Questions tagged [recreational-mathematics]

Puzzles, curiosities, brain teasers and other mathematics done "just for fun".

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81 views

Find the missing number: $3+5+7=152181$, $4+5+6=202461$, $3+4+7=122172$, $9+4+5=364518$, $8+6+8=?$ [on hold]

Given $$ \begin{align} 3+5+7&=152181 \\ 4+5+6&=202461 \\ 3+4+7&=122172 \\ 9+4+5&=364518 \\ 8+6+8&=\;? \end{align}$$ I am able to get the first two numbers but for third ...
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2answers
75 views

What is the geometry of points inside a square, that are equally distant from the square [on hold]

Image: A visual of the problem For the square ABCD, what would be the geometry of points that are equally distant (distance r) from all points of the square? How would the shape written out by r ...
3
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0answers
95 views

Smallest number not expressible using first $n$ powers of $2$ (exactly once each), with $+$, $-$, $\times$, $\div$, and parentheses?

Motivation Solution to this problem is a lower bound for a more general problem. Problem Given first $n$ powers of two: $1,2,4,8,16,\dots,2^{n-1}$ that all need to be used exactly once per number ...
5
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1answer
75 views

Swedish mathematical competition problem for pre-tertiary students

In a class of students, one student is given a bag of 2014 coins whilst none of the other students in the class recieve any coins at all. Every time two students meet, if they have an even amount of ...
3
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1answer
52 views

Constructing a point arbitrary close to the Mandelbrot set

This question is motivated by the coloring schemes of the Mandelbrot fractal, namely that the color is determined by the points outside the set, and is proportional to the number of iterations $n_c$ ...
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0answers
35 views

Why does the square root of -1 squared have two answers? [duplicate]

If $$(\sqrt{-1})^2 =$$ $$(\sqrt{-1})(\sqrt{-1}) =$$ $$\sqrt{(-1)^2} =$$ $$\sqrt{1} = $$ $$1$$ Does 1 = -1? What is wrong?
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1answer
42 views

Vanishing cuboid problem

Consider 28 $19\times44\times29$ cuboids in a box. {19, 44, 29} {7, 2, 2} pack in a {133, 88, 58} box. {44, 29, 19} {3, 3, 3} pack in a {132, 87, 57} box. By reorienting the cuboids, one can be ...
3
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6answers
215 views

A curiosity on a first three natural numbers

Let's review a triple of numbers, $1, 2, 3$, it is a curiosity that $$1+2+3 = 1\times2\times3 = 6$$ Are there another triples (or not necessary triples) such that their multiple equal to their sum? ...
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0answers
671 views
+100

Limit associated with a recursion

If $z_n < 2y_n$ then $y_{n+1} = 4y_n - 2z_n$ $z_{n+1} = 2z_n + 3$ Else $y_{n+1} = 4y_n$ $z_{n+1} = 2 z_n - 1$ Consider the following limit: $$\lim_{n\rightarrow\infty} \frac{1}{n}\left(z_{n+1}...
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277 views

How many consecutive integers can be made using: addition, subtraction, multiplication, division?

If you are first time reading this, you may want to skip "solution summary..." section. That is, read the "problem", "example", "definitions",... and other sections below it first. Solution summary, ...
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2answers
115 views

100 blue eyed islanders - how to prove the waiting time cannot be less than 100?

Here is the famous blue eyed islanders puzzle and here is the traditional solution. It has been also discussed here in StackExchange. What I don't understand is how do we know that for $n$ blu-eyed ...
1
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2answers
79 views

Explanation of the solution to Fitch Cheney's $124$-card trick

In Peter Winkler's Mathematical Puzzles A Connoisseur's Collection, he posed Fitch Cheney's card trick problem as follows. His solution for the last question concerning the $124$-card, rather than ...
3
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0answers
91 views

Generalization of Four Fours puzzle - optimal set of quadruplets?

Four fours is a math puzzle whose goal is to build numbers out of mathematical expressions using four fours, and a restricted set of mathematical operations and symbols. Problem I'm interested in ...
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1answer
59 views

Novel power series [closed]

Has anyone collected such novelties: $1+5+25+125=156=12\cdot13$; $1+2+4+8=15=3\cdot5$; $1+2+4+8+16+32=63=7\cdot9$; $1+18+324=343=7^3$; $1+3+9+27+81=121=11^2$?
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0answers
22 views

A graphical representation of the final digit resulting from iterations of the product of the two numbers formed from the alternating digits of n

Sorry for the extremely long title. I did not know exactly what to put in the title since the question I want to pose is rather vague (and so I shall also accept vague answers, hehe...). The purpose ...
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2answers
52 views

Quick way to estimate powers of $9$?

