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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Puzzle: Calculate amount of combinations of "houses" [closed]

I've found this enigma in a french book and I've tried using combinatorics to find the answer but I've been unsuccessful, could you help me out? Here's what's given: "A child wants to build "...
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9 votes
1 answer
181 views

Collatz conjecture but with $n^2-1$ instead of $3n+1.$ Does the sequence starting with $13$ go to infinity?

Let's consider the following variant of Collatz $(3n+1) : $ If $n$ is odd then $n \to n^2-1.$ $1\to 0.$ $3\to 8\to 1\to 0.$ $5\to 24\to 3\to 0.$ $7\to 48\to 3\to 0.$ $9\to 80\to 5\to 0.$ $11\to 120\to ...
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How to calculate the orthogonality error between sine and cosine wave?

As the picture below(assume the magnitude is the same),the zero-crossing points of the SIN and COS signals do not occur at the precise distance of 90°.So I want to figure out the φ which is φx-φy. ...
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What is a sublinear function? Is $y'(x) = x^5(e^{4-y^2}-1)$ sublinear?

May I have any simplest definition of sublinear function? I tried reading through Wikipedia but couldn't understand it well. Moreover, how can I check whether any function follows sublinearity or not? ...
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1 vote
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What is the smallest unseen number in an iid sample? (From "A number NOBODY has thought of - Numberphile")

In this Numberphile video, the question: "What is a number nobody has thought of?" is addressed. The method is as follows: Estimate a number $N$ as the number of times humans have thought ...
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-1 votes
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Whats 7z + 8y if z = 4 and y = 9 [closed]

What's 7z + 8y if z = 4 and y = 9 . I need help with this so if you can what's 7z + 8y if z = 4 and y = 9 . Just post a comment on what the answer is.
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2 votes
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What is the probability that three living people in the same family will celebrate their birthdays on exactly the same day.

I celebrate my birthday on the same day as one of my grandchildren. Just wonder how rare it would be for three people in the same family to celebrate their birthdays on the same day.
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Calculate an ambiguity score based on LDA topics and Hellinger distance

I am trying to calculate some sort of ambiguity score from text based on topic probabilities from a Latent Dirichlet Allocation model and the Hellinger distance between the topic distributions. Let’s ...
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1 vote
1 answer
80 views

Magic squares for everybody: for statesmen and pedestrians

The book [1] is a book focused mainly in Franklin's magic squares but it has very interesting and suggestive sections and paragraphs that accompany this topic (summarizing is a jewel). I refer that ...
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-2 votes
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Feller formula for the probability of a pair of heads or tails coming out in succession.

Referring to https://en.m.wikipedia.org/wiki/Feller%27s_coin-tossing_constants In the example given, If we toss a fair coin ten times then the exact probability that no pair of heads come up in ...
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When the sum of two sins a constant for any odd n

I'm trying to solve an interesting trig/optimisation problem that I've never seen before. Thought I'd reach out for inspiration. I need to figure out a solution (value of x & y) to the below ...
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5 votes
0 answers
101 views

N lousy shooters in a gunfight

$N$ players are in a gunfight. Starting from player 1, each player takes turns to act in the order of $1,2,...,N,1,2,...$ In their turn, a player randomly chooses one of the other remaining players as ...
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8 votes
3 answers
429 views

Connecting $\sqrt{i \sqrt{i \sqrt{i \sqrt{i \dots}}}}$ to an infinite process?

I just watched this YouTube video by Michael Penn about the expression $$\sqrt{i \sqrt{i \sqrt{i \sqrt{i \dots}}}}$$ and how to evaluate it. At the end of the video, he mentions that if we do not ...
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1 vote
2 answers
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General formula for the upper bound of pi involving nested square roots (circumscribed perimeters of regular polygons)

The formula for the lower bound of pi involving nested square roots looks like this: $p_{2^m} = 2^m\sqrt{2-\sqrt{2+\sqrt{2+ \sqrt{2+...}}}}$ where there are $m-1$ nested square roots. For example, ...
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2 votes
1 answer
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Minimum amount of points in $\mathbb{R}^{n}$ which satisfies certain conditions

The problem that I'm working one is the following Let $S$ be a finite set of points in $\mathbb{R}^{n}$. A stamp set $S$ is a set such that every point in $\mathbb{R}^{n} \setminus S$ is an irrational ...
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2 answers
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Can every number be written as $2^{a_1}+\cdots+2^{a_n} + 1$?

I am reading an algorithm that calculates $x^y$. Basically it is about an implementation of a function $power(x, y)$ where $x$ is the base and $y$ is the exponent i.e. the power. The algorithm uses ...
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1 vote
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Formal application of Pigeonhole principle on voting and candidates

I am reading about the Pigeonhole principle and the following problem under that section: A state has $7$ counties. In one year, three candidates run in a statewide election. Is it possible that in ...
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1 vote
1 answer
78 views

Are there any non-trivial stable tournaments?

