2answers
229 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:...
3
votes
3answers
376 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:...
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 ...
11
votes
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 ...
4
votes
2answers
232 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 ...
78
votes
18answers
14k 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 ...
30
votes
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 ...
5
votes
1answer
385 views

What is a Composite Word™?

Here are some Composite Words™: ...
4
votes
1answer
189 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, ...
1
vote
1answer
175 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 ...
-9
votes
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 ...
61
votes
24answers
19k 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 ...
19
votes
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 $3 \times 3$...
28
votes
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 ...
11
votes
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 ...
7
votes
3answers
482 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, ...
3
votes
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. ...
30
votes
1answer
726 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....
16
votes
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 ...
10
votes
1answer
483 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 ...
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 ...
6
votes
8answers
521 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 ...
5
votes
1answer
245 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 ...
7
votes
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 ...
6
votes
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 ...
5
votes
4answers
849 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 ...
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 ...
4
votes
2answers
345 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 ($-$).
3
votes
2answers
143 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? ...
1
vote
1answer
213 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 ...