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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"Tadpole Moe stored all the coins of R$1,00 that he got by selling cupcakes..." Brazilian Math contest problem. [closed]
Tadpole Moe stored all the coins of $R\$ 1,00$ that he got by selling cupcakes. To store the money that he would receive from the sales, he made sacks with sixteen, seventeen, twenty-three, twenty-...
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Finding The Value of CD [closed]
In triangle ABC, D be a point in BC, ∠BAD=30°, ∠CAD=90°, BD=1=AC, DC=?
a.∛2 b.√3 c.⁵√3 d.√2
Here, I've been trying solving the problem by using trigonometric ratios but am not able to get ...
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1
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Probability Theory Math [closed]
A pair of fair six sided dice are rolled. Assume all of the possible outcomes are equally likely. Let 𝐴 = {the sum of two dice is equal to 4} 𝐵 = {the numbers on both dice are the same}. What's 𝑃(𝐴...
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Get the days for consumption for each product of inventory
I try to calculate the days of consumption for every product when I have the consumption per week for a total of 26 weeks. As you can see in the table below.
Product
Consumption Week 1
Consumption ...
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2
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1
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51
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Value of the First Fermat Pseudoprime for each base
A Fermat pseudoprime for base $b\geq 2$ is a composite integer $n$ such that $$b^{n-1} - 1\equiv 0\: \left(\mathrm{mod}\:n\right)$$ Looking at the list of Fermat pseudoprimes for bases $b \leq 1024$ ...
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1
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Efficient way of choosing lottery ticket numbers
You want to buy a given number $n$ of lottery tickets, in each of which you have to guess exactly 6 numbers from 1 to 100. If you get 0 or 1 numbers right, you don't get any money. If you get $i>1$ ...
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Is there a winning strategy in the card removal game?
A and B play a game with blue, red, and green cards. They start with an even number of $n$ blue cards ("stacks"). Player A starts, and the players take turns. Only two moves can be made:
(i) ...
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Football betting. How can I calculate the stake values for 2 teams to give the same profit whichever team wins taking into account both stakes?
I am trying to implement this formula using a spreadsheet (currently Libre Calc) but any mathematical advice will be gratefully accepted.
Let's say, as an example, there are two teams with the ...
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3
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A generalization of Pythagoras'
Consider a quadrilateral inscribed in a semicircle of diameter $d$, as in the picture below, then $$d^2 = a^2 + b^2 + c^2 + \frac{2 a b c}{d}$$
Notes:
If one of the $a$, $b$, $c$ equals $0$, we get ...
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The book you'll need when someone came up with a random car plate number
I believe I've seen this book in a bookstore somewhere: it was a directory of integers, in ascending order, annotated with why each integer is interesting in certain aspects. e.g. the smallest (...
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How to find the time of a sine wave based on the y position?
I'm trying to find an equation that allows me to find the time of x of the sine wave based on the y position.
For example:
The y position of 0.25 on the sine wave should equal to 0.0804 in x time
y0....
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Successor Function in high numbers
Recently, I tried creating a Peano-based googology model. I started with 10^3003, which I understand is named a millillion. I then defined a millillillion - or, for short, a thousand 2ill (there are 2 ...
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Two questions about emirp's
Define $r(n)$ to be the reverse of a positive integer , that is the number emerging if the decimal expansion is written down in reverse order. Emerging leading zeros are of course omitted , but this ...
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Is it possible to calculate the size of the sum of the first Graham's number terms of the harmonic series?
I am curious about the size of the sum of the first Graham's number terms of the harmonic series. The harmonic series is a well-known mathematical series, and Graham's number is an incredibly large ...
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How to factor the following
what is the expansion from
$$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\cdots + xy^{n-2}+y^{n-1})$$
if for example I take the value of $n=\dfrac{1}{3}$ does it still work?
$$x^{1/3}-y^{1/3}=(x-y)(x^{1/3-1}+x^{1/...
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Possible cases with set of four non negative integers
Consider set of four non negative integers $S=\{s_1,s_2,s_3,s_4:s_i\in\mathbb{Z^+\cup\{0}\}\}$. Then there are the following possible broad cases:
$1$. All four integers are equal.
$2$. These integers ...
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Loop (topology)
Why doesn't the given function $f:[0,1] \rightarrow \mathbb{R}$ create a loop on the graph?
$f(t)=\frac{1+t^2}{1+t}$, $0 \le t \le 1$. Even it is define in the unit interval, $\mathbb{R}$ is a ...
