# Questions tagged [recreational-mathematics]

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

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### What is the geometry of points inside a square, that are equally distant from the square

Image: A visual of the problem For the square ABCD, what would be the geometry of points that are equally distant (distance r) from all points of the square? How would the shape written out by r ...
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### Maximum run in binary digit expansions

For numbers between $2^{k-1}$ and $2^{k}-1$, how many have a maximum run of $n$ identical digits in base $2$? For instance, $1000110101111001$ in base $2$ has a maximum run of 4. See picture below ...
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### Neat expressions that equal 1

I would like to see beautiful and elegant expressions involving elementary and non-elementary functions, transcendental numbers, etc. that equal 1. Be creative!
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+100

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### Quick way to estimate powers of $9$?

I was trying to estimate how much is $9^{12}$, and I came with my own formula that does the work with powers of $10$, that is : $$9^n\approx|10^n-n \cdot 10^{n-1}|$$ but the larger the $n$ (integer) ...
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### What is the maximum number of arcs that intersect at a single point (not including the center of the circle)?

Consider $n$ points evenly spaced around a circle. Connect pairs of points that are the furthest away from each other with $k$ arcs connecting each pair of points (these arcs could be pictured as ...
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### How many times are these three quantities simultaneously prime?

For recreation, I would like to count the number of times the path length, "area above," and area below the prime counting function, are simultaneously prime. The following function is used to count ...
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### If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$.

If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? Update: No ...
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### Understanding the following two equations

I'm reading through some texts and came across the following two lines. I'm curious how they should be read. Volume = $4/3\enspace π r^3$ Surface Area = $4\enspace π r^2$ In the example of Volume: ...
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### any Math game for kids?

I need to plan a "game" for kids about 10-11 years old that involves mathematics and some physical activity or game. It must be short-time and not very difficult because it's a stage of a big game. ...
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### I can't solve this modulo operation

I'm trying to find an easy and fast way to get the result of $$353094232 \mod 721.$$ I would solve this by dividing manually the terms until I get the remainder of dividing, but I was wondering if ...
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Let's define a mikado configuration $m$ as a countable collection $\{T_j\}_{j \in \mathbb{N}}$ of disjoint subsets of Euclidean 3-space $(\mathbb{R}^3,\cdot)$. Each $T_j$ is a "tube of radius $R>0$"...
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### maximal minimum distance between points in a rectangle of size $17\times 32$

I am training in problem solving and this one greatly resists to my understanding: Given a rectangle with specified length and width, you have to place 5 points in the rectangle such that you ...
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### Need any kind of insights about a strange function

The function in question is $f(x; b) = \sum_{k=0}^\infty \{ b^k x \}\cdot b^{-k}$ where the base $b$ is equal to $2$ here. Also, $x\in [0, 1[$. The curly brackets denote the fractional part function. ...
### Curves in $\overline{\Bbb{F}}_p$
Let $k$ an algebraically closed field, for example $k = \Bbb{C}$ or $k = \overline{\Bbb{F}}_p$, $a \in k$, $k(x)$ the field of rational functions, fix an algebraic closure $\overline{k(x)}$, let \$...