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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7
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1answer
169 views

Primes made from alternating factorials

While messing around with factorials, I noticed this: $$3! - 2! + 1! = 6 - 2 + 1 = 5$$ $$4! - 3! + 2! - 1!= 24 - 6 + 2 - 1=19$$ $$5! - 4! + 3! - 2! + 1! = 5! - 19 = 101$$ $$6! - 5! + 4! - 3! + 2! - 1! ...
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23 views

Learning about probabilities

Assume that there is a raffle that randomly selects a number from 0 to 9. You can choose a single number to place a $1 and gain <...
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0answers
28 views

Triangle game on $\mathbb{R}^2$: Can Alice always seek to construct equilateral triangles of length 1?

This is a game came to my mind last month. I have thought a lot and have searched the literature, only to find nothing much related. Alice and Bob are playing a triangle game on the Euclidean plane $\...
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2answers
81 views

Why was it valid to invent the imaginary unit, $i$?

I understand that imaginary numbers have turned out to be very useful but aren't there rules in mathematics that prevent you from inventing objects which, as far as I can see, contradict some ...
5
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2answers
91 views

6 Heads Followed by 6 Tails (Coin Flipping)

You throw a fair coin one million times. What is the expected number of strings of 6 heads followed by 6 tails? The answer given is: There are $1,000,000 - 11$ possible slots for the sequence to ...
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1answer
54 views

Conjecture related to Lychrel numbers

My conjecture is as follows: Take any number. It will become a palindrome eventually through the same reversal process used for Lychrel numbers except if the term (first term is excluded) starts with ...
4
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0answers
135 views
+50

Closed forms of special cases of $\sum\limits_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b)}$ with the Incomplete Gamma function.

$$\Large{\text{Goal:}}$$ One goal is to find better ways of expressing: $$\sum_{n=0}^\infty \frac{(pn+q)^{rn+s}Γ(An+B,Cn+D)}{Γ(an+b)}$$ which reminds us of the Product Logarithm/ W-Lambert function: ...
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2answers
43 views

how triangle inequality works here?

In a Canadian Math Olympiad Problem's solution, They used triangle inequality . The problem is as follows. the problem is the fourth problem from the 2019 Canada national olympiad. Let $n$ be an ...
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0answers
36 views

Counting distinct number of elements

I have three 10-tuples : $(a_1,d_2,d_3,d_4,....,d_{10})$, $(d_1,b_2,d_3,d_4,....,d_{10})$ and $(d_1,d_2,c_3,d_4,....,d_{10})$. Suppose I put a $*$ at the first and second place in the First and ...
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0answers
188 views

Intriguing moves with the $3\times3\times3$ Rubiks cube

Preface This is a recreational problem I kinda figured out during my free time playing around with the Rubik cube, so I hope everyone will take it as chilling as I did. Okay, so for those of you who ...
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26 views

An open problem concerns primorials plus one having square factors. What about primorials squared plus one?

No primorial plus one has ever been found that has square factors. However, when I started looking at the square of primorials plus one, I found: (23#)^2 + 1 = 29^2 * 53 * 1116604864937 I've only ...
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1answer
44 views

How to check if a number is within a range without $<$, $>$?

I know we can simply check if $x$ is within $(\min,\max)$ by the following function: if (x > min && x < max) then x lies inside of the range. Can I ...
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1answer
107 views

How many bishops can be placed on a $m \times n$ chessboard?

I came across this question and since I did not find it answered anywhere I do not know if my thoughts are correct, but I went like this: We know, that if we want to place bishops so they cant attack ...
3
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2answers
113 views

Edgematching tiles

Consider a 3×3 grid. Now, look at the patterns which generate 1 to 7 dots around the edges, taking into account rotations and reflections. Turns out there are 49 patterns, as seen in the set below ...
2
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1answer
72 views

On the evaluation of $\sum\limits_3^\infty \frac1{\ln\Gamma(n)}$

Motivation: This question will be inspired from: Evaluation of $\sum\limits_{n=1}^\infty \frac 1{\text G(n)}≈ 3+\frac{\,_0\rm F_2(2,3;1)}2 $ with the Barnes G function? and Evaluating $\sum\limits_{x=...
-1
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0answers
52 views

How to convert any number within a range to an even number in the same range [closed]

Suppose we have a range of positive integer numbers from a to b. I'm trying to find a function that can convert any given number (x) from this range to an even number in the same range. The only ...
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0answers
55 views

What are non-trivial dissections of a $45^\circ-60^\circ-75^\circ$ triangle into smaller $45^\circ-60^\circ-75^\circ$ triangles?

At this link, a square is divided into $45^\circ-60^\circ-75^\circ$ triangles in various ways. To solve that problem, several people built databases of shapes that could be built with that triangle. ...
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1answer
350 views

A way to express 100 by using the first four natural numbers [closed]

Is there a way to express 100 by using the first four natural numbers in order? The numbers 1, 2, 3, 4 can be linked by using $+$, $-$, $\times$, $\div$, $($ $)$, $!$ and exponents are also allowed. ...
5
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1answer
122 views

Rigid nonagons and rational solutions of a hyperelliptic equation

Here is a rational bracing of the regular nonagon of side $1$ with $12$ extra rods (extensions of the nonagon sides don't count), fulfilling a long-held dream of mine: I found it in a similar manner ...
3
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1answer
66 views

What are the chances of losing a game of hangman?