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What is the smallest set $A$ upon which $f(a)$, $a \in A$ must be specified to fully determine $f(x)$, $x \in \mathbb{R}$ if $f(2x)=f(x)$?
Let $f: \mathbb{R} \to \mathbb{R}$ satisfy $f(2x)=f(x)$ and let $A \subset \mathbb{R}$ such that if I then specify $f(a)$, $\forall a \in A$ then $f(x)$ will be defined $\forall x \in \mathbb{R}$. ...
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Is the "reverse" of the $33$ rd Fermat number composite?
If we write down the digits of the $33$ rd Fermat number $$F_{33}=2^{2^{33}}+1$$ in base $10$ in reverse order , the resulting number should , considering its magnitude , be composite.
But can we ...
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Prove That an Approximation of $\sin(x)$ via Euler's Formula Approaches $\sin(x)$
I'm trying to approximate the trigonometric functions for a code library, and I want to ask if this is a good way to go about it. I'm aware of the Taylor series approach, but I wanted to go with ...
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$N^2$-ball Bingo Game with $N \times N$ Bingo Card Theorem
In this modified version of bingo game, there are total of $n^2$ bingo balls, half of the bingo balls will be randomly rolled out (in the case of odd number of balls, round up to nearest integer). ...
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3
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A number greater than one billion, with deep mathematical meaning.
The most famous large number that have appeared in mathematical papers is the Graham number. However, the Graham number is an upper bound, and its largeness has no meaning.
I would like to know about ...
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How to calculated the expected value of the overlap between groups of numbers?
I try to implement a lookup or join like functionality in code, which leads me to the following combinatorial challenge which I fail to tackle in the proper way:
Lets assume I have 50 million numbers, ...
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3
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Number of solutions of $y^2=x^3$ in $\mathbb{Z}_{57}$
The title explains the question. It was one of the 25 questions of a 3 hour olympiad, so I hope it is not too hard. The olympiad is for undergraduate students, so I also hope it doesn't use any "...
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Finding a maximizing partition $0<t_1,\dots,t_N$ and a basic function $f(t_1,\dots,t_N)$ when $\sum_{i=1}^Nt_i=C$ for a given $C>0$
Let $C>0$ be given and call a function $f:\mathbb{R}^N\to\mathbb{R}$ basic if it contains only addition, multiplication or exponentiation. So e.g.
$$f_1(x_1,\dots,x_N) = \sum_{i=1}^N x_i$$
$$f_2(...
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Let $\{a_n\}$ satisfy $a_1=1, a_{n+1}=\sin(a_n)$, find $\lim_{n\to\infty}\frac{\log(a_n)}{\log(n)}$.
This is question 24 of a Brazilian Olympiad multiple-choice test (for undergraduate students) that you need to pass in order to qualify for the main national math Olympiad. The problem is as follows: ...
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Counting the pentagons and hexagons on the surface of a football (aka, soccer ball)
For the Americans: I'm talking about soccer :-)
As you might know, a football consists of a collection of regular pentagons and hexagons, but how much of which ones?
The construction is simple: for ...
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Formulas I created for the area of any curved closed shape
Basically the main idea behind it is this: you take a random curved closed shape like this:
Then, since this can't be a function because for one x-value there's more than one y-value, you use the x-...
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Hexagon to Rectangle dissection: 3 pieces minimal?
A hexagon can be divided into 3 pieces to make a rectangle.
Can we prove 3 pieces is minimal?
For a equilateral triangle to square dissection, it's thought that 4 pieces is minimal. We can prove that ...
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How would you formally describe a common aspect of several recreational river crossing problems?
There are a few river crossing problems that I have seen that share some common aspects. The cannibal and missionary problem is typical. All these problems involve moving everyone from one side of ...
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When is "do-almost-nothing" a good idea in CHOMP?
Now asked at MO:
The proof by strategy-stealing that CHOMP on a rectangular board is a first-player win involves player 1 taking the top-right square on their first move. Of course given the proof-by-...
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Help me find the inverse of this function
While working on a recreational math problem I have come to the point were I need to find the inverse of one of the two following functions;
$$\frac{W_0(x)(W_{-1}(x)+1)}{(W_0(x)+1)W_{-1}(x)}$$
$$\frac{...
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Analyzing a specific function
I was trying to solve a proving question and I finally arrived at this step where I am stuck. The statement is proved if there exists infinitely many ordered pairs $x_a$ and $x_b$ such that: $$(x_a-...
