Questions tagged [recreational-mathematics]

Mathematics done just for fun, often disjoint from typical school mathematics curriculum. Also see the [puzzle] and [contest-math] tags.

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46 views

Flatland and dimensions

Isn't flatland assuming that the 2d objects have sides because otherwise there's no way they can even see lines? Even further, how come their eyes exist on sides, when they only exist in the 2d ...
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What's the minimum number of keystrokes needed to type n asterisks?

Suppose you're editing a plain txt file and need to input a line of $n$ asterisk characters. You could press the * key $n$ times (assuming you have a numpad with a dedicated * key) but there is ...
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Show that the set of factors of 12 under divisibility form a lattice. [closed]

Show that the set of factors of 12 under divisibility form a lattice.
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If a continuous unimodal function intersects a weakly decreasing function from above, must it be after it reaches its apex?

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, unimodal function$^{1}$ and $g:\mathbb{R}\to\mathbb{R}$ be a continuous, weakly decreasing function. Suppose that $f$ intersects $g$ from above at $z\...
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Accounting for Chance Changes in Averages

Moderators: I really don't know what to name this or what tags to put so feel free to edit this I'm a software developer, and in my spare time, I wanted to create a computer program to play a card ...
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Expected Value Of a Random Grid That Contains Multipliers

I am a software developer and recently came across this problem when working on a hobby project, and my knowledge of probability is too small to solve this. I have a bag that contains 5 negative ones, ...
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2answers
59 views

Sum of Partial Sums of Geometric series

Suppose we have a Geometric Sequence {a,r} a being the initial term and r being the common ratio. Is there a condensed formula for Sum of upto nth Partial sums of the terms in Geomtric series. $$ S = \...
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1answer
127 views

1/109 and Fibonacci

I saw the image at the end of this post and wanted to check it. With the Mathematica code Abs[N[1/109, 106] - Total[Table[Fibonacci[n] 1/10^(109 - n), {n, 1, 89}]]] ...
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Evaluating $\mathrm{\sum_{x=2}^\infty \left(\frac1{π^2(x)}+\frac1{π\left(x^2\right)}\right)}$

It can be seen that: $$\sum_{n=1}^\infty n^{-2}=\frac{\pi^2}{6},\sum_{n=2}^\infty \frac{(-1)^n}{π(n)}=1$$ Where π(x) is the prime counting function with $π(0\le x<2)$=0 hence the starting index at ...
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What's the optimal strategy in the following board game?

The following is not a homework problem. I just found it on a Chinese online community. In the following variant of the board game Risk, suppose you and your opponent both start with armies of equal ...
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Functions with “Unstable” Antiderivatives in Terms of Elementary Functions

I recently came across the following claim on this Wikipedia article: I assumed that this is due to a small change in one of the coefficients of the polynomial yielding a large change in the roots, ...
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Team Balancing / Score Normalisation for gamification

This is the first time I post in Mathematics as I'm normally in programing. But for this question, I think I can get better help here. I'm trying to create a gamification system where different teams ...
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42 views

Can we approximate the occurrence of an arbitrary structure in a subset of $\mathbb{N}$ from the growth rate of that set?

I'm wondering if it makes sense to use the growth rate of a subset of $\mathbb{N}$ to approximate when an arbitrary structure or construction that is defined independently from the set is likely to ...
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38 views

How to calculate win probability for each team in a Round of 16 knockout tournament

Let's say there are 16 teams in a standard Round of 16 bracket. The initial matchups are known (A plays B, C plays D, etc.). We know the probability for each possible matchup (A has a 54% chance of ...
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Pushing Nontransitive Dice to the Limit

Nontransitive, sometimes intransitive or non-transitive, dice are a fascinating concept in probability. It concerns dice such that, in head to head matches, instead of having a neat ranking of "...
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2answers
50 views

Explanation for this method to find incenter of triangle

Euclidea is a mobile game that requires you to construct certain geometric structures using only a straightedge and a compass. One of the levels requires you to consruct the incenter of a scalene ...
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Functions with dense singularities.

$\underline{\text{Motivation}:-}$ I started thinking about this question a few days ago while checking on my old questions. I thought about a similar question to this one, but instead of roots this ...
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Finding the maximum number of members in the math club.

I asked a question few days ago which was from a local math contest in my city. The question and the solution seems interesting to me and I am interested in solving the generalization of the problem. ...
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Jacobsthal numbers occurring in reduced cobweb plot for the Collatz Problem

I have been working on the Collatz problem for a while now, and have made this efficient cobweb plot function for it, where it automatically does all the dividing by two and always returns an odd ...
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What is the function that defines this set? How can I go about finding this?

