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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0answers
29 views

Does this histogram approximate any known distribution?

Here is the histogram of the data: What could be a distribution density function for the data of the histogram?
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1answer
19 views

Weight after milling of a cylinder [closed]

I have a counterweight, a cylinder, that weighs 2kg, is 8cm tall and 6cm is the diameter. I want to mill it and reduce the diameter to 5cm. how much will it weigh afterward? thanks
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9answers
222 views

Integral of $1/x^2$ without power rule

I was wondering if it was possible to evaluate the following integral without using the power rule for negative exponents \begin{equation*} \int \frac{1}{x^2} \; dx \end{equation*} When using ...
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How Does Jacob Barnett Calculate the Number 32? [closed]

First of all, I would like to state that I wrote this article with Google Translate since I do not speak English. I was very curious about the subject below and I saw that the only sharing was made ...
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2answers
45 views

Covering paired squares on a $10 \times 10$ board [closed]

In a game, the board is a $10 \times 10$ table (in the style of a chessboard). Two players make moves in turns. In a single step, a player covers two neighboring free cells of the table by $2\times 1$...
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Internal Resistance of a circuit [closed]

A torch bulb has a power supply of two 1.5 volt cells connected in series, the potential difference across the bulb is 2.2 V and it dissipates energy at the rate of 550mmw, Calculate a) the current ...
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89 views

What is the smallest polyomino that can't surround a $1\times 1$ hole?

Given a polyomino $P$, we can ask if it is possible for disjoint copies of $P$ to surround a single cell in the square grid - i.e., for the complement of their union to have a connected component of ...
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1answer
40 views

Points in space

Let $P =\left \{ A_{1}, A_{2} \cdots A_{n} : A \in \mathbb{R}^{3}\right \}$ (where $A_{0}=A_n, A_{1}=A_{n+1})$. By middling of a point $A_{i+1}$ we mean setting it's value to a rectangular projection ...
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determining the formula of equations similar to ${2}^{x-1}$

the formula for $1,2,3,4,5,6,7\space\text{is}\space{x}\\ 1,2,4,7,11,16,22,29\space\text{is}\space\frac{x\left(x-1\right)+2}{2}\\ 1,2,4,8,15,26,42,64,93\space\text{is}\space\frac{x\left(x\left(x-3\...
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42 views

Prove that the points X,Y, B, C lie on a circle and determine the center of this circle. [closed]

Let I be the incenter of the triangle ABC. Suppose that the incircle touches the sides AC and AB at P and Q respectively. The lines BI and CI meet the line PQ at X and Y respectively. Prove that the ...
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2answers
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How to find the perimeter of a shaded region when one side isn't given?

The problem is as follows: Use the figure from below to find the orange shaded regions. $\begin{array}{ll} 1.&16(4+\pi)\,\textrm{in}\\ 2.&8(12+\pi)\,\textrm{in}\\ 3.&4(16+\pi)\,\textrm{...
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1answer
87 views

How to derive a formula with interacting variables (with example problem)?

Imagine you are an enterprising child: and instead of starting a lemonade stand, you open a lemonade bank. People can deposit lemons or sugar. In return, they get credit at your bank which they can ...
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71 views

Searching for treasure in a Penrose tiling: How can we navigate to a given pattern of tiles?

While there are an uncountably infinite number of rhomboid Penrose tilings following the rules given here, all such tilings have the interesting property that any legal bounded region appears ...
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1answer
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Recreation math : Fill the box type problem with equal sum of row and column

Today I solve an interesting recreational math problem which took some time. First I will briefly explain the situation of the problem. Look above figure. (1) a,b,c,d,e,f,g,h,i,j are one of 1,2,3,4,...
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1answer
45 views

Pigeon Hole Principle question, proving it is possible to select a number of points on a plane such that they are all interior to a circle

Thirteen points are given in the plane so that among any three of them there is a pair whose distance apart is less than 1. Prove that it is possible to select seven of the points so that they are all ...
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1answer
86 views

Is there a pattern/formula to this?

I came across a nice "sequence" on a mathematics Facebook page and it intrigued me somewhat. This is not homework or anything of the sort, just something fun to start the year off I guess. ...
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2answers
38 views

How to find the perimeter of a rectangle and an scalene triangle?

The problem is as follows: The alternatives given in my book are as follows: $\begin{array}{ll} 1.&\textrm{21 cm}\\ 2.&\textrm{39 cm}\\ 3.&\textrm{28 cm}\\ 4.&\textrm{27 cm}\\ \end{...
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3answers
35 views

How to find the length of an unfolded piece of paper which has been folded by its diagonal?

