# number-theory's questions - English 1answer

146 number-theory questions.

### 17 A partition of 1000 into nine parts

1 answers, 448 views number-theory graph-theory arithmetic
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. ...

### 20 A Tour Around a Triangle

1 answers, 662 views number-theory graph-theory arithmetic
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 ...

### 11 What number follows up next? Part 2

2 answers, 919 views number-sequence number-theory
I'm trying to figure out what number follows next in this sequence. Can you help me? 5, 21, 341, 5461, 1398101, 22369621

### 5 Six sisters on the ski lift

1 answers, 308 views number-theory arithmetic
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 ...

### 10 Four Marathon Runners

3 answers, 1.141 views number-theory arithmetic
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 ...

### 1 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), ...

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

2 answers, 202 views formation-of-numbers number-theory
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 Product of Factorials

5 answers, 323 views mathematics number-theory
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 ...

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

6 answers, 320 views mathematics number-theory
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 ...

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

13 answers, 4.288 views mathematics number-theory
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 ...

### 7 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 ...

### 10 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 ...

### 8 To Plunder Treasure Islands

1 answers, 228 views mathematics word-problem number-theory
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 Finding unique number properties

2 answers, 111 views mathematics number-theory
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 What number follows up next?

1 answers, 147 views number-sequence number-theory
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 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 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 Find the value of $\bigstar$: Puzzle 9 - Options

1 answers, 185 views mathematics number-theory
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 ...

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

1 answers, 153 views mathematics number-theory
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 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 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 Largest odd factors summing to a square

2 answers, 554 views mathematics number-theory
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 ...

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

2 answers, 203 views mathematics number-theory
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 Find the value of $\bigstar$: Puzzle 6 - Enclosed Operations

3 answers, 344 views mathematics number-theory
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:...

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

2 answers, 195 views mathematics number-theory
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 Find the value of $\bigstar$: Puzzle 4 - In Between

2 answers, 151 views mathematics number-theory
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 Find the value of $\bigstar$: Puzzle 3 - Substitution

3 answers, 163 views mathematics number-theory
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 Find the value of $\bigstar$: Puzzle 2 - Switch-a-roo

3 answers, 186 views mathematics number-theory
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 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 ...

### 15 Optimal Money-Saving on the NYC Metro

5 answers, 1.923 views mathematics no-computers number-theory real
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 My forgotten PIN

3 answers, 473 views number-theory arithmetic
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 My five daughters

5 answers, 4.238 views number-theory arithmetic
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 A unique partition of 200 into 6 parts

2 answers, 713 views number-theory arithmetic
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 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 Expressing numbers as sum of two squares and two powers of 2

2 answers, 108 views formation-of-numbers number-theory
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 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 Another loop of integers with consecutive terms adding to a square

1 answers, 135 views formation-of-numbers number-theory
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. ...

### 29 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....

### 18 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 $3 \times 3$...

### 14 Mother and Daughter

4 answers, 3.194 views calculation-puzzle number-theory
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 ...

### 7 Prime sums in a 4 x 4 board

2 answers, 677 views calculation-puzzle number-theory
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.

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

1 answers, 125 views mathematics visual number-theory
Find the divisor and all the digits of the sum. source : New scientist Magazine

### -3 Find number of divisors

1 answers, 99 views mathematics number-theory
If xy is a two digit number with three divisors, how many divisors would each of the following numbers have? xyxyxy yxyx xy4xy

### 6 The wizened mathematician

1 answers, 416 views mathematics number-theory
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 ...

### 15 Find out the rule, then solve it

2 answers, 1.229 views mathematics visual no-computers number-theory
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

### -1 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 ...

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

3 answers, 263 views mathematics visual number-theory
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 ...

### 7 3d x 1d = 2d x 2d

5 answers, 2.247 views mathematics number-theory
Find the largest integer that is a product of three-digit number and a one-digit number and also a product of two two-digit numbers. For example, 200 x 1 = 10 x 20. Of course, 200 is not the largest.