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Finding or parametrizing integer solutions to $pq(p^2-q^2)=rs(r^2-s^2)$
Background: The order-3 magic square of squares problem (MSS3) is a well-known open problem that involves finding eight separate arithmetic progressions of three squares (APSs). In particular, two ...
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Half angle identities for complex trig forms
We have developed interesting properties of adding vectors with equal magnitudes. Similar properties can be written in terms of complex numbers. Let $z_1$ and $z_2$ be complex numbers with $\left|z_1\...
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A problem involving vector form of law of cosines
Example 1.22. Monica and Linda traveled from Shanghai, China (east $121^{\circ}$, north $31^{\circ}$ ), to Albany, New York (west $73^{\circ}$, north $42^{\circ}$ ), to visit their friend Hilary. ...
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Maths challenge question: Is my reasoning behind this question correct?
Problem:
(From Hamilton Maths Olympiad 2015)
Some boys and girls are standing in a row, in some order, about to be photographed. All of them are facing the photographer. Each girl counts the number ...
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Concatenation of square numbers is a square?
Just a curiosity of mine.
If I define the $n^{th}$ concatenation number (denoted $Q_n$) to be the the concatenation of digits from the $1^{st}$ square number to the $n^{th}$, can $Q_n$ ever be square ...
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Help on a simple-seeming modular arithmetic question [duplicate]
I'm trying to solve this problem:
Suppose $a$ and $b$ are positive integers such that $a^2 + b^2$ is even and $a^3 + b^3$ is a multiple of 3. What is the largest positive integer that must divide $a^...
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N people in a pit are required to press their kill buttons one at a time, what percentage of the initial population is expected to live on?
I posed this problem for myself based on a simpler problem I saw on reddit, here is the more detailed version of my problem:
The game master traps N people in a pit and equips them with a sort of kill ...
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having trouble finding the explicit function for a pursuit problem
I was trying to solve the following problem. A mouse is located at the origin of the coordinate plane and a cat is directly 10 units below it. Then the mouse starts traveling at a constant speed of 6 ...
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What is a combination of the numbers 1, 2, 3, 4, and 5 that yield 170, using only basic operations (+, -, *, /, ^) and the factorial (!)?
Rules:
Only basic operations (+, -, *, /, ^) and the factorial (!) are allowed.
No concatenation (i.e. 34, 12, 125, etc).
Parentheses are allowed.
All numbers must be used (omitting numbers is not ...
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Carmichael-numbers with only one odd digit
Here I ask for a third Carmichael number with only odd digits in their decimal expansion. Far more Carmichael numbers seem to exist with the property that in the decimal expansion there is only one ...
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What day(s) of the week cannot be the first day of a century non leap-year that is a perfect square?
This question is from contest our in school:
What day(s) of the week cannot be the first day of a century non-leap year that is a perfect square year?
My attempt
A Century year is a non-leap year ...
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A recreational question: Exploring some assumptions. Have I made an accurate conclusion?
I am not a mathematician (so I apologize if this is difficult to parse) but I have some questions that I am struggling to give a satisfactory answer to. I'll begin with some underlying assumptions.
...
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Where do I start learning more Maths for fun and that which is useful to my domain Computer Science and useful for Physics - purely out of interest?
So, basically, I am quite comfortable with mathematics in general. I enjoy tackling problems and thinking about them day-in and day-out when solving the mind boggling one's. Additionally, I also would ...
3
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Ridiculously Large Derivatives
Assuming $f(x)$ as the position of an object, $f'(x)$ would be the speed of an object. The second derivative would be acceleration, the third jerk. I can only find up to snap, crackle, and pop for the ...
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Patterns of 1-gon numbers
The general pattern of the n-gonal numbers is that the mth n-gonal number is equal to
$$\left(\frac{n}{2}-1\right)m^2-\left(\frac{n}{2}-2\right)m .$$
For instance, the formula of the triangular ...
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alternative trigonometric proof to inequality problem
This is the solution by using the inequality provided; However, I seem to not understand how this $$
\begin{aligned}
P & =14+5 \tan ^2 a+9 \cot ^2 a+\left(\tan ^2 b+4 \cot ^2 b\right)\left(1+\tan ^...
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Moving average on a list of numbers while keeping same total value
I need to replace a list of numbers, [1,0,3,2,1,4,1,2] for instance, by the moving average with a window of 3. The second element, 0 in this case, should become (1+0+3)/3.
The problem I have is that I ...
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