I have the following set: x1 | x2 | x3 | Nx 30 | 50 | 20 | 83 30 | 20 | 50 | 79 45 | 15 | 40 | 74 65 | 35 | 45 | 72 50 | 20 | 30 | 77 55 | 25 | 30 | 79 $$...
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Finding the formula for the ulam spiral starting with $0$ as a bijective function $U:\mathbb{N}\rightarrow\mathbb{Z×Z}$

$\underline{\text{Introduction}:-}$ For the last few days I've been wondering about the question below. I don't think that my approach is an elegant approach but this is the best I can do. (I am a ...
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Number of solutions of $\sum_{i=1}^{n} a_i \exp(b_i x)=0$, for $x\in\mathbb R$

Let $a,b \in \mathbb R^n$. I am interested in the following equation: $$ \sum_{i=1}^{n} a_i e^{b_i x} = 0 $$ where $x\in\mathbb R$, $a = (a_1,\dots,a_n)$ and $b = (b_1,\dots,b_n)$. Edit: $b_i\neq b_j$ ...
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How many people that are vaccinated should be hospitalized in a city that has 20% of the population vaccinated? [closed]

Lets say, in a city where 20% of the population is vaccinated for COVID-19, 3 vaccinated people are hospitalized and 8 unvaccinated are hospitalized; a total of 11 people are hospitalized. How would ...
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Is there an exact solution for $\large\int \frac{dx}{\tan^{-1}(x)}$?

Do you know about the inverse tangent integral function? It is defined as: $$\mathrm{Ti_2(x)=Ti(x)=\int_0^x\frac{tan^{-1}(x)}{x}dx=-\frac1x\sum_{n\ge 1}\frac{(-1)^nx^{2n}}{(2n-1)^2}}$$ Expanding the ...
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1answer
31 views

Is it possible to find the i'th largest number using only min and max functions?

If we are only able to use min and max which each take two integer arguments and return the lowest and highest integer, ...
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1answer
183 views

How many paths exist on an $n\times m$ grid?

How many paths exist on an $n\times m$ grid? The Path requirements are You start at the bottom left and end at the top right The Path can not intersect it's self Each Line has to start and end on a ...
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1answer
52 views

Is $2p_ip_j\lt p_{j+1}^2$ for $i\lt j$?

Initially, I was wondering is the following statement true or not- Let $p_k$ be the $k^{\text{th}}$ prime. Then, $2p_ip_j\lt p_{j+1}^2$, where $i\le j$ But then I realized that the case $i=j$ has ...
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1answer
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Is it possible to factor the risk ratio from this equation?

I have this weighted risk ratio equation here: $$\frac{a(b/e+d/f)}{b(a/g+c/h)}$$ I am interested in factoring this out of the equation, i.e. the unweighted risk ratio: $$\frac{a(b+d)}{b(a+c)}$$ Is ...
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2answers
188 views

Circle from (2D) random walk

A method is presented to create circles from sort of random walks. The hobby project is based upon two earlier topics: circles from $n-$gons with circumference: $C=1$ SE or area: $A=1$ SE. The $n-$gon ...
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1answer
69 views

Could Fermat's Little Theorem be expanded to include some semiprimes for p?

Fermat's Little Theorem states that a^p%p=a. For example, if a=2, then 2^p%p=2 if p is any prime. That's all the theorem says. However, I tried plugging in 3p instead of p, and I got an interesting ...
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Analytically characterizing the set of integers at which the floor and ceiling of two affine functions with the same slope are equal.

Assume that $a>0$, $b_0>0$, $b_1>0$ and that $-2<b_1-b_0\leq -1$. Define $f_0(x) = \lfloor b_0-ax \rfloor$ and $f_1(x) = \lceil b_1-ax \rceil$. Finally, define the following set: $$\...
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1answer
191 views

Evaluating $\mathrm{\int_0^1 ?(x)dx}$

This function interestingly shown as ?(x) is dubbed the Minkowski Question Mark Function. It looks very similar to x. Wolfram Alpha can even plot the derivative of this apparently smooth function. ...
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63 views

How 2 special constants are related?

