# 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.

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### 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), ...
418 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 ...
215 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 ...
1k 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
282 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 $n$ ...
178 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 ...
185 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!
1k 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 ...
270 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 ...
273 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, I recommend you solve Puzzle 1 ...
160 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: ...
6k 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, "...
599 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 ...
203 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 ...
290 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 ...
265 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 ...
392 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:...
265 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:...
204 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 ...
289 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 ...
356 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 ...
570 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 ...
313 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 ...
355 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 ...
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 ...
634 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?
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?
838 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 ...
213 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 ...
125 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 ...
294 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?
146 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. ...
803 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....
3k views

### Mother and Daughter

A mother (not yet a centenarian) and her daughter (who happens to share her mother's birthday) are both a prime number of years old. Moreover, in their lifetimes there have been at least a dozen other ...
2k views

### Prime sums in a 4 x 4 board

Place the integers 1 to 16 in the sixteen cells of a 4 x 4 board so that the sum of any four numbers in a row or column is a different prime.
130 views

### Find the divisor and all the digits of the sum

Find the divisor and all the digits of the sum. source : New scientist Magazine
105 views

### Find number of divisors

If xy is a two digit number with three divisors, how many divisors would each of the following numbers have? xyxyxy yxyx xy4xy
428 views

### The wizened mathematician

Once I walked up a hill, and on the summit sat a very old man cross-legged, calmly taking the air and observing the view below. I asked him if he was alone and he said: "No. I have five cousins who ...
187 views

### How many ways to iterate? [duplicate]

There are 9 coins, out of which one is odd, i.e. its weight is either less or more than that of the other 8 coins. How many iterations of weighing using a pan balance are required to find the odd coin ...
276 views

### Replace each fruits and animals with numbers and operators

Replace each fruit with numbers 5 to 10. Replace each animal with operators $[+,-,×,/]$. So all the math operations vertically and Horizontally correct. Ignore BODMA rule (order of operations DO NOT ...
2k views

### Fun with numbers: solve A to E

A, B, C, D and E are five distinct non-negative integers no greater than 99. Can you figure out their values based on the following simple clues? Not all of the five numbers have the same parity A + ...
818 views

### Change $102 \to 201$

Suppose you start with the number $102$ and, at each step, you are allowed to double the number, add the number's digits, multiply its digits, or add 1. Your goal is to go from $102$ to $201$ in as ...
370 views

### Looking for another partition of 100

The sum of ten, not necessarily different, positive integers is 100. If placed adequately on the vertices of this graph, two of them will be joined by a line if, and only if, they have a common ...
398 views

### Arrange numbers to the triangles, so the sums are equal to S

Rules: Arrange numbers from 1 to 11, and each number can appear max twice. The numbers are arranged, so that any 3 numbers which forming an up-triangle sum to the desired number (S). A equal sign ...
1k views

### Find out the rule, then solve it

First, Find out the rule from the example, Then solve the puzzle without computer. The answer must be unique (just 1 valid answer). Example Solve This
322 views

### Arrange numbers to 3x3 square with concentric (Addition and Difference)

2 Similar questions: I. Difference Arrange numbers 1 to 8 to replace the letters so: A = abs (B – D) C = abs (B – E) F = abs (D – G) H = abs (E – G) To eliminate duplicates by rotation and ...
2k views

### Math logic Triangle

Rules Arrange numbers from 1 to 15 to the white triangles. A op B = result Green triangle means 2 numbers of its side are adjacent numbers. (-) means find the difference (/) means divide the bigger ...
852 views

### Delete a digit then sum

Take a number $(x)$, then create the complete list of the numbers formed by deleting single digits from its base ten representation $(d_1,d_2,...,d_n)$. If the sum of those new numbers equals $x$ we ...