# Questions tagged [mathematics]

A puzzle related to mathematical facts and objects, whose solution needs mathematical arguments. General mathematics questions are off-topic but can be asked on Mathematics Stack Exchange.

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### I am thinking of a number

I am thinking of a number between 1 and 5000. It is the product of two primes. I tell my brother the larger of those primes, and to my sister the smaller. You are allowed to ask each of us questions, ...
1answer
130 views

### Add or subtract 212 in octal to get a palindrome

The puzzle is as follows: Suppose you have a three-digit number in the octal system. If you add or subtract 212 (also in the octal system) from that initial number, you get a three-digit palindromic ...
2answers
134 views

### ⌈3n/2⌉−2 weighings are required to find the lightest and the heaviest coin among n coins. Prove/disprove this

We are given n coins, all of which have a different weight. Prove/disprove that ⌈3n/2⌉−2 weighings are required to find the heaviest and the lightest coins among them using a 2 pan balance scale. You ...
1answer
156 views

### Solving Kordemsky's Prime Cryptarithm and proving uniqueness

This is a cryptarithm from Kordemsky's The Moscow Puzzles, problem 273 to be precise. Each digit is a single-digit prime ($2,3,5,$ or $7$). Find a solution and prove that it is unique. \begin{array}{...
0answers
90 views

### First digit of 2021^2021 [duplicate]

Can you find the first digit of $2021^{2021}$ without a computer? Good luck!
1answer
449 views

### Stop the robot by changing value of variable Z

You're a computer security expert (i.e a hacker who doesn't break law). Your neighbour is infamous professor MacGoofy himself. He specializes in designing programming languages and programming ...
1answer
435 views

### Consecutive integers with digit sum divisible by 19

What is the smallest positive integer N, such that the digit sum of N and N+1 are both divisible by 19?
3answers
2k views

### What makes this polynomial a square number?

For which integer values of $x$ is $x^4+x^3+x^2+x+1$ a square number? Please include a proof that the polynomial cannot be a square number if $x$ is not one of your answer(s). Source: a math olympiad ...
3answers
1k views

### An ant's walk in the Cartesian Plane

An ant lives in the origin of the Cartesian Plane. Every morning, at 6 am, it sets out on a 16-hour walk which gets her back home precisely at 10 pm. In the first hour the ant walks exactly one unit, ...
2answers
995 views

### How many descendants can this spaceship crew produce?

A spaceship is on a very long voyage. It starts with a crew of 4 women and 4 men, none of whom are related by blood. How many descendants at most can this 8-person crew produce without inbreeding? ...
1answer
148 views

### My sixteen graph theory students

I will have sixteen students in my graph theory course this semester. In our first session I asked each of them with which of the other 15 students in the class they were already acquainted before the ...
3answers
155 views

### Maximum number of triangles formed in a pentagon with equal area

All diagonals of a convex pentagon are drawn, dividing it in one smaller pentagon and 10 triangles. Find the maximum number of triangles with the same area that may exist in the division. The best I ...
1answer
180 views

### More primes and squares, in a summation triangle

Place a different prime number or perfect square in each of the twenty-one disks that make up the triangle below, so that the number in any disk that lies on two others is precisely the sum of the ...
0answers
194 views

### Infected Cylinder and Torus

A variant of the well known Infected Checkerboard problem. If we've a 𝑛x𝑛 square, then we fold it along top and bottom row to form a cylinder. A cell in this cylinder becomes infected if at least ...
2answers
793 views

### Can this equation be solved with whole numbers?

$(x+y)(x+z)(y+z) = 33...3$ (A 333 digit number that consists only of 3's)
1answer
77 views

### Fill big box with smaller boxes

Let's say we have a big box with inner edges with the lengths 2m, 1.5m, 1.4m. Can we fill this with smaller boxes with the edge lengths of 3dm, 5dm and 1m, without any gaps?
1answer
99 views

### Identify heavier weights

[This is from one of my friends, a year ago] We have 6 weights: 2 red, 2 black, 2 blue. Of each color, there is one that's 100g and one that's 101g. How can you identify which ones are 101g, in two ...
3answers
197 views

### Missing numbers in triangular sequence

I am not sure if this is an appropriate puzzle for this site, but I searched and found 1 or 2 other types of puzzles like this. This is an original puzzle. Find the missing numbers in rows 8 through ...
3answers
280 views

### A Circle of numbers

Just saw a Circle of numbers on my Whats App message (source not listed) which is as following Arrange numbers 1 to 32 in a circle such that any two adjacent (neighboring) numbers add up to a perfect ...
1answer
186 views

### Moving coins in a grid

Here is a great puzzle from Ed Pegg Jr. Place two coins in the center cell of the following grid. Now you can choose a coin X and move the second coin Y one cell in the direction of the arrow under ...
1answer
205 views

### Half… only sometimes

In a great world of numbers Heard some people cry, I dunno why. They lost a lot Mainly the biggest ones But a lot of them Got into a new world Dunno why. They were first imprisoned Then the confused ...
2answers
780 views

### Time… to be prime?

A standard analogue clock face has numbers 1 to 12 around the edge arranged sequentially, which is nice for telling the time, but not especially interesting. It is possible to arrange the numbers in ...
1answer
497 views

### A special number

Here is a nice puzzle from my friend. Can you find a number that is a product of two consecutive primes and when multiplied by its own reversal produces a palindrome? The answer may surprise you. No ...
1answer
83 views

### Painting a plane!

Paint the points on a plane with three colors, so that the points on each line are a maximum of 2 colors, and all three colors are used. (Math Festival 1990)
3answers
470 views

