Puzzles, curiosities, brain teasers and other mathematics done "just for fun".

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3
votes
4answers
702 views

Is there an algorithm to recover a crossword grid based on the clues alone?

Suppose that we have access to only the clues of a crossword puzzle along with the number of letters that the answers are supposed to be. Is there an algorithm that we can use to reconstruct the ...
3
votes
2answers
3k views

How to construct magic squares of even order

Could someone kindly point me to references on constructing magic squares of even order? Does a compact formula/algorithm exist?
5
votes
2answers
385 views

Famous Finite Sets [closed]

What are the most famous (or most beautiful, IYO) finite sets in mathematics? I'm especially looking for 'large' sets that contain more than $2^{10} \approx 1000$ but fewer than $2^{20} \approx ...
1
vote
1answer
89 views

Measuring how monotonically “staircase-like” a set of values is

A bit of a bizarre question here -- I'm looking for assistance in generating a robust metric to measure how monotonically "step-wise" a series of values is. The set must not start or end at a specific ...
13
votes
3answers
986 views

The Math behind rotation puzzles?

In the game Machinarium, there is the following puzzle where the goal is to get all of the green points on the green area by rotating them along any of the 3 circles engraved on the background plate. ...
19
votes
1answer
460 views

Extracting individual race results from Mario Kart final scores

In Mario Kart, one "cup" involves 4 races, and after every race each racer gets points awarded based on what place they came in (better rank means more points). After playing it enough I grew curious ...
4
votes
1answer
441 views

Decomposing a circle into similar pieces

Is it possible to decompose a circle into finitely many similar disjoint pieces, one of which contains the circle's center in its interior?
18
votes
4answers
2k views

Is chess Turing-complete?

Is there a set of rules that translates any program into a configuration of finite pieces on an infinite board, such that if black and white plays only legal moves, the game ends in finite time iff ...
7
votes
5answers
856 views

Cauchy-Schwarz inequality and three-letter identities (exercise 1.4 from “The Cauchy-Schwarz Master Class”)

Exercise 1.4 from a great book The Cauchy-Schwarz Master Class asks to prove the following: For all positive $x$, $y$ and $z$, one has $$x+y+z \leq 2 \left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + ...
2
votes
2answers
717 views

In Towers of Hanoi (with 3 sticks and n disks without backtracking), do all legal sequences of moves reach the solution?

Updated Question : How to show that in TH we never reach a state where there are no paths to the solution? ( without reversing moves, as if reversing is allowed this becomes trivial ) Edit : Thanks ...
4
votes
1answer
665 views

Social Golfer Problem - Quintets

I wrote an article on the Social Golfer Problem, which has questions like: Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with ...
14
votes
2answers
868 views

What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
4
votes
0answers
177 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...
16
votes
4answers
971 views

Fun math for young, bored kids?

For 6 months, I'll be organizing, as part as my volunteer work in an NGO, math classes with small groups (~10 students, aged 16 or 17). These classes are not compulsory, but students willing to stay ...
6
votes
1answer
530 views

Solving $n$-queens with determinants

I keep reading about a proposed method of finding solutions to the $n$-queens problem using determinants, but I can't find any specific details anywhere. Can somebody explain to me how to find ...
10
votes
2answers
297 views

Zombie Survival: What is the optimal way to place seven entities on an infinite grid to reduce number of adjacent pairs?

I am designing a zombie-survival type scenario in a tabletop RPG game. My system is going to work in such a way that the players take damage at the start of their turns based on how many adjacent ...
14
votes
2answers
503 views

The Farmyard problem

Problem: There is a farmer who has a $1\text{ mile}\times 1\text{ mile}$ square piece of land. He knows that there is a completely straight pipe underneath some part of his property, but it could ...
160
votes
5answers
6k views

Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?

My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question: I am thinking of a number... It ...
5
votes
1answer
421 views

Is there an analytical solution to this nonlinear ODE?

Is there an analytical solution to the nonlinear ODE $$\frac{dx}{d\theta} = -\sqrt{\frac{x^2}{4\cos^2\theta} - \cos^2\theta}$$ over $\theta \in [0, \pi/2]$ with initial condition $x(0) = 2$? Using the ...
37
votes
6answers
2k views

A variant of the Monty Hall problem

Everybody knows the famous Monty Hall problem; way too much ink has been spilled over it already. Let's take it as a given and consider the following variant of the problem that I thought up this ...
5
votes
2answers
267 views

How to suggest new entries to David Wells' “Book of Curious and Interesting Numbers?”

This book. I'm sure many here, if not most, have read it. If not, I recommend it. It's great fun. Is the author even alive? I'd like to suggest a few entries that are not in the latest (1997) ...
8
votes
2answers
829 views

All possible permutations on a Rubik cube ($3\times3\times 3$) can be reached from the initial state?

If I were to represent a state in the Rubik cube as a permutation of the colors on the 9 tiles per side on all sides of the cube, could I reach all possible states (i.e. colorings) by the permutations ...
0
votes
3answers
887 views

Triangle whose height and sides are consecutive integers

This is probably a old puzzle,and maybe you have seen it somewhere else before.Imagine a special triangle. The height and the three sides of this triangle are 4 consecutive integers.Can you figure out ...
3
votes
1answer
107 views

Eight queens problem, wondering about the non-unique solutions

I've done the code that generates all the solutions. But know I am suppose to filter out any redundant solutions based on symmetry and rotations. I have code for vertical symmetry, horizontal ...
3
votes
1answer
382 views

Does solving a Rubik's cube imply alignment?