In graph theory, a tournament is a graph where every pair of vertices are connected by exactly one directed edge. In this problem, each tournament represents the possible outcomes of a two-player ...
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Math talk with constraint on words (e.g. only use 6-letters words)

I am trying to find a math talk on YouTube, which I had seen a while ago. The only thing I remember is that it was an entertaining talk (like a fun challenge) where the speaker had a specific ...
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2 votes
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$\frac{2(1+4a^2)}{(12x-1)^3}\leqslant \frac{(1-a)^4}{[12x-(1-a)^2]^3}+\frac{(1+a)^4}{[12x-(1+a)^2]^3}$ for $0\leq a<\frac13$ and $x>\frac{(1+a)^2}8$

How to prove the inequality below? $$\frac{2(1+4a^2)}{(12x-1)^3}\leqslant \frac{(1-a)^4}{[12x-(1-a)^2]^3}+\frac{(1+a)^4}{[12x-(1+a)^2]^3}$$ holds for all $0\leqslant a<\frac{1}{3}$ and $x>\frac{(...
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5 votes
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Which objects can be Minkowski halved?

The Minkowski Sum of two subsets $A,B \subset \mathbb{R}^n$ is $$A \oplus B = \{a + b | a \in A, b \in B\}$$ For a given $A$, is there some condition that tells me when I can find a $B$ such that $A = ...
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2 votes
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What is the probability that I live in 2 different houses with the same door number at different points in my life?

The first house I used to live in at the door number 486. After a few years time, I lived in another house with the same door number 486 which I thought was incredibly lucky. I was curious what is the ...
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4 votes
1 answer
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When is the best date to switch from civilian to military factory production in Hearts of Iron IV?

The video game Hearts of Iron IV is a WWII grand strategy game. One of the most important aspects of this game is industry. There are three types of factories: civilian, military, and dockyards. ...
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-2 votes
0 answers
322 views

Ec primes: a pattern? [closed]

Ec numbers are numbers formed by concatenating in base ten two neighboured Mersenne numbers, ec(2)=31 for example, ec(3)=73. ec(69660) and ec(92020) are primes. The first thing I noticed is that: $...
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2 votes
1 answer
72 views

Doubt about Gardner' solution to bug chase problem

A famous Martin Gardner problem goes something like this. Four bugs are placed on the corners of a 10-inch side square. Each bug is looking at the bug to its right and starts to walk towards it. ...
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1 answer
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Calculating damage per second that can stack and have stack limit, with variety of durations

Lets say, a plane attack a ship once every 12.80 seconds, each attack throw 6 bombs, and each bomb have 25% chance of giving the ship 1 stack of fire. When the ship get 1 stack of fire, it will deal ...
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3 answers
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At any given point in time two players have finished the same number of games

I am reading the following problem, which falls under the Pigeonhole principle. A chess tournament has $n$ participants and any two players play one game against each other. Then it is true that at ...
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How to invert $ \binom{9}{x} \binom{x}{\lfloor \frac{x}{2} \rfloor} = \sum_{k=0}^{3} f(x,k) \: 2^k $ in relation to Tic-Tac-Toe?

"Tic-Tac-Toe" is a game played on a 3x3 grid where 2 players alternate taking turns placing tokens (X's and O's) onto the grid. The game is completed when one player has reached 3 of their ...
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33 votes
2 answers
806 views

Does Fermat's Last Theorem imply $\sqrt{2} \not \in \mathbb{Q}$?

A well-known overkill proof of the irrationality of $2^{1/n}$ ($n \geqslant 3$ an integer) using Fermat's Last Theorem goes as follows: If $2^{1/n} = a/b$, then $2b^n = b^n + b^n = a^n$, which ...
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How to draw the function $y'(x) = y(x)(y(x)-1)^{1/3}$ qualitatively?

Let $k \in R$, consider the following Cauchy problem $$y'(x) = y(x)(y(x)-1)^{1/3} $$ $$y(0) = k $$ To draw the graph of solutions, defining the domain, studying the monotonicity, the convexity, and ...
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0 votes
1 answer
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How is the diagonal constraint in lattice path needed for the Catalan proofs?

I have been reading about the Catalan numbers and how they are they appear in many problems such as: lattice paths valid pair of parenthesis mountains with up/downstrokes non-crossing handshakes ...
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  • 1,507
0 votes
1 answer
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Formula to maintain distance along the x-axis between an outer object and an inner object while scaling

Formula to maintain distance along the x-axis between an outer object and an inner object while scaling: ...
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  • 415
3 votes
1 answer
115 views

Group of 9 people, with(out) 3 people who all know each other

Problem: In a group of nine people, one person knows two of the others, two people each know four others, four each know five others, and the remaining two each know six others. Show that there are ...
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0 votes
2 answers
57 views

Integration of $\int_0^\frac{\pi}{4} (\sin^6 2x+\cos^6 2x) \cdot\ln (1+\tan x) dx$

I've found an integration problem from Molodova Matholympiad 2008. The problem is as follows. Find the Integration of $$\int\limits_0^\frac{\pi}{4} (\sin^6 2x+\cos^6 2x) \cdot \ln (1+\tan x)\ \mathrm ...
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0 votes
2 answers
27 views

Place a nonzero digit in each space so that the equation is true.