I was trying to estimate how much is $9^{12}$, and I came with my own formula that does the work with powers of $10$, that is : $$9^n\approx|10^n-n \cdot 10^{n-1}|$$ but the larger the $n$ (integer) ...
4
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2answers
86 views

Maximum run in binary digit expansions

For numbers between $2^{k-1}$ and $2^{k}-1$, how many have a maximum run of $n$ identical digits in base $2$? For instance, $1000110101111001$ in base $2$ has a maximum run of 4. See picture below ...
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0answers
25 views

What is the maximum number of arcs that intersect at a single point (not including the center of the circle)?

Consider $n$ points evenly spaced around a circle. Connect pairs of points that are the furthest away from each other with $k$ arcs connecting each pair of points (these arcs could be pictured as ...
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2answers
60 views

For what natural $n$ does there exist a square composed of $n$ squares?

I recently stumbled upon a cute puzzle involving squares: For what natural $n$ does there exist a square composed of $n$ squares? For example, $1,4,$ and $6$ are valid: But $2$ and $3$ are not ...
3
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3answers
72 views

Are there any notebooks of famous mathematicians, through which we can understand their thinking and learning process?

Please name some published/preserved notebooks of famous (or not so famous) mathematicians, which you think reflect their learning or thinking process. Notebooks which contain mistakes are highly ...
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0answers
149 views

Pattern related to sums of alternating binomial coefficients

The Pattern $$\begin{array}{} +1\\ -1\\ +3\\ \end{array}$$ $$\begin{array}{} +5 & \color{black}{-2} \\ -21 & \color{black}{+4} \\ +99 & \color{black}{-10} \\ \end{array}$$ $$\begin{...
3
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2answers
54 views

Find every $n$: $n^2 + 340 = m^2$

Let $n$, $m \in N$. The problem asks to find every natural number $ n $ such that: $ n^2 + 340 = m^2 $ I tried to solve the equation like this: $ n^2 - m^2 = 340 $ $ (n + m)(n - m) = 2^2 * 5 * 17 ...
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1answer
60 views

How many times are these three quantities simultaneously prime?

For recreation, I would like to count the number of times the path length, "area above," and area below the prime counting function, are simultaneously prime. The following function is used to count ...
2
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0answers
110 views

If $n = 18k+5$ is composite, there are at least 9 divisors of $\phi(n)$ which do not divide $n-1$.

If $n$ is a composite of the form $18k+5$, there at least 9 divisors of $\phi(n)$ which do not divide $n-1$. Is this true in general or if not, what is the smallest counter example? Update: No ...
2
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2answers
59 views

Dispersion of points on a 4D sphere

In ℝ4 surface S is defined as defined as all points 1 unit from the origin. Taking the 5 points P1, P2... P5, that all lie on the surface S, move the points along S such that the distance between ...
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2answers
49 views

Understanding the following two equations

I'm reading through some texts and came across the following two lines. I'm curious how they should be read. Volume = $4/3\enspace π r^3$ Surface Area = $4\enspace π r^2$ In the example of Volume: ...
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0answers
31 views

Centuries where the Revised Julian calendar coincides with the Gregorian one

The Revised Julian calendar has the rule that centuries that are 200 or 600 mod 900 are leap years, but other centuries are not. With the above rule, the year 2800 will be the first year where the ...
3
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1answer
118 views

How do I prove that $x^n+x^{n-1}+…+x^2-nx+1=0$ ($n>2$) has one and only one root in $(0,1)$

How do I prove that $f(x)=x^n+x^{n-1}+...+x^2-nx+1=0$ ($n\in\mathbb Z_{>2}$) has one and only one root in $(0,1)$? My idea is assume there are two roots, $a,b$, then $f(a)-f(b) = 0$, then $\sum_{i=...
20
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1answer
420 views

Interesting patterns in $f(k,n)=k\pi-\sum\limits_{x=1}^n\tan^{-1}\left(\frac1{\sqrt[k]x}\right)$

Motivation: If we draw a right-angled triangle $A_1$ with sides $1,1$ then the hypotenuse is of length $\sqrt2$. If we draw a right-angled triangle $A_2$ with sides $\sqrt2,1$ attached to $A_1$ then ...
3
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1answer
75 views

Spectacular approximation to $\sum_{k=1}^n \{k \log_2 3\}$ where the curly brackets represent the fractional part

As many of the questions that I have asked recently, this is related to my investigations in finding a standard mathematical constant that has 50% of its binary digits equal to zero. My approximation ...
2
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1answer
97 views

How to prove: $11=10^{12}+10^{7}-45\sum_{n=1}^{999}\csc^4\frac{n\pi}{1000}$ [closed]

How to prove: \begin{equation} 11=10^{12}+10^{7}-45\sum_{n=1}^{999}\csc^4\frac{n\pi}{1000}\;. \end{equation} This identity appears on my clock:
5
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1answer
72 views

