Questions tagged [recreational-mathematics]

Applications of mathematics for the design and analysis of games and puzzles

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126 votes
67 answers
46k views

Math puzzles for dinner [closed]

You're hanging out with a bunch of other mathematicians - you go out to dinner, you're on the train, you're at a department tea, et cetera. Someone says something like "A group of 100 people at a ...
113 votes
54 answers
53k views

Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in the game's structure, optimal strategies, practical strategies, analysis of the game ...
112 votes
25 answers
36k views

Examples of math hoaxes/interesting jokes published on April Fool's day?

What are examples of math hoaxes/interesting jokes published on April Fool's day? For a start P=NP. Added 2023-04-01 Anything new in 2023?
86 votes
27 answers
19k views

Which popular games have been studied mathematically?

I'm planning out some research projects I could do with undergraduates, and it struck me that problems analyzing games might be appropriate. As an abstract homotopy theorist, I have no experience with ...
78 votes
49 answers
43k views

Examples of interesting false proofs

According to Wikipedia False proof For example the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as ...
74 votes
11 answers
26k views

Does War have infinite expected length?

My question concerns the (completely deterministic) card game known as War, played by seven-year-olds everywhere, such as my son Horatio, and sometimes also by others, such as their fathers. The ...
Joel David Hamkins's user avatar
63 votes
8 answers
14k views

Fair but irregular polyhedral dice

I am interested in determining a collection of geometric conditions that will guarantee that a convex polyhedron of $n$ faces is a fair die in the sense that, upon random rolling, it has an equal $1/n$...
Joseph O'Rourke's user avatar
63 votes
2 answers
3k views

Guessing each other's coins

I recently thought about the following game (has it been considered before?). Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $(A_n)$, and then chooses an ...
Guillaume Aubrun's user avatar
56 votes
28 answers
11k views

Nontrivial question about Fibonacci numbers?

I'm looking for a nontrivial, but not super difficult question concerning Fibonacci numbers. It should be at a level suitable for an undergraduate course. Here is a (not so good) example of the sort ...
51 votes
2 answers
18k views

Is there winning strategy in Tetris ? What if Young diagrams are falling?

Question 1 Is there a winning strategy (algorithm to play infinitely) in Tetris, or is there a sequence of bricks which is impossible to pack without holes? Consider generalized Tetris with Young ...
Alexander Chervov's user avatar
49 votes
1 answer
2k views

Exploding primes

Suppose every prime $n$ could "explode" once. An explosion results in $\lfloor \alpha \ln n \rfloor$ particles being uniformly distributed over the integers in a range $n \pm \lfloor \beta \ln n \...
Joseph O'Rourke's user avatar
48 votes
8 answers
5k views

A sudden smiley? :-)

This is a vague question, and I will no doubt be (properly!) chastised for posing it. I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which has the ...
Joseph O'Rourke's user avatar
47 votes
4 answers
11k views

Volumes of n-balls: what is so special about n=5?

I am reposting this question from math.stackexchange where it has not yet generated an answer I had been looking for. The volume of an $n$-dimensional ball of radius $R$ is given by the classical ...
Andrey Rekalo's user avatar
38 votes
3 answers
4k views

Parabolic envelope of fireworks

The envelope of parabolic trajectories from a common launch point is itself a parabola. In the U.S. soon many will have a chance to observe this fact directly, as the 4th of July is traditionally ...
Joseph O'Rourke's user avatar
35 votes
1 answer
3k views

"The Two Sheriffs" puzzle

This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler. Two sheriffs in neighboring towns are on the track of a killer, in a case involving eight ...
Alexey Ustinov's user avatar
34 votes
34 answers
11k views

Titles composed entirely of math symbols

I apologize for burdening MO with such a vapid, nonresearch question, but I have been curious ever since Suvrit's popular October 2010 Most memorable titles MO question if there were any "$E=mc^2$...
34 votes
2 answers
4k views

Who wins two player sudoku?

Let's say players take turns placing numbers 1-9 on a sudoku board. They must not create an invalid position (meaning that you can not have the same number in within a row, column, or box region). The ...
Christopher King's user avatar
31 votes
9 answers
9k views

Recreational mathematics: where to search?