Halfway through a game of hangman (the word guessing game) I was playing, I wanted to calculate the chances of losing the game. I have $15$ letters remaining, and $3$ spaces to fill-up (no letter ...
2
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0answers
177 views

Evaluation of $\sum\limits_{n=1}^\infty \frac{(-1)^n\text{Ei}(n)}{n!}$

Motivation: This sum came up in a sum of a central gamma function problem: Evaluation $$\sum_{-\infty}^{-1} \Gamma(n,n)= \pi\left(\frac1e-1\right)i+ \sum_{n=1}^\infty \frac{(-1)^n \text{Ei}(n)}{n!}+\...
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1answer
513 views

Can we find sums for $\int_1^{\infty }e^{\pi i t} \left(t^{1/t}-1\right) \, dt?$

The MRB constant is the sum of the series https://oeis.org/A037077. cf. https://mathworld.wolfram.com/MRBConstant.html The MRB consstant (CMRB) $$=\sum _k^{\infty } \left((2 k)^{\frac{1}{2 k}}-(2 k-1)^...
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2answers
44 views

A BDMO functional equation problem.

The problem is as follows. $f:\mathbb{Z}\rightarrow \mathbb{Z}$ $f(f(x+y))$= $f(x^2$$)$+$f(y^2)$ $f(f(2020))$=$1010$ Find $f(2025)$ I approached by substituting $f(2020)$= $\alpha$ Then I substituted ...
6
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2answers
162 views

Determining sufficient area of a 2D annulus such that it can fit 48 circles each with radius of 35

I'm helping my mother with a project she wishes to complete as part of her job as a kindergarten teacher. She wishes to create a $2D$ annulus (doughnut) and fill it with $48$ Circles as part of a ...
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1answer
57 views

Solving quartic equation with square roots

I am trying to solve the following equation for $d_2$ (from this paper here): $$ \frac{c{\sqrt{(d_1+d_2)}}-\sqrt{d_1}z_1 -( \alpha)(d_2)\big(p_1(1-p_1)\big)^{1/2}}{\sqrt{d_2}}= -z_{\beta}$$ I've tried ...
2
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1answer
211 views

Range $[n,2n]$ containing smallest positive integer $x$ such that $x^x$ contains digits $2016$ in a row

How do I find a range $[n, 2n]$ such that it contains the smallest positive integer $x$ such that $x^x$ contains the digits $2016$ in a row (consecutively), up to sufficiently high probability? I ...
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0answers
30 views

Long-term behavior of closed two-player Elo system

A closed two-player Elo system depends on constants $k$ and $c$ in the following way. The two players begin with Elo rating of 0, and each match is zero-sum (w.r.t. Elo). If they enter with ratings of ...
3
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2answers
154 views

Series Representation of the Glasser function: $\text G(x)\mathop=\limits^\text{def} \int_0^x \sin(t\sin(t))dt\sim2\sqrt{\frac x\pi}$

Here is an uncommon special function called the Glasser function as referenced by Wolfram Mathworld. which is defined as: $$\text G(x)\mathop=^\text{def} \int_0^x \sin(t\sin(t))dt\sim2\sqrt{\frac x\pi}...
2
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1answer
66 views

Big Matrix Small Determinant

From a programming competition: Construct a square matrix with $N$ rows and $N$ columns consisting of non-negative integers from $0$ to $10^{18}$, such that its determinant is equal to $1$, and there ...
2
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0answers
46 views

The phenomenon of exceptions for small values

I am teaching a Linear Algebra class for first year students and one problem we are discussing in class is the fact that the sum of all elements in a finite field $F$ is $0$ if $|F|>2$. This result ...
3
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1answer
41 views

Unusual definition of the sphere $S^n$

In a paper I’m reading, the author defines the sphere $S^{m(k-1)-1}$ ($k \geq 2$) as the set of $m \times k$ matrices $(a_{ij})$, with $a_{ij} \in \mathbb{R}$ and satisfying the following two ...
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0answers
45 views

diagram of inner product - generalization?

Consider a sort of graphical language of linear algebra...tensor network diagrams. So consider the space of $n$ non-intersecting paths connecting a node $A$ to a node $B.$ This "picture" is ...
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19 views

Is it possible for a nonnormal magic square to have the same magic constant as a normal magic square of the same order?