Place a nonzero digit in each space so that the equation is true. 0.2_ * 7._ = 2._ Here is the work I've done so far: 2/7=0.285714... Then I did some guessing and checking and got 0.28 * 7.5 = 2.1. ...
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22 votes
6 answers
4k views

What is meant, exactly, by nonrepeating when talking about irrational numbers?

My question is referring to the exact definition mathematicians use when describing the decimal expansions of irrational numbers as "nonterminating and nonrepeating." Now, I understand, at ...
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12 votes
1 answer
271 views

Representing graphs by an arrangement of chess rooks

Consider a potentially infinite chessboard on which a number of rooks has been placed, under the restriction that any 2x2 square containing at least 3 rooks must contain a 4th rook. This way, for ...
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2 votes
2 answers
97 views

The limit of $\lim_{n \to +\infty} a_0+a_1+a_2+.......a_n$ where $a_p=\sum_{i=0}^p (-1)^i\frac{\binom{p}{i}}{(i+2)(i+4)}$

I have taken this question from molodovian national MO 2008 The question is as follows The sequence $(a_p)_p\ge 0$ is defined as $$a_p=\sum_{i=0}^p (-1)^i\frac{\binom{p}{i}}{(i+2)(i+4)}$$ Now let's ...
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2 votes
1 answer
112 views

Why are the catalan numbers giving the unique/correct patterns from all the combinations?

I am reading about catalan numbers and they are considered to represent the number of valid pair of parentesis, mountains etc. Although the number checks out correct when comparing against specific ...
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1 vote
1 answer
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How many different ways can you solve the cracker barrel peg game? And is there an optimal starting arrangement of pegs?

The cracker barrel peg game is a strategical table game that is mockingly cocky with its confidence that the player won't win. Let me explain... Here is the game set up: The game itself is a ...
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1 vote
0 answers
62 views

Force Of Attraction Without Bounce

When programming animations controlled by mouse / touch gestures, I like to add some smoothness to the interaction. usually I work with position $q$ and velocity $p$ for the animation, and there is an ...
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-1 votes
1 answer
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Is there a simple test for divisibility by seventeen in base-twelve? [duplicate]

I am investigating math in the dozenal (a.k.a. duodecimal, base-twelve) system. As part of this, I am compiling a list of tests for divisibility. (All numbers in this post are dozenal, not decimal, ...
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0 votes
1 answer
60 views

How do you create an expression using $\tan$ and $\arctan$ that evaluates to an integer?

I was looking at one of those cool/nerdy clocks that has math expressions in place of numbers. For the number $3$, they had the expression $$79\tan\left(\frac{\pi}{8}-\frac{5}{2}\arctan\left(\frac{1}{...
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1 vote
1 answer
42 views

Definition of aperiodic tiling

I think I got confused with the definition of aperiodic tiling. Look at the following example: First, try to find a "1-dimensional aperiodic tiling". Start with the string 0, then make the ...
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4 votes
2 answers
228 views

Is there a simple test for divisibility by sixteen in base-twelve?

I am investigating math in the dozenal (a.k.a. duodecimal, base-twelve) system. As part of this, I am compiling a list of tests for divisibility. (All numbers in this post are dozenal, not decimal, ...
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  • 751
0 votes
0 answers
37 views

A math logic problem [duplicate]

Say there are 12 people on an island, 11 weigh the same. One is either lighter or heavier. There's a beam balance to measure the weights. Is it possible to find the heavier OR lighter person in only 3 ...
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7 votes
3 answers
143 views

Mathematics hiding in plain sight

What are some basic math facts (say, secondary or early undergraduate level) that somehow went unnoticed by you for a long time, and when you realized they made you wonder how you could have missed ...
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0 votes
2 answers
45 views

Finding all numbers such that this algorithm calculates their square

A relative of mine found an algorithm on TikTok that could supposedly calculate the square of any two digit number. The number 35 was used as an example, so I shall use it to explain how it works: ...
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2 votes
1 answer
69 views

To show that there are two people in the tournament with the same starting hand and position in a Texas Hold'em poker tournament.

A Texas Hold'em poker tournament has $1557$ players, in $9$ distinct positions at $173$ tables. Each table has a standard $52$ card deck, and each player is dealt $2$ cards. Hands are considered "...
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-1 votes
1 answer
80 views

Psychic predict all hard-to-predict games in the first round of the NCAA "March Madness'' tournament.

A psychic calls everyone on a list of $275$ sports bettors with tips about who will win the "hard-to-predict" games in the first round of the NCAA "March Madness'' tournament. This ...
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