# 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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### Mice and their relations

In a colony of $(m n+1)$ mice, must at least one of the following statements be true? If so, why? There is a set $A$ of $(m+1)$ mice none of which is a parent of any other in the set. There is an ...
107 views

### Find all solutions to a sum of fractions

Find all the solutions to: $$\frac{1}{x}+\frac{2}{y}-\frac{3}{z}=1$$ where $x, y, z$ are positive integers.
97 views

### How high does the ladder reach up the wall?

A ladder of length $l$ rests against a vertical wall. Suppose that there is a rung on the ladder which has the same distance $d$ from both the wall and the (horizontal) ground. Find explicitly, in ...
1 vote
54 views

### What size square grid can you tile?

A tiling of an n × n square grid is formed using 4 × 1 tiles. What are the possible values of n? A tiling has no gaps or overlaps, and no tile goes outside the region being tiled.
117 views

### Show there is always a pair at least 16 apart

The integers 1 to 4 are positioned in a 6 by 6 square grid as shown and cannot be moved. The integers 5 to 36 are now placed in the 32 empty squares. Prove that no matter how this is done, the ...
1k views

Puzzle The letters $a, b, c, d, e$ and $f$ represent single digits and each letter represents a different digit. They satisfy the following equations: $$a+b=d, \quad b+c=e \quad \text { and } \quad d+... 4 votes 1 answer 239 views ### Bertrand's Ballot Theorem A total of X voting papers for candidate A and Y voting papers for candidate B were cast in one section during the election. Where X > Y At the end of election day, the voting papers were counted ... 1 vote 2 answers 271 views ### Good Rectangles and Evil Numbers Rectangles and Squares Good Rectangle We define a good rectangle as a rectangle in which  \frac lw = 3  where  l  is the length of the rectangle and  w  is the width of the rectangle. Tiling ... 4 votes 2 answers 217 views ### Can the Spider catch the Fly? The spider and the fly are both on an infinite line and the spider is hungry. He can move twice as fast as the fly, however his vision is very bad (he can only see the fly when he is 1 meter or less ... 7 votes 3 answers 222 views ### How fast do you need to run to catch the bus? You are running to catch a bus that is going on a horizontal road. The bus is at point (0,0) while you are at point (s,k) where s and k are constants. The bus moves at some velocity v and you have to ... 5 votes 1 answer 226 views ### Problem XXX (Squared) - Number Stile Happy weekend folks, The intersections of the three circles divide their interiors into seven "regions" By complete non-coincidence, we have seven tiles A "valid" placement is one ... 4 votes 1 answer 375 views ### Six Different Rectangles a) Six different rectangles, none a square, have all integer sides chosen from a, b, c, and d. If I take any two of these rectangles with no common side (there are three ways of doing this), the ... 4 votes 2 answers 246 views ### Math Crossword with a Twist This puzzle was inspired by this one Who will be the first to solve this tricky math crossword puzzle completely? Across 2: Half of ({20 Across} x (smallest perfect number) + 446) 3: (Number of ... 6 votes 0 answers 261 views ### A bunch of circles and squares The grey region is simply the region you see below. The answer will give you 10 letters. US's: TSDFCK RTOYLG AWQMRZ PUWSTU SRNXEL Hint 1: 19 votes 4 answers 1k views ### Moving around a plane A small plane went through some heavy turbulence and all its passengers ended up in the wrong seat. Now they need to get back to their assigned seats. The image below shows the map of the plane. The ... 1 vote 2 answers 107 views ### What's the correct intuition behind the expected number of die rolls until a 6 if all previous are even? [duplicate] I came across a puzzle on YouTube recently (spoiler). You roll a fair dice until you get a 6. What is the expected number of rolls, including the roll of 6, conditioned on the event that all previous ... 15 votes 4 answers 2k views ### The Game of Barranca Barranca is played with sixteen cards, numbered 1, 2, ... , 16. Two players alternately choose a card, until each has eight. The winner is the one who has a (sub)set of numbers whose product is 220, ... 4 votes 2 answers 187 views ### Splitting the integers 1 to 36 Split the integers 1 to 36 into two sets, A and B, such that any number in set A has a common divisor greater than 1 with no more than two other numbers in A, but for every number in B there are at ... 11 votes 3 answers 515 views ### Find the 3 Famous Numbers There are 3 clues hidden in the poem below. Each clue reveals a famous number. Find the 3 famous numbers and explain your answer in detail. By. This. I. reveal three clues displaying The relentless ... 12 votes 2 answers 509 views ### The mower's challenge Weeds have taken over the roads. If mowed, they don't grow back, but unmowed weeds spread at speed 1 along the road. What's the minimum speed of the mower to get rid of all weeds? Roads are connected ... 2 votes 1 answer 290 views ### How many distinct pentominos can be placed on a 8×9 board? Upon proving optimality of an 8-pentomino solution for an 8×8 board, I was curious to see whether there is a 9-pentomino solution for an 8×9 board, namely a way to arrange 9 distinct pentominos within ... 14 votes 3 answers 2k views ### How many distinct pentominoes are possible to place on an 8 x 8 board? Rules Place some pentominoes into an 8 x 8 grid. They do not touch each other. They can touch only diagonally (with corner). Pentominoes cannot repeat in the grid. Rotations and reflections of a ... 4 votes 4 answers 959 views ### Six positive integers Find six different numbers (positive integers) such that each of them has a common divisor with precisely three of the other numbers. How small can the largest of the six numbers be? What if 2n, n&... 1 vote 2 answers 343 views ### How many different tiles are there when each corner may have 0-6 dots, each of which may have 0-6 dots? There are four corners to each tile. Each corner can be empty, or contain an arrangement of dots (1-6) like the sides of a dice. Within each of these dots can be a further arrangement of 1-6 dots, or ... 