I am not sure I can strictly define recreational mathematics. But we all feel what it is about: puzzles, problems you can ask your mathematical friends, problems that will bother them for a couple of ...
31 votes
1 answer
9k views

Reconstructing the argument that yields Graham's number

Graham's number achieved a kind of cult status, thanks to Martin Gardner, as the largest finite number appearing in a mathematical proof. (It may no longer hold that record, but that is not my concern ...
Timothy Chow's user avatar
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31 votes
5 answers
2k views

You pass X people and Y people pass you: how relatively fast are you?

This question occurs to me every time I go jogging. I suspect every runner probabilist in the world must have thought of it (though I'm no probabilist), but I could not specifically find it online. I ...
R Hahn's user avatar
  • 2,721
28 votes
6 answers
2k views

How fast are a ruler and compass?

This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO. Consider the standard assumptions ...
John Watrous's user avatar
27 votes
1 answer
979 views

The lion and the zebras

The lion plays a deadly game against a group of $N$ zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the $N$ zebras may ...
Eric's user avatar
  • 2,601
26 votes
2 answers
2k views

Runner's High (Speed)

I find the following mind-boggling. Suppose that runner $R_1$ runs distance $[0,d_1]$ with average speed $v_1$. Runner $R_2$ runs $[0,d_2]$ with $d_2>d_1$ and with average speed $v_2 > v_1$. I ...
Dominic van der Zypen's user avatar
25 votes
5 answers
2k views

Covering a Cube with a Square

Suppose you are given a single unit square, and you would like to completely cover the surface of a cube by cutting up the square and pasting it onto the cube's surface. Q1. What is the largest ...
Joseph O'Rourke's user avatar
25 votes
1 answer
1k views

Is there an open subset $A$ of $[0,1]^2$ with measure $>\frac{1}{100}$ that satisfies this property?

This is a crosspost from MSE. Can we find for any given $\varepsilon>0$ an open subset $A\subseteq[0,1]^2$ with measure $>\frac{1}{100}$ such that, for any smooth curve $\gamma:[0,1]\to\mathbb{R}...
Saúl RM's user avatar
  • 7,916
25 votes
1 answer
2k views

Bouncing a ball down the stairs

In a nutshell, the question is whether it can be faster to bounce a ball down an infinite flight of stairs than to bounce it down a ramp with the same slope. To be more specific: this is a $2$ ...
Jeff Strom's user avatar
  • 12.5k
25 votes
1 answer
2k views

Who wins this two-player game based on the sandpile model?

Given a connected graph $G$, two players, Blue and Green, play the following game: initially, all vertices are unclaimed. Players alternate turns. On her turn, Blue adds a token to either an ...
JBL's user avatar
  • 1,723
24 votes
6 answers
5k views

Neutral tic tac toe

I heard this puzzle from Bob Koca. Suppose we play misere tic-tac-toe (a.k.a. noughts and crosses) where both players are X. Who wins? That particular puzzle is easy to solve, but more generally, ...
Timothy Chow's user avatar
  • 78.1k
24 votes
6 answers
2k views

Number of collinear ways to fill a grid

A way to fill a finite grid (one box after the other) is called collinear if every newly filled box (the first excepted) is vertically or horizontally collinear with a previously filled box. See the ...
Sebastien Palcoux's user avatar
24 votes
1 answer
3k views

Can an odd number of marbles jump to infinity?

Loosely inspired by the game Abalone, I've encountered the following simple problem I cannot solve. Suppose that we are given a finite set of marbles on an infinite chessboard. One move consists of ...
domotorp's user avatar
  • 18.3k
23 votes
2 answers
1k views

Evaluating an integral using real methods

This is a bit of recreational integration. The following, rather attractive integral is quite straightforward via residues: $$\int_0^1 x^{-x}(1-x)^{x-1}\sin \pi x\,\mathrm{d}x=\frac{\pi}{e}$$ ...
ocg's user avatar
  • 453
22 votes
5 answers
3k views

Irreversible chess

Suppose we play a chess-variant, where any finite number of pieces are allowed, and the board is as large as we wish, but only two kings in total. And there is no 50 move-rule, no castling and no ...
GM2001's user avatar
  • 223
22 votes
2 answers
1k views

$x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?

This question was asked on MathStackexchange here, but there was no answer, so I am asking it here. Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...
user avatar
22 votes
2 answers
2k views

How to get rich in a Hilberts Hotel?

