# 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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### 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 ...
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 ...
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 ...
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 ...
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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### 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 ...
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 ...
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: ...
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. ...
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 ...
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 ...
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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### 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 ...
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 ...
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 ...
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### 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 ...
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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### 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 ...
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 $366$...
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 ...
137 views

### 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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### 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 ...
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### 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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### 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 ...
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### 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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### 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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### 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 ...