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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Finding if there relationship between numbers
I have a challenge. This may be little tricky or even not possible but wanted to check if anyone has any thoughts on this?
PS : This question is in general and not related to only to R. May be I can ...
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A game of identifying real coins and weighing them.
Here is my problem as follows.
There are 2 counterfeit coins among 5 coins that look
identical. Both counterfeit coins have the same weight and the other
three real coins have the same weight. The ...
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Weighted count of Egyptian fraction representations
This question emerged during an activity I ran for some middle school students this week; basically, it's about a way to "count" - with an appropriate kind of weight - the Egyptian fraction ...
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Maximum amount of iterations to get to an empty list from repeatedly taking the smallest complement of a list
I was trying to solve Iterative Smallest Complement on the Code Golf website, and thought of an interesting question.
A basic run down of what the smallest complement of a list is:
... ...
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Dependence of Events - Proof
I've been learning for my probability exam checking some proofs in book and I found the following question I can't really answer. Any ideas or help?
Let's assume that events $A$ and $B$ are NOT ...
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Eliminating red balls in a jar with up to four colors
In a jar, there are $r$ red balls, $b$ blue balls, $y$ yellow balls, and $g$ green balls, for a total of $T=r+b+y+g$ colored balls. Let $(r,b,y,g;T)$ or equivalently $(r,b,y,g)$ represent this state.
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prove that one of the players will have two cards with the same numbers
Here's the problem:
There are 25 people sitting around a table and each person has two cards. One of the numbers 1,2,..., 25 is written on each card, and each number occurs on exactly two cards. At a ...
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An un contains cards numbered 2 . 100 Let X be the least number on the 50 cards drawn randomly without replacement from the urn . Then P ( X
An un contains cards numbered 1.2 ... 100 . Let X be the least number on the 50 cards drawn randomly without replacement from the urn . Then P ( X =>5/2) is
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Is it true that for sequences that satisfy the property-type in Goldbach's conjecture, there is an integer which cannot be expressed in a unique way?
Let $A\subset \mathbb{Z}$ be such that $\exists\ c\in\mathbb{Z}$ such that $\forall\ n\geq c: \exists\ a,a'\in A$ with $a+a' = n.$ In other words, $A$ satisfies sort-of Goldbach conjecture, but for ...
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On projecting a sphere onto a cylinder, and preserving horizontal arcs
You take a sphere, and inscribe it in a cylinder (so the edges are touching). You project the sphere onto the surface of the cylinder through rays emanating from the axis of the cylinder outward, ...
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A number is called "a super prime number" if the sum of its digits is equal to a prime number. How many two digit "super prime numbers" do we have? [closed]
I know that I can write all numbers from 1 - 100, but I'm curious if there is another way like using a formula to solve this kind of question.
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Does this autonomous system of two first-order ODE have a solution?
Let $f:[0,1]^2\to\mathbb{R}$ be continuous and let $\delta>0$. Fix initial condition $\vec{x}_0=(x_{10},x_{20})\in [0,1]^2$; time $t$ takes values in $[0,\infty)$. Consider the following system of ...
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What is the probability the statement is true given A and B contradict each other??(Slight twist in a very unique question)
The textbook question goes like this
A and B are independent witness in a case. The probability that A speaks the truth is 'x' and that of B is 'y'.If A and B agree on a certain statement, how to find ...
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A potter makes pots and arranges them in rows for drying. How many pots does the potter make?
A potter makes more than $100$ but less than $300$ pots and arranges them in rows, with each row consisting of the same number of pots, for drying. He finds that if he places $6$ pots more per row, he ...
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Why the degree of $\pi$ does not increase
What is the underlying reason, that $\pi$ is of degree $1$ in the volume formula for an $n$-ball of radius $r$,
i.e. the perimeter of circle is $$2\pi \cdot r$$
its area is $$\pi \cdot r^2$$
the ...
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A quadrilateral version of fagnano's problem.
