Questions tagged [recreational-mathematics]

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

684 questions with no upvoted or accepted answers
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Mondrian Art Problem Upper Bound for defect

Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles? ...
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614 views

Can we remove any prime number from this strange process?

This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post. The process goes as follows: Start with the ...
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849 views

Mathematics for the afterlife

Paul Erdős said this about the $3n + 1$ conjecture: Mathematics may not be ready for such problems. Similarly, there are parts of mathematics that I am not yet ready for. Some things, however, I ...
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597 views

Pattern in Pascal's triangle

Known pattern In general, numbers in pascals triangle produce sierpinski-triangl-like fractals. Some examples can be seen in this answer and are explained in this answer. If we take the last digit ...
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This should be a piece of cake… right?

You probably know the following problem: We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only ...
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416 views

The Complexity of “The Baby Shark Song”.

This question is just for fun. I hope it's received in the same goofy spirit in which I wrote it. I just had the pleasure of reading Knuth's "The Complexity of Songs" and I thought it'd be hilarious ...
14
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154 views

Numbers whose decimal digits are the coefficients of its continued fraction form

A curious question recently crossed my mind. And that is can we construct decimal numbers of the form $$\text{"a.bcdefghij..."}$$ where each letter represents a digit $0-9$ (the number may or may ...
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722 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
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371 views

Strategy for board game 2

In this question the following was asked: Alice and Bob are playing the following game: They have a $4 \times 4$ empty grid and take turns coloring one square each, starting with Alice, both using ...
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490 views

Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
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198 views

Generalization of Gray codes

A friend of mine asked me if it was possible - physical difficulties aside - to generate all 32 combinations of raised/lowered fingers by changing status of a fixed number of fingers at every step. ...
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287 views

Number of circles in configuration

Consider the $n^2$ lattice points $(i, j)$, where $1 \leq i, j \leq n$. Let the number of circles that pass through at least 3 points of this set be $C(n)$. What is a good way to count this? Is there ...
12
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1answer
329 views

Mathematics of the Ice Bucket Challenge

I've been considering the mathematics of the now global ice bucket challenge. Simple model In the simplest incarnation, there is one original seed, who then nominates 3 others, each of which take ...
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369 views

What turmite runs the longest before becoming predictable?

When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable ...
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203 views

Take an m x n grid, and in each box pick two opposite corners at random to connect. What can be said about the resulting pattern?

Inspired by the upcoming book 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior. Here's an example: (Note that I'...
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1answer
471 views

A non-composite sequences

Can you provide a counterexample for a claim given below? Inspired by Puzzle 937 I have formulated the following claim: For any $n > 0$ let $B = p_1 \cdot p_2 \cdot .... \cdot p_n$ be the ...
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200 views

Frogs jumping on trees

The frog game on a Tree: Default start of the game is placing one frog on each node (vertex). The goal is to move all the frogs to one single node. A single move consists of moving all $n$ frogs from ...
10
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1answer
216 views

What would you see inside a spherical mirror?

Image to build a huge spherical shell made of semitransparent glass, and to cover the internal part with a reflecting material. In such structure some light can enter, and an observer inside it (e....
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188 views

Rep-tiles of order 4 and order 9

Rep-tiles are figures which are dissected by the same figures as itself. As you can see, the rep-tile of order 4 is also a rep-tile of order 9 in the above figures: Compare the L-shaped figure at ...
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305 views

Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the digits $0$ through $9$, each used at ...
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403 views

$5 \times 5\;$ “square additive set”

Problem: IBM Research - Ponder This - January 2019 monthly contest (which was closed few days ago) leads to the problem: Find sets $A = \{a_1,a_2,\ldots,a_n\}$, $B = \{b_1,b_2,\ldots, b_m\}$ such ...
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Largest Numerator of Sum of Egyptian Fractions

What is the largest possible numerator when put in reduced form over all sums of the form $$\sum_{k=1}^n\frac{c(k)}{k}$$ where $c(k)\in\{-1, 0, 1\}$? An easy bound is to consider what happens when we ...
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1answer
394 views

leaf-labelled unordered, rooted binary trees and perfect matchings

While playing with findstat.org, I noticed the following coincidence: The number of leaf labelled unordered rooted binary trees with $n+1$ leaves $\{1,\dots,n+1\}$, with the leaf labelled $1$ at ...
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232 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all positive integers $m$ such that the ratios $$ \frac{2(5^m+5)}{3^m+1}\quad\text{and}\quad \frac{9^m+1}{5^m+5}$$ are both integers. Attempt at a solution: If the ratios are both ...
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Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they ...
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Heesch numbers in 3D

At the Tiling Database there are 3, 20, 198, 1390 (A054361) non-tiling polyominoes of order 7 to 10. In 3D, solids with a particular Heesch number don't seem to be well known. Glenn Rhoads found ...
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How can we formalize Jorge Luis Borges' Aleph?

