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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.

13
votes
1answer
712 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
922 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
66 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
491 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 ...
55
votes
24answers
17k 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
199 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
12answers
8k 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
966 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
13k 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
200 views

Just another simple math problem

$4+5=9$ $7+9=13$ $11-5=9$ $17+29=\,?$ Find the value of "?"
4
votes
0answers
168 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
477 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
180 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 ...
21
votes
1answer
698 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
464 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. ...
10
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
214 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
4k 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
5answers
340 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
193 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
934 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
245 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
129 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
147 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
552 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
232 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
192 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, "...
13
votes
2answers
573 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
155 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
221 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
213 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
357 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:...
4
votes
2answers
214 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:...
6
votes
2answers
156 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
3answers
173 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 to the following rules: Each symbol ...
5
votes
3answers
195 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
6answers
345 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 ...
4
votes
1answer
305 views

A certain partition of 130

Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than ...
5
votes
1answer
327 views

Six sisters on the ski lift

The sum of the ages of six sisters known to me is 92. Though there is no single whole number greater than 1 that simultaneously divides the ages of any three of them, I did notice this morning, while ...
15
votes
5answers
2k views

Optimal Money-Saving on the NYC Metro

You are on vacation in New York City. You didn't bring your car, and it's currently around $-50^\circ C$, so it's probably a good idea to take the NYC metro subway to move around. You need a metro ...
8
votes
3answers
511 views

My forgotten PIN

I´ve forgotten my PIN, a four-digit number. All I remember is that it is a perfect square, and that it has at least one digit in common with every other four-digit square number. What is it?
13
votes
5answers
4k views

My five daughters

The sum of the ages of my five daughters is 43. The ages of any two of them have a common factor greater than 1. How old are my daughters?
8
votes
2answers
746 views

A unique partition of 200 into 6 parts

The sum of six positive integers is 200. If placed appropriately on the vertices of this graph, two of them will be joined by an edge if, and only if, they are not relatively prime, that is, if they ...
5
votes
1answer
160 views

Three positive integers whose sum is 120

The sum of three positive integers is 120. Pairwise, exactly once (out of three possible pairs) are the numbers relatively prime (i.e. they have no common divisor greater than 1). What are the three ...
-5
votes
2answers
115 views

Expressing numbers as sum of two squares and two powers of 2

It has been shown that the smallest integer, greater than 1, that cannot be represented as a sum of two squares and at most two powers of 2 is 535,903. Show how to express 535,902 as the sum of two ...
4
votes
1answer
289 views

How many numbers can we choose so that no two differ by 2 or 5?

We choose some numbers from the set $\{1, 2, ..., 100\}$. What is the largest possible number of numbers from the set that can be chosen so that no two of the chosen numbers vary by 2 or 5?
3
votes
1answer
138 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. ...