Today, I got my hands on a Rubik's cube with text on it. It looks like this: Now, I would love to know whether solving the right cube will also always correctly align the text on the cube or ...
4
votes
1answer
226 views

Expressing any given number in the form of $x^y + y^x$

I was told by one of my friends that any given positive integer can be expressed in the form of $x^y + y^x$ where x & y are integers. For example: 17 = $2^3+3^2$ Surprisingly,this could be done ...
3
votes
1answer
2k views

Mathematical model for solving minesweeper situations

Suppose there's a minesweeper board like the following: 1 1 1 A B C Where A, B, C is an unrevealed square which could contain a mine. This can be represented ...
10
votes
4answers
2k views

Ten soldiers puzzle

This is a puzzle from one popular book called "The Man Who Counted: A Collection of Mathematical Adventures",author is Malba Tahan. How to arrange ten soldiers in five lines in such a way that each ...
20
votes
1answer
1k views

Expert Minesweeper Probability Question

This is just a question I thought of while playing minesweeper. I think that finding the solution might be kind of fun, so I'm sharing it with you guys. If you have no concept of what minesweeper is, ...
6
votes
4answers
317 views

How many ways can we place these ships on this board?

I want to find out how many ways we can arrange these ships on this field. I just have no idea how to go about solving this. So I bring it to the Pros! The board is an 8 by 8 Board. There are 5 ...
15
votes
2answers
902 views

Minimally inconsistent Sudoku puzzle

A sudoku puzzle is a partially filled $9\times 9$ grid with numbers $1,\ldots,9$ such that each column, each row, and each of the nine 3×3 sub-grids that compose the grid does not contain two of the ...
1
vote
4answers
369 views

Proof for divisibility rule for palindromic integers

I am studying for a test and came across this in my practice materials. I can prove it simply for some individual cases, but I don't know where to start to prove the full statement. Can you help me? ...
6
votes
1answer
7k views

How are Blackjack Basic Strategy tables calculated (What is the maths behind them)

Lets assume a very basic set of rules and table for them, these rules are unlikely to be seen in any casino and the reason is clear, there is only a 0.04% edge in favour of the casino, this could be ...
7
votes
5answers
9k views

How many bananas can a camel deliver without eating them all?

This is a fun puzzle I was assigned on the first day of highschool (over a decade ago). I just dug it up randomly from under my bed and thought I'd share it with the SE community. At the time, I ...
1
vote
1answer
388 views

The Game of Nim

A position in Nim consists of some piles of coins. Two players alternate, with each move removing a portion of one pile. The winner is the player who takes the last coin. Suppose that the starting ...
5
votes
2answers
250 views

Finding the largest set of integers over an interval where the sum of any 'k' elements is unique

Consider the set $(s_1, ..., s_N) \in S$, where all $s_i$ are positive integers selected from some interval $[M, L]$ and the sum of any $k$ integers in $S$ is required to be unique and to have a ...
4
votes
1answer
296 views

The “beach problem”: does anyone know it? or know how to solve it?

The following problem was given some years ago in the German computer-science contest for pupils ("Bundeswettbewerb Informatik"). It was originally wrapped in a story which I will briefly translate ...
5
votes
1answer
504 views

Space-filling polyhedra (or honeycomb) survey?

Is there a survey anywhere of space-filling polyhedra? MathWorld's article, space-filling polyhedron, mentions about 400 being seen in pre-1981 books and papers. Wikipedia mentions 28 convex uniform ...
4
votes
3answers
1k views

What is the highest number that can be got from 4383 by moving exactly 2 matches?

What is the highest number that can be got from 4383 by moving exactly 2 matches? Number 1 has got 2 matches, so I thought it will be 47831 as I remove two matches from second number (3), but it ...
1
vote
3answers
123 views

compare which two cube is the same

I am solving following problem: The problem states that on figure 1 there is shown a cube with three facets on which there is drawn three section(length). This cube was put on other facet and ...
2
votes
1answer
96 views

How determine largest reflected number

I was trying to determine maximum number from list of given integer in problem 8 here (page 5). So as you see, there are 5 written numbers on paper, and on the wall there is a hanging mirror. We ...
1
vote
3answers
231 views

Deconstructing $0^0$ [duplicate]

Possible Duplicate: Zero to zero power It is well known that $0^0$ is an indeterminate form. One way to see that is noticing that $$\lim_{x\to0^+}\;0^x = 0\quad,$$ yet, ...
2
votes
3answers
271 views

Squares on a checkerboard

How many squares of all sizes arise using an $n$-by-$n$ checkerboard? How many triangles of all sizes arise using a triangular grid with sides of length $n$ ?
3
votes
4answers
681 views

Conway's game of life variations

Is there any known two-dimensional Conway's game of life variation where each cell can not be just on/off but able to hold more states, maybe 4 or 5?
1
vote
1answer
126 views

Need help on proceding a paper about estimating numbers of sudoku

I was reading a paper that I found via spiked math (http://spikedmath.com/comics/424-the-numbers-quiz-solutions.png): http://www.afjarvis.staff.shef.ac.uk/sudoku/sudoku.pdf. I have problem ...
0
votes
1answer
78 views

Game statistics: Extracting interesting patterns out of users and level

I made a small game and in course of time collected fair amount of data between users and level The level chart is long (120 levels) but looks somewhat like this $$ \begin{array}{|c|c|} ...
78
votes
4answers
4k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
3
votes
1answer
228 views

What is known about functions of type $f(n+1) = f(n)^ {f(n)}$?

Update : having looked at Knut's Double Arrow Notation ( Thank you DJC), it seems that this question is nothing more than a frivourless wondering that should be undertaken by whom ever wonders it, it ...
2
votes
1answer
178 views

Evenly dividing candy bar into $n$ pieces

I have 2 friends. We have one candybar and we want to divide it evenly. Unfortunately we don't have any way of accurately measuring and cutting the candybar. Therefore we are looking for a method ...