Suppose you can make infinitely many copies of yourself. Each of them starts his/her life in a Hilberts Hotel, where each room is labeled by an element in the free group with two generators, and ...
Sune Jakobsen's user avatar
22 votes
3 answers
2k views

An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights. Call a ...
Chris Taylor's user avatar
21 votes
4 answers
2k views

Fairest way to choose gifts

Suppose that a parent brings home from a trip $2n$ gifts of roughly equal value for his/her two children. The children get to choose one at a time which gifts they want. What is the fairest way to ...
Richard Stanley's user avatar
21 votes
2 answers
1k views

Why are this operator's primes the Sophie Germain primes?

I was seeking a binary operator on natural numbers that is intermediate between the sum and the product, and explored this natural candidate: $$x \star y = \lceil (x y + x + y)/2 \rceil \;.$$ Then ...
Joseph O'Rourke's user avatar
21 votes
2 answers
3k views

Walking to infinity on the primes: The prime-spiral moat problem

It is an unsolved problem to decide if it is possible to "walk to infinity" from the origin with bounded-length steps, each touching a Gaussian prime as a stepping stone. The paper by Ellen Gethner, ...
Joseph O'Rourke's user avatar
21 votes
2 answers
5k views

Surface equivalent of catenary curve

A catenary curve is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is: What is the shape taken by an ...
Joseph O'Rourke's user avatar
21 votes
1 answer
803 views

Who wins the Rubik's cube game?

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube (to make the problem easier). Solver and Spoiler take turns making 90 degree twists (starting with Solver). ...
Christopher King's user avatar
20 votes
7 answers
3k views

Are there results in "Digit Theory"?

Results about numbers that are related to their decimal representation are usually confined to recreational mathematics. There I have seen mainly questions about individual numbers, like finding a ...
user avatar
20 votes
4 answers
2k views

Does the set of happy numbers have a limiting density?

A positive integer $n$ is said to be happy if the sequence $$n, s(n), s(s(n)), s(s(s(n))), \ldots$$ eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$. For example, 7 is ...
Dave R's user avatar
  • 856
20 votes
3 answers
677 views

Escaping from infinitely many pursuers

The fugitive is at the origin. They move at a speed of $1$. There's a guard at $(i,j)$ for all $i,j\in \mathbb{Z}$ except the origin. A guard's speed is $\frac{1}{100}$. The fugitive and the guards ...
Eric's user avatar
  • 2,601
19 votes
4 answers
3k views

Is it possible to formulate the axiom of choice as the existence of a survival strategy?

Consider the following situation: There is an infinite set $G$ of giraffes. A lion comes and announces a set $C$ of all possible colours and an infinite cardinal $\kappa$. The hungry lion tells ...
Joonas Ilmavirta's user avatar
19 votes
2 answers
1k views

Does this number exist?

Does there exist $x\in\mathbb{R}$ such that $\lfloor 10^nx\rfloor$ is a prime number for all $n\in\mathbb{N}$?
Dattier's user avatar
  • 3,737
19 votes
1 answer
1k views

A game on sets of reals

A 2 player game on $\mathcal{P}(\mathbb{R})$: Players take turns playing uncountable sets of reals. Each play must be a subset of the previously played set. Player 1 wins if the intersection of all ...
Monroe Eskew's user avatar
  • 18.1k
18 votes
1 answer
1k views

Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?

Does a function from $\mathbb{R}^2$ to $\mathbb{R}$ which sums to 0 on the corners of any unit square have to vanish everywhere? I think the answer is yes but I am not sure how to prove it. If we ...
Ivan Meir's user avatar
  • 4,782
18 votes
2 answers
4k views

Turning pants inside-out (or backwards) while tied together

An entertaining topological party trick that I have seen performed is to turn your pants inside-out while having your feet tied together by a piece of string. For a demonstration, check out this ...
Tony Huynh's user avatar
  • 31.5k
18 votes
3 answers
647 views

Tic-tac-toe with one mark type

Parameters $a,b,c$ are given such that $c\leq\max(a,b)$. In an $a\times b$ board, two players take turns putting a mark on an empty square. Whoever gets $c$ consecutive marks horizontally, vertically, ...
pi66's user avatar
  • 1,199
18 votes
1 answer
1k views

Knight's tours in higher dimensions

I wonder if Knight's Tours have been explored in higher dimensions, using the following definition of a knight move. In dimension $d=2$, the knight moves left/right and forward/back one step and two ...
Joseph O'Rourke's user avatar

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