For the following, how could I prove the statement $(1-i) \text{CMRB}=\text{M2}-\text{part}$ is true? $$f(x)=e^{i \pi x} \left(1-(x+1)^{\frac{1}{x+1}}\right),$$ $$\text{CMRB}=\sum _{n=0}^{\infty } ...
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How do I calculate exponent (x^n) of a certain number (x) without self multiplication and without computers? Any efficient methods?

I have a number 'x' raised to 'n', and I want to calculate the x^n without x.x.x.x....(n times). How do I do that? Is it possible to do it without the tedious self-multiplication? (I mean to do it ...
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Probability of a mini game in “Mario Party 4”

There's a minigame in Mario Party 4 called "Hide and Go Boom". 4 players must compete in the game. There are 4 cannons labeled A,B,X, & Y. 3 players are "hiders", and 1 player ...
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Simple Geometrical Proof

I know this will seem like an obvious question to many of you, but: Consider an acute $\triangle ABC$ with $AB=AC$. Also let the midpoint of $BC$ be $M$. Is it always true that $A$, the circumcenter ...
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How to play a winning move at this Nim board?

assuming that I have this situation at Nim: 1,6,12,34,45,23,56,101,212 If I understand right, the is a winning situation because the Nim number is: ...
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Numbers that are equal to the concatenation of the positions of its primes

Given a natural number $N\in\mathbb N$, we can construct another number as follows: Let $p_1\leq p_2\leq\ldots\leq p_n$ the prime factorisation of $N$, where duplicates can occur. Let $\pi$ be the ...
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229 views

How can I get started doing math for fun?

My math teacher says that they do math for fun, and I wonder how they do that? I want to do math in my free time. Where can I find problems to solve that will engage me and will take a while to ...
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I love maths but I am lazy to read maths undergraduate textbooks in the least [closed]

I started developing passion for mathematics back in 2013 when my Visual Basic Programming language tutor asked me and two other students to write a program to compute the factorial of any positive ...
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175 views

Is $404$ a palindrome in base negative $31$?

I was browsing on a website and I accidentally clicked on a link (Here is the link, but it may not show the same thing). The following was written there: $404$ is also a palindrome in base negative $...
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1answer
414 views

Classifying all numbers $n$ with the property that if $p$ is a prime, then $p \mid n \iff p-1 \mid n$

If a number $n$ has the property that if $p$ is a prime, then $p \mid n \iff p-1 \mid n$, we call $n$ a nice number for brevity. A recent question on MSE (edit: now deleted) asks to prove that $1806$ ...
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309 views

What is the maximum “alive density” of cells in Conway's Game of Life when played on a torus?

I've read that Conway's Game of Life (CGOL) can have unbounded growth from a finite initial number of alive cells (e.g. a glider gun). However, if CGOL is played on a torus, space (the number of cells)...
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3answers
54 views

How can we count the number of combinations without casework in this problem?

This is an interesting problem that I remembered today: If a billboard can be painted either red, orange, or yellow, and it is never painted red for 3 days in a row, then how many paint-sequences ...
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1answer
86 views

Number of Niven numbers less than or equal to n

Niven (or Harshad) numbers are known as the numbers that are divisible by the sum of their digits. The sequence of Niven numbers begins as $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, ...
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1answer
30 views

How to reduce differential operator to a 4th order ODE of the standard Euler type?

I read this article https://mae.ufl.edu/~uhk/STOKES-DRAG-FORMULA.pdf How do we actually arrive to a 4th order ODE below?
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240 views

How many numbers are there such that its number of decimal digits equals to the number of its distinct prime factors?

Problem A positive integer is said to be balanced if the number of its decimal digits equals the number of its distinct prime factors. For instance, $15$ is balanced, while $49$ is not. How many ...
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Strange parking problem on my street - looking to solve for the probability spots are effectively filled

This my first post here, so forgive me if the style is not to spec yet. The street I live on has the following parking structure: On the 15th of every month the cars switch sides they are legally ...
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66 views

Would it be possible to create a Lucas-Lehmer-like test for primes of the form$ 3^{n}-2$

I was thinking maybe this could be a proof formed around $(3+\sqrt7 )(3-\sqrt7)$ instead of $(2-\sqrt3)(2+\sqrt3)$, but I don't know. I'm not an expert on this by any stretch, just curious.
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1answer
109 views

Find the number of unbeatable integers between $256$ and $1024$ inclusive

Problem Mugdho and Snigdho are playing a game against each other. Mugdho has a red computer. If the computer screen displays the number $x$, Mugdho can make a move and choose to: Add $1$ to $x$. ...

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