The problem is as follows: The alternatives given in my book are as follows: $\begin{array}{ll} 1.&\textrm{68 cm}\\ 2.&\textrm{130 cm}\\ 3.&\textrm{63 cm}\\ 4.&\textrm{112 cm}\\ \end{...
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2answers
52 views

How to find the perimeter of a piece of paper which has been cut?

The problem is as follows: From a rectangular piece of paper, $4$ straight cuts have been made. These cuts are parallel to the diagonals of the rectangle. After making the cuts, the 4 pieces are ...
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5answers
46 views

How to get the maximum perimeter in a set of straight squares?

The problem is as follows: Rudy has a toy set which consists of $11$ squares. All of them are equal in size. The lenght of each edge is equal to $\textrm{1 cm.}$ Assuming he's been given the ...
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1answer
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Interesting cyclic infinite nested square roots of 2 and cosine values

It is interesting to note that any angle between 45° to 90° satisfying $1\over4$ < $p \over q$ <$1\over2$ where $ p \over q$ is of form $p = 2^n $ and $q$ is an odd number satisfying $2^{n+1} &...
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3answers
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How few $(42^\circ,60^\circ,78^\circ)$ triangles can an equilateral triangle be divided into?

This is the parallel question to this other post with many answers already, in the sense that the $(42^\circ,60^\circ,78^\circ)$-similar triangles form the only non-trivial rational-angle tiling of ...
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139 views

Analytic solution to $\alpha , \beta \in \mathbb{R}$ such that $\cos \alpha \cdot \sin \beta = \cos \bigl(\sin (\alpha \cdot \beta)\bigr)$?

Are there any $\alpha , \beta \in \mathbb{R}$ such that $$\cos \alpha \cdot \sin \beta = \cos \bigl(\sin (\alpha \cdot \beta)\bigr)?$$ The trivial solutions are $(\alpha, \beta)=(0, \dfrac{\pi}{2})$. ...
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3answers
219 views

Alignment of multiple bodies on circular paths

Below is a problem that I came across in which I don't know the answer. It's as follows: $\\$ Problem: Suppose there are $n-2$ planets currently at rest (stationary) - all positioned on the positive ...
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3answers
165 views

What is something special about $2021$ as a number? [closed]

Looking for special properties that very few numbers including 2021 as an integer/real numbers have. Something like special equation, or seemingly unrelated to 2021.
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1answer
72 views

Prerequisite to Stevin's decimal expansion construction - it's a complete totally ordered set.

Update: The theory here has been checked by Brian M. Scott and with the New Year in mind I am changing the solution-verification tag to recreational-mathematics and adding this question, Given any ...
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0answers
200 views

Maximum number of permutations not repeating smaller permutations

There are $n$ soldiers, lining up every morning for their military service. The commander demands that the morning lineup of these soldiers be arranged differently for every next day according to the ...
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61 views

Can we further simplify the closed solution to the Goat Problem?

This year, there was published a closed-form solution to the Goat Problem. The problem seeks the radius of a circle (teal) with center on the edge of a unit circle (green) such that the first circle ...
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3answers
92 views

Product of random numbers.

Find the constant $p$ such that the product of any (positive) number $N_0$ multiplied by successive random numbers between $0$ and $p$ will, on average, neither diverge to infinity nor converge to ...
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3answers
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A $2021$ problem: $20\sim 21$ and $43\times 47$

Notice that $2021$ is a concatenation of consecutive integers: $20\sim 21$ Also $2021$ is a product of consecutive primes: $43\times 47$. What is the next number with both of these properties? $...
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1answer
201 views

Where is the flaw in my logic in this collatz conjecture idea?

Using this article from math stack exchange ( open problem - What does proving the Collatz Conjecture entail? - Mathematics Stack Exchange fourth answer down that starts "I think, for the ...
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1answer
97 views

How to prove that following algorithm is correct?

Suppose I have some $n$ numbers which are powers of $2$, say $a_1,a_2,a_3.....a_n$ which are not necessarily all distinct. I have option to give them any sign. I have to find if I can make their sum ...
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1answer
64 views

An infinite sequence of inscribed squares alternate black and white. Find the black area as a fraction of the first square's area. [duplicate]

There is an infinite sequence of squares such that vertices of every next square lie in the center of the sides of the previous square. Odd-numbered squares are filled with white color, and even-...
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170 views

Is the fact that $6!7!=10!$ a pure numerical coincidence?