### Put line segment the way they cover end points

Is it possible to place 1000 line segments on the page so that the two ends of each line segment are on the inner points of other line segments? (By the inner point of the line segment, we mean a ...
1answer
147 views

### Cut the string!

There are five pieces of blue string on the table with different lengths, the total length of which is 30 cm. There are also five pieces of red string with different lengths, the total length of which ...
2answers
236 views

### Place 4 players to make 6 distances between pairs

Is it possible to place 4 players on a football field in such a way that the 6 distances between every pair of them are 1, 2, 3, 4, 5, 6 meters? Source: Moscow Math Olympiad 2001 (Look Inside to Page ...
2answers
358 views

### Rack 'Em Up! 🎱

In a game of English eight-ball pool, a set of 15 balls are arranged or 'racked' in the shape of an equilateral triangle. In order for the balls to be racked fairly, they must be arranged like so: <...
1answer
722 views

### How many circles needed to pass through each of 5x5 lattice points?

You are given a 5x5 set of lattice points. What is the minimum number of circles, which pass through each of the 25 points at least once?
1answer
243 views

### An intriguing game construction

I found that the book "Amusements In Chess," by Henry Ernest Dudney, has been put onto the web in the form of a website. In the section "Various Chess Puzzles, I came across this ...
3answers
1k views

### The maximum period of dancing program

Sixteen people named A, B, ..., P are standing in line in the order ABC...P. They "dance", or swap places, according to some predefined instructions. ...
1answer
162 views

### Visiting streets, not houses

The section points are houses and lines are streets, all with one unit length. What is the fewest number of units you must travel to visit every street at least once?
1answer
250 views

### Summing numbers and then increasing used numbers

I was making a mathematical game (viewable here: www.michaelprimo.it/beta) where you have a 3x3 grid, one central yellow button with a generated number and the other eight blue buttons with clickable ...
1answer
354 views

### An X-Mensa dq question

Which of the numbers should fill in the ?. 4, 3, 4, 7, 0, 45, 7, 15, 64, 0, 1111, 55, 5, 5, 0, 20, 32, 1, ? a) 2 b) 0 c) 20 d) 1 Hints: 1.
1answer
104 views

### Splitting figures on the Cartesian plane

What is the minimum number of lines to separate the sets? a) 2 b) 3 c) 4 d) 1 e) 5 Observe the graph below, it is possible to separate linearly with a line at least two of the classes? a) Yes, just ...
3answers
250 views

### A square in the plane with 4 vertices of the same color

Every point in the plane is colored either red or blue. Is it necessarily the case (i.e., is it true for all such colorings) that there exist some four points of the same color that are the vertices ...
2answers
1k views

### The Transmogrifier

The city had a new transmogrifier. I was willing to go in, but it would be boring, I was told. Meanwhile, some people were bragging about how they would come out the transmogrifier. One said, "My ...
1answer
146 views

### Four rational numbers whose product is 10 and whose sum is 0

It has been shown that every positive integer is the product of four rational numbers whose sum is 0. Thus: ...
3answers
881 views

### 8 soldiers lining up for the morning assembly

There are 8 soldiers, gathering and lining up every morning for their military service. The commander at the head of these soldiers demands that the morning lineup of these soldiers be arranged ...
1answer
139 views

### Why, a math riddle for you!

I am not real, but I am part of a complicated thing. Two is negative? What am I?
2answers
141 views

### Generalized rectangular tilings with no “fault lines”

I recently came across this question: One rectangle, indivisible The goal is, by tiling 2x1 rectangles, to create a larger rectangle that cannot be split into 2 smaller rectangles. But my question is ...
0answers
115 views

### Does anybody know the history of this star/triangle puzzle?

Does anybody know the history of the puzzle posted by Mario Bilotti? Who created this puzzle? Who deserves the credit? Find 10 triangles in a five pointed star using two straight lines
5answers
2k views

### What fraction of the larger semicircle is filled?

What fraction of the larger semicircle is filled? The two smaller semicircles are of equal size. This is a puzzle originally set by Catriona Agg, who is a puzzle setting genius.
1answer
303 views

### Long division cryptarithm - KLMN/HIJ = K

The problem statement: Solve the following long division problem. Each letter represents a unique digit (0-9) ...
4answers
2k views

### Make 38 using the least possible digits 8

This is a follow up question to this one, which was: Make the number 1998 using the minimum amount of digits 8. Your allowed operations are +, -, *, /, ^, % (percent). You need not use only integers ...
1answer
134 views

### a 17X17 grid filled with trominoes of three different colors

Let's have an 17x17 grid. We can fill this grid with 96 trominoes of three different colors, 32 trominoes of each color. On this particular grid the empty single square is the position A1. By visual ...
1answer
129 views

### A function. Really?

If you give me a 5, I give you a 21. If you give me a 6, I give you a 2. If you give me a 7, I give you a 12. If you give me a 10, I give you a 101. What am I processing to get the result? Hint 1 ...
2answers
223 views

### Number Sequence (yet again?)

This number sequence has numbers rounded to the nearest tenth. 1, 2, 4.5, ?, 26, ? Find the pattern and the answers to the question marks rounded to the nearest tenth.
1answer
112 views

### A simple two-operation calculator. How to obtain any natural number from any other?

Imagine that you possess a very simple electronic calculator. It has a screen and ten buttons from 0 to 9 to enter natural decimal numbers (positive integers). However, it can perform just two unary ...
1answer
265 views

### Alice and Bob playing Neighboring Sums Game

Alice and Bob are playing the neighbouring game which is originally single game to get the highest point at the end. You start with an empty 4x4 grid. At each turn you can choose an empty cell and ...