Background. Jorge Luis Borges was a post-modern short-story writer of the 20th century, whose stories often invoke a healthy dose of surrealism. One of his works is called The Aleph. In this book, ...
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1answer
118 views

Reversing the digits of $2^n$ to yield a prime power

This question asks about the existence of an $n\in\Bbb N$ such the number obtained reversing de digits of $2^n$ is a power of $7$. The same question with $5$ instead of $7$ was asked by Freeman Dyson, ...
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324 views

Separating Heavier from the Lighter Balls

This was posted Here and received a good answer, solving the general questions in either $n$ or $n+1$ moves, which is by just half a move on average "less good" than my manual solutions here. Classic ...
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242 views

Limit approximation for $\pi$ in the four fours puzzle?

The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
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Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in $\...
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Solution manual to Larson's “Problem Solving through Problems”

I am working through Larson's "Problem Solving through Problems" (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) but many of the problems have neither solutions nor sources included. Does ...
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120 views

How many consecutive integers can be made using: addition, subtraction, multiplication, division?

Update: "Weak" lower exponential bound was established: $f(n)\ge 3^{n/5}$, and $f(5)$ improved. Problem Up to how many consecutive integers $\in\mathbb N$, can be made using only the four basic ...
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63 views

Borromean Weaving

Here's a picture by Rashmi Sunder-Raj. Is this topologically equivalent to Borromean Rings?
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Determining finitude or infinitude from a simple geometric construction

Playing with a pencil on a checkered sheet I encountered this construction: 1) take a point $A$ on the grid and a point $B$ that is distant from $A$ $n=2,3,4...$ horizontal steps and $1$ vertical ...
7
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1answer
109 views

Expected area “sweeped” on an infinite Minesweeper board

Given an average density (x/y) of x mines in y squares, is it possible to calculate the expected number of squares you can "sweep" (i.e. identify whether there is a mine or not) on an infinitely sized ...
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1answer
176 views

Packing problem - how to fit things nicely with just the aspect ratio of the objects

Suppose you have a rectangle that is w units wide and h units tall (bounding rectangle). You also have an even ...
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202 views

The Heegner Polynomials

What is special about $x^3- 6 x^2 + 4 x -2$? The 24th power of the real root - 24 is curiously close to two other numbers, one being the Ramanujan constant. There are more of these polynomials ...
7
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1answer
188 views

On prime numbers of the form $7\times10^n+69$ and the lights out puzzle

Consider those natural numbers $n$ such that $7\times10^n+69$ is a prime number. The first $15$ such numbers are $1$, $2$, $3$, $6$, $7$, $8$, $10$, $12$, $13$, $21$, $46$, $68$, $91$, $153$, and $366$...
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172 views

Magic Cubic Curve Permutations

The permutation $(-2,9,-4,7,-6,5,-8,3,1)$ can be considered magical. With their negative values diametrical to $0$ at $(0,0)$, a placement of integers begins so that all zero-sum triples form straight ...
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How are Trott constants found; are there mathematical results?

While reading Steven Finch's wonderful book Mathematical Constants, I encountered the Trott constant which was presented as the real number such that the digits of its decimal expansion are the digits ...
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Sequential square packings

There are various studies for packing sequential squares of size $1$ to $n$. We can try to find the smallest square they will pack into, as in tightly packed squares. We can find the smallest square ...
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417 views

Oblongs into minimal squares

Consider $a(n)$, the minimal number of squares into which the oblong of size $(n+1)\times(n)$ can be divided. What is the behavior of $a(n)$? The first 379 terms of the oblong square packing sequence ...
7
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1answer
115 views

For what numbers is $a_{b}= b_{a}$? (Reference?)

A student recently asked me about solutions to the equation $$a_{b} = b_{a},$$ where the subscript notation $a_{b}$ denotes interpreting the digits of $a$ in base $b$. It turns out there are tons of ...
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2answers
168 views

Determining the strength of an abstract army

Lets imagine we have two armies, represented by lists of pairs of positive numbers, like this: [($attack1$,$defence1$),($a2$,$d2$)...($an$,$dn$)] face each other in combat. The rules of combat are ...
7
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1answer
177 views

From prime to prime by squaring the digits

I took prime $131$, squared digits of it and wrote them in natural order as they appear, from left to right, and obtained $191$, then I obtained $1811$ by the same procedure, and then $16411$ and then ...
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63 views

Is this kind of “Gerrymandering” NP-complete?

Consider the following simplified form of "Gerrymandering": You have $n^2$ squares arranged as an $n\times n$ matrix. Each square is marked with either $0$ or $1$ which means a "voter preference" ...
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121 views

What's up with the cycloid-shaped pot in Melville's Moby Dick?

People interested in the intersection between mathematics and fine literature may be familiar with the following quote from Herman Melville's famous novel Moby Dick: It is a place also for profound ...
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161 views

People's preferences assigning them to a group using Excel

I would love some assistance with the following problem. Each year for school camps we take the students preferences 1-6 of who they'd like to bunk in with for the week. We usually sort this out ...
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134 views

Does inscribing a circle, then a triangle, then …, inside of an initial triangle telescope to some “center” of that triangle?

Start with a triangle, and construct its inscribed circle. Take the three points where the inscribed circle is tangent to the triangle, and construct a new triangle with those points as the vertices. ...