6 votes 2 answers 327 views ### How to prove Yin-Yang alternating 2 by 2 is not allowed Yin-Yang is a puzzle where one needs to fill each cells with either black circle or white circle following these rules: Each color's circles must be connected to one another according to four-way ... 2 votes 1 answer 144 views ### Hitting twice with different choices This game of two players has public parameters an integer n\ge2, and a probability p with 1/n<p\le1. E.g. n=4, p=1/3. In the first phase of a game, a player secretly decides n ... 1 vote 1 answer 261 views ### Can you escape from two lions? You're at the center of a circular arena. A pair of lions are at the border, planning to catch you. One of them moves as fast as you, but the other moves slower than you. The three of you are confined ... 3 votes 1 answer 195 views ### Geometric game on a n*n chessboard You can get famous (OK, Warhol-15 minutes-famous :-)! First a few definitions. Of course, two rooks of the same colors don't attack, but since two colors are needed, "attacking" here means &... 6 votes 2 answers 385 views ### Multiplication puzzle with trios of numbers I created a puzzle that I was curious what its properties are, and how it could be determined it is solvable or not. It consists of 6 rows of 3 of the same numbers, which in each row in order are 1, 2,... 15 votes 2 answers 730 views ### Can the lion protect the sheep from the wolves? In a closed arena, three wolves are on the vertices of an equilateral triangle at the border. The sheep and his lion friend are at the center. The wolf eats the sheep if their distance is 0, and ... -2 votes 1 answer 196 views ### Summa Cum Laude - Tile and Error For any set X, {X} denotes the sum of its elements. "I divide {A}" means all items in that set are divisors of the sum of A's elements. Arrange the tiles so that four are in each set and the ... 0 votes 4 answers 175 views ### Solve for 69 using only 1 9 9 2 in that order [closed] You must use all the digits of 1 9 9 2 in that order to come up with 69 as the answer. For example  - (1 + \sqrt {9})! + 92 = 68  but you must solve for 69. 1 vote 0 answers 67 views ### Riddle: Irregular math [duplicate] A numbery riddle: When I take five and add six, I get eleven, but when I take six and add seven, I get one. What am I? What is the answer? 2 votes 1 answer 223 views ### SETI message received At 03:39:18 UTC, the Allen Telescope Array detected a series of distinct signals from the direction of the Procyon star system. There is speculation that these signals considered together could carry ... 14 votes 3 answers 2k views ### Palindrome from the first 20 numbers Can you concatenate the numbers from 1 to 20 in some order, such that the resulting 31-digit number is a palindrome? Bonus: if there are multiple such palindromes then what is the smallest one? 2 votes 1 answer 162 views ### How do you produce the number 5, using only 3 eights? How do you produce the number 5, using only 3 eights? The 4 operations (+ - x and division) + square root and factorial are allowed but no extra numbers (no squaring). -1 votes 1 answer 122 views ### How to arrange the colored cells in game grid? Puzzle: In a game grid some cells are missing. Each line has only one colored cell with a label (a number greater than zero). This is an example grid and the number of columns/rows can be less than ... 2 votes 1 answer 176 views ### Probability that there will be no mutual best friendships? [closed] Here is a problem: There are two groups of n users, 'A' and 'B'. Each user in A is friends with those in B, and vice versa. Each user in A will randomly choose a user in B as their best friend and ... 7 votes 1 answer 221 views ### O princess, where art thou? In the highest tower of her castle, a princess has N bedrooms which are arranged in a circle. She never sleeps in the same room on two consecutive days. Every morning she moves to another room by ... 5 votes 2 answers 201 views ### Ernie and the Disguised Donors I was spending the morning helping Ernie dig up the potato patch when he got an emergency phone call from the Society for Absent-Minded Mathematics Professors – it appeared that they had “A bit of a ... 0 votes 0 answers 70 views ### A house with 100 lights and 100 switches [duplicate] There is a house with 100 lights. In the basement there are 100 switches for the lights. Sadly, you have forgotten which switch is connected to which light. Currently they are all on. You go down and ... 1 vote 1 answer 155 views ### What are the exact times an analog clock with two identical hands and its vertically mirrored image show the same time? Suppose you have a clock with two identical hands (there is no second hand). What are the exact times when this clock and its vertically mirrored image are identical? 2 votes 1 answer 250 views ### What's the next term in this sequence? I created this simple number sequence puzzle, I don't know if anyone used the same idea before. Find the term at the question mark in the sequence.$$ 25, 42, 56, 97, 176, 232, 251, 269, 394, 518, 572,... 164 views

### A simple number sequence puzzle

This is a simple number sequence puzzle, you are given 6 related number sequences, observe them, find the pattern; Your task is to give the first sixteen terms in the seventh sequence. 1d5, 2d3, ... 363 views

### Determine the highest possible sum of all faces of cube

There is a non-negative integer written on all six faces of a cube. Integers on all possible three common adjacent faces (all faces are adjacent to each other) are multiplied and then they are added. ...
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### Tic-Tac-Collatz

Have you ever heard of the Collatz conjecture? Just in case you haven't, I'll summarize it for you! Take any positive integer $n$, if it is even then simply divide it by $2$; however, if it is odd, ...
1 vote
242 views

### 4x4 grid with the shortest longest path

This is an extension of this beautiful puzzle. This time your task is to find the hardest 4x4 grid. In particular, find a 4x4 grid containing every number from 0 to 9 at least once, such that the ...
492 views

### Find out the longest path being alive [closed]

Start from any integer. Move horizontally or vertically (not diagonally), and if you come across the same integer more than once you will die. Moving diagonally is not allowed. What is the longest ...