Here is a problem from the german national math olympiad 2021. This problem looks very similar to fagnano's problem except it is quadrilateral.
The problem is as follows.
Let $P$ on $AB$ ,$Q$on $BC$ ...
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Umbrella puzzle with several places
You all know the Markov umbrella puzzle about professor having umbrellas at home and at office and taking one whenever it rains. But how about if we have more than two places? Of course then the ...
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General equivalence of differential equation and difference equation
Let $g$ be a continuous function (not necessarily smooth). Let $\Delta g=\frac{g(x+\Delta x)-g(x)}{\Delta x}$.
Consider the difference equation:
$F(\Delta\Delta g,\Delta g,g)=K$
Question: Under ...
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Calculate remaining filament based on rotation
This might not be possible at all but I can't get my head around it, not really mathematically minded.
A 3D printer spool obviously depletes as it is used. Knowing the diameter of the filament, the ...
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What is the average minimum value in a set of 100 integers chosen at random uniformly and independently in the range 0-99 inclusive?
Obviously the smallest possible value is 0, but 0 isn't guaranteed to be in the set. I'd assume that most of the time the minimum value would be either 0 or 1, which would make the average minimum be ...
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Exploring the divergent series and busting misconceptions! [closed]
I require clarifications on some things.
$1 + 2 + 3 + \cdots + n = \frac{n(n+1)}{2}$.
To get this I used the formula: $S_{n} = \frac{n}{2}(2a + (n-1)d)$, or, $\frac{n}{2}(a + l)$ where $n$ is the ...
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Clock's min-hand overtakes hour-hand in $x$ mins
This was the question I faced:
The minute-hand of a clock overtakes the hour-hand at intervals of $62$ minutes of a correct time. How much in a day does the clock gain or lose?
And since I wasn't ...
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Is there such a thing called an imaginary angle? [duplicate]
Basically if $\theta = i$ just like if $\theta = 360$ or $90$ or $1$.
If such a thing exists, how can we visually express it?
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Belarus 1998 Math Olympiad Inequality - Can someone help me to fill in missing details?
Can someone help me figure out to fill in the missing details?
I have got this far
Belarus 1998 Math Olympiad Inequality and my working steps as far as I got so far
Thank you
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Percentage symmetry
I am currently reading Investing in the Unknowns and Unknowable by Richard Zeekhauser.
On page 6, 3rd paragraph, there is following lines:
...
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How to find the enveloping curve of this family of polynomials?
I was studying the Rule 90 cellular automaton and came across a family of polynomials defined by
\begin{equation}
D_n(x)=\begin{cases}
\displaystyle\sum_{k=0}^{m}(-1)^{m+k}\binom{m+k}{m-k}x^{2k}\ , &...
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The N-barrel problem. From some national math olympiad.
Someone told me in a math olympiad a very interesting problem, it goes like this:
Suppose you have a circular platform with $N$ barrels arranged in a "nice" order, each barrel will be the ...
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Multi-variable Rational Function that is always between the largest and second largest of the arguments.
This is a problem I came up for HMMT a few years ago, but I cannot find a solution.
Question: Does there exist a rational function $f(x_1,\cdots, x_n)$ such that for any positive $x_i$, $f(x_1,\cdots,...
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Quantitative measure of a good (musical) temperament
Is there actually a quantitative measure to know if a temperament is "good" or not? The motivation for my question is that there are quite a few temperament for the 12-tone systems (which ...
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Area of a special triangle in a quadrilateral
As is the case with triangles, I am sure that there is an entire ocean of not-so-well-known theorems about Euclidean quadrilaterals. In particular, I am interested in the following problem and suspect ...
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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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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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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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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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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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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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$N$ 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 ...
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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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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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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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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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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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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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$\frac{2(1+4a^2)}{(12x-1)^3}\leqslant \frac{(1-a)^4}{[12x-(1-a)^2]^3}+\frac{(1+a)^4}{[12x-(1+a)^2]^3}$ 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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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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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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