I was reading about factorials recently, and I happened to come across the curious, almost pseudo-Pythagorean-seeming fact that $$6!7!=10!$$ I was greatly intrigued by this, but couldn't think of any ...
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1answer
94 views

How to find the last page of a book when some numbers are skipped?

The problem is as follows: Julia enumerated consecutevely the $204$ pages of her diary, starting from $1$, excluding those numbers where the digits $1$ and $7$ appear together in any order. For ...
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4answers
66 views

Given positive real numbers a and b, prove that $\frac{2}{\frac{a}{x}+\frac{b}{y}} \leq ax + by, x>0, y>0$

I am working through Problem Solving through Problems by Larsons. I am stuck on question 1.8.5b) which is as follows: " Given positive real numbers $a$ and $b$ such that $a + b = 1$, prove that $$...
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1answer
13 views

How to calculate overall conversion percentage?

Let's a say there's an exam which has a 10% passing percentage from Prelims to Main then 15% passing percentage from Main to Interview and 2% passing percentage from interview to final selection. Then ...
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1answer
43 views

Calculate the $F^{−1}$ of a function

Let $V:=\{x \in \mathbb{R}:-5 \leq x \leq 5\} .$ Consider the function $F: V \rightarrow V$ defined by $$ F(x)=\left\{\begin{array}{ll} -1, & \text { if }-5 \leq x \leq 1 \\ 2, & \text { if } ...
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37 views

How long before we all pay $100 rents?

A local administration has passed a new law to curb excessively high rents in gentrified areas. Under the new rules, landlords cannot charge more than the average price per square foot in their area. ...
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2answers
58 views

Probability of seeing each face except one in $n$ throws of a die with 4 faces

Imagine that we throw a die with 4 faces $n$ times. The faces have probabilities of $(0.1, 0.4, 0.2, 0.3)$ of showing up. What is the probability that we see all faces except a specific, pre-...
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2answers
64 views

Addition vs multiplication as binary operators

It is known that arithmetic operations of addition and multiplication are binary operations that take in two inputs and give out a single output. However, consider the following scenario: Suppose I ...
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45 views

How many n-cliques are in a co-normal product of graphs?

I was wondering how one would go about counting the n-cliques in a co-normal product of 2 simple undirected graphs $G$ and $H$, given that you know the number of vertices, edges, and cliques of all ...
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3answers
352 views

What is the chance that an infinite series of random numbers will, at some point, repeat all previously generated numbers in that order? [duplicate]

The main reason of this post is to provide a writeup of what I think is a correct solution to this problem from the r/math subreddit. I'll quote the poster to phrase the problem: What is the chance ...
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3answers
365 views

Solving Dudeney's “A Question of Cubes”: sum of consecutive cubes is a square

The following is a puzzle of Dudeney: Professor Rackbrane pointed out one morning that the cubes of successive numbers, starting from $1,$ would sum to a square number... He stated that if you are ...
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24 views

Describing the distributive property of multiplication using flow charts

I'm trying to visualize basic multiplication using flow charts, I've shown some pictures below On the left side, I have depicted how to show addition as taking in two numbers and outputting a third. ...
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1answer
83 views

Is there a “nice” rep-tile of order $6$?

A planar set is said to be a rep-tile if it can be tiled by congruent shapes, each similar to the original. If there are $k$ such shapes, each scaled down by a factor of $\sqrt{k}$, it is said to be ...
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1answer
35 views

Determine the final stage of a game

Peter and Chris are playing a game. At the beginning, each of them has some balls, where each ball is red, blue, or green. There is also a ball pool which contains more than enough balls of each ...
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2answers
56 views

Christmas Tree Problem: String around Cone

This problem came up when I was trying to figure out how long of a string of lights I need for my Christmas tree. It was easy to estimate and obviously this isn't for use in the real world, but I ...
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1answer
49 views

Limits of the digits of Squared Numbers

Hey I don’t know what to call the post I think the name doesn’t reflect the question very well (considering limits are a separate thing in maths), but the question, I think, is very intriguing. When a ...
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3answers
42 views

Reducing $2018^{2018}\pmod6$ via $\mod2$ and $\mod3$

I've been tasked to reduce $$2018^{2018}\pmod6$$using the results of reducing$$2018^{2018}\pmod2$$and $$2018^{2018}\pmod3$$ I have deduced that $$2018^{2018}\equiv0\pmod2$$and $$2018^{2018}\equiv1\...

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