I've been programming some quick and fast solutions to 4x4 magic square problems, and I've come across an assumption I've been using: The only squares with the magic constant 34 are the normal 4x4 ...
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0answers
21 views

Nested/intersecting causal diamonds

A maximally symmetric causal diamond is a solution to Einstein's equation with a cosmological constant. Consider a causal diamond in a maximally symmetric spacetime for a ball-shaped spacelike region $...
5
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3answers
155 views

If $\frac{(1+x)^2}{1+x^2}=\frac{13}{37}$, then find the value of $\frac{(1+x)^3}{1+x^3}$

Let $x$ be a real number such that $\frac{(1+x)^2}{1+x^2}=\frac{13}{37}$. What is the value of $\frac{(1+x)^3}{1+x^3}$? Of course, one way is that to solve for $x$ from the quadratic $37(1+x)^2=13(1+...
2
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2answers
80 views

Strange/Unexpected behavior of an Infinite product

Some friends and I were playing around with this continued fraction: We noticed when writing it out for each next step, the end behavior went either to 1 (when there was an even number of terms) or ...
4
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1answer
66 views

Minimum area ellipse passing through the vertices of an isosceles trapezoid

An isosceles trapezoid has its four vertices as follows: $A(0, 0), B(10, 0), C(7, 5), D(3, 5)$. I want to find the ellipse passing through the four vertices and having the minimum possible area. ​ ...
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0answers
45 views

How does linear transformation preserves solution of equations?

I was reading a solution to a problem of IMO 1982. Here is the problem Prove that if n is a positive integer such that the equation $x^ 3 − 3xy^2 + y^ 3 = n$ has an integer solution (x, y) then it has ...
2
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2answers
103 views

Existence of a certain composition series for the group $2^{\mathbb Z / 2^n \mathbb Z}$

This question is essentially to come up with a solution for this problem in the case where there are $2^n$ coins. Let $C_n$ be the group $\{f : \mathbb Z / 2^n \mathbb Z \to \mathbb Z / 2 \mathbb Z\}$ ...
17
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1answer
406 views

Closed-form for $B=\lim_{n\to\infty}\sum_{a_1=1}^{\infty}\frac{1}{a_1^2}\sum_{a_2=1}^{a_1}\frac{1}{a_2^2}\cdots\sum_{a_n=1}^{a_{n-1}}\frac{1}{a_n^2}$?

Introduction This problem came to my mind few years ago when I first learned about limits and infinite sums. I saw sums, double sums, triple sums etc, but never an infinite sum basically an infinite ...
3
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1answer
125 views

What is a negative inversion?

In a solution to the problem, IMO 2015 #3 Evan Chen and Telv Cohl used the notation of negative inversion which looks like Homothety: Here , $\angle$$AML$=$\angle ABL=\angle AKL$. Also,$H, J$ are the ...
3
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2answers
110 views

How is AC an angle bisector of $\angle PAB$?

Here is a problem involving tangent circles. Let $\omega$ be a circle with a diameter $PQ$. Another circle $t$ is tangent to $\omega$ at $M$ and also tangent to $PQ$ at $C$. Let $AB$ be a segment ...
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0answers
60 views

Simplest expression for cos(1°) predating calculus

I was wondering what are the simplest formulas (or iterative process) to compute $\cos(1°)$ using only methods which predate calculus. To give a more precise meaning to my question, let me just recap ...
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0answers
27 views

Recursive definitions and roots of polynomial equations

The following problem is given in Exercise 1.11 chapter 1 of "Galois Theory" by Ian Stewart. Let $P(n)$ be the number of binary strings of length $n$ in which consecutive ones occur in ...
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0answers
136 views

How these circles are congruent?

Here is a problem involving curvilinear incircles and mixtilinear incircles. Let a triangle$\triangle$$ABC$ have circumcircle $\gamma$.It's A-Excircle tangency point at side$BC$ is $D$ Let $\gamma_1$ ...
2
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0answers
129 views

Finding all natural $n$ such that $n^5+79$ has all digits same [closed]

Find all $n\in\mathbb{N}$ such that $n^5+79$ has all digits same. I worked with mod 3 and mod 9, because all the digits are same, but I couldn't move further. Although, I know $n=2$ is a trivial ...
3
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2answers
88 views

HH vs. THTH Coin Toss

Alice tosses a fair coin until either she or Bob wins. Alice wins if she gets two consecutive heads (HH) and Bob wins if he gets Tails and Heads twice in a row (THTH). What is the probability that Bob ...
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0answers
89 views

Visually intriguing unsolved problems which are easy to explain

I have come across a list of visual proofs which are wrong (Visually deceptive "proofs" which are mathematically wrong) visual proofs which are not wrong (Proofs without words) visually ...
0
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1answer
24 views

Fitting a sine curve given four known boundary conditions

I would like to be able to calculate the four parameters (A, B, C, D) for the sine curve $y=A\sin(Bt-C)+D$ given the generic boundary conditions that the value of the curve and its first derivative ...
1
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1answer
82 views

Size of a subset of $\mathbb{N}$

A subset $S\subset\mathbb{N}$ contains the numbers $5$ and $8$. If $n\in S$, then $7n-3$ and $7n+5$ are also in $S$ If $n\in S$, and $n=5m+4$, then $11m+9$ is also in $S$. Numerically, $|S\cap[1,N]|=...

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