Arbitrarily long palindromes in two consecutive number bases

Is it possible to construct an arbitrarily long double palindrome? The double palindrome of length $d$ is a number that is palindromic (digits are the same when reversed) in two consecutive ...
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1answer
77 views

Interesting series involving factorials and logarithms

Not sure if the following is easy to prove or not, I obtained the result using probabilistic arguments exclusively. Define $x_1 = 2; x_2 = 4/3; x_3 = (8\cdot 6) / (7\cdot 5); x_4 = (16\cdot 14 \cdot ...
1
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1answer
85 views

I can't solve this modulo operation

I'm trying to find an easy and fast way to get the result of $$353094232 \mod 721.$$ I would solve this by dividing manually the terms until I get the remainder of dividing, but I was wondering if ...
2
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0answers
60 views

Kuzco pots stack problem

I get idea from the emperor's new groove video game. https://www.youtube.com/watch?v=DtClbzUd4QI#t=160 At 2.40, he go to about 3-levels height. The question is if Kuzco want to go to 10-levels ...
4
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2answers
71 views

maximal minimum distance between points in a rectangle of size $17\times 32$

I am training in problem solving and this one greatly resists to my understanding: Given a rectangle with specified length and width, you have to place 5 points in the rectangle such that you ...
6
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1answer
91 views

Need any kind of insights about a strange function

The function in question is $f(x; b) = \sum_{k=0}^\infty \{ b^k x \}\cdot b^{-k}$ where the base $b$ is equal to $2$ here. Also, $x\in [0, 1[$. The curly brackets denote the fractional part function. ...
2
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0answers
70 views

Curves in $\overline{\Bbb{F}}_p$

Let $k$ an algebraically closed field, for example $k = \Bbb{C}$ or $k = \overline{\Bbb{F}}_p$, $a \in k$, $k(x)$ the field of rational functions, fix an algebraic closure $\overline{k(x)}$, let $...
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4answers
319 views

Please let me know why 2/16 has a remainder of 2. Thanks [closed]

I would like to know why this division 2/16 has a remainder of 2. I understand remainders from this division 10/6 = 1 remainder is 4. But I can't figure out why 2/16 has a remainder of 2. Thanks
5
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0answers
62 views

What is the minimal area of a fully-fledged Dyson sphere?

A Dyson sphere is a hypothetical megastructure made up of a finite number of thin sun-screens in free-fall orbit around a central star. The goal is to harvest as much of the sun's radiation power as ...
0
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1answer
47 views

How is the quadratic equation formed?

Recently I was going through a problem. Below is the problem. Each evening after the dinner the SIS's students gather together to play the game of Sport Mafia.For the tournament, Alya puts candies ...
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0answers
45 views

Infinite volume but finite surface area, in higher dimensions?

In Infinite Volume but Finite Surface Area a question is asked whether there is some shape in $\Bbb R^3$ that have an infinite volume, but a finite surface area, sort of the opposite of Gabriel's Horn....
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0answers
45 views

Having trouble understanding how symmetry is used in the triomino problem

Given below is the classic combinatoral geometric problem of whether 8x8 board can be covered by 21 straight triominos. It is taken from Solomon W. Golombs Polyminoes and Puzzles. Now I am having ...
4
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1answer
65 views

Full Change Graphs

The following graph has vertices with total 183. There are 183 ways to remove a set of one of more connected vertices so that the remaining vertices are connected. These 183 sets have distinct ...
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0answers
39 views

Expected number of calls for a Bingo win

I have read a few similar questions (see this question and this one), but cannot figure out how to adapt the solutions to fit my question. Unfortunately, my understanding of math is extremely limited, ...
3
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1answer
102 views

Simplest Turing Machine for a particular binary string

At the Bank of England is a proposed £50 note. Alan Turing was born on the 23rd June 1912. 23061912 in decimal is 1010111111110010110011000. Starting from a blank tape, what is the simplest Turing ...
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1answer
28 views

What can be a function that take integers as input values,and give output as odd Numbers?

I want a function that can get input as integers and give result as odd numbers. for example if x=1,y=1 then (2,3) (3,5) (4,7) (5,9) (6,11) expressing y in terms of x.
3
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1answer
71 views

Is the coin fair

So, I told my friend a story ... Probability professor assigned a homework to his students. The assignment was to record a 200 tosses of the fair coin. After the assignments were handed, the ...
4
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1answer
190 views

A magician places $n$ coins on a table and walks down off the stage.

A magician places n coins on a table and walks down off the stage. A volunteer comes, turns over whichever coins he wishes, selects one coin and whispers its number to the apprentice. Then the ...
3
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2answers
50 views

An idealized $\infty$-body problem: is an infinite and regular configuration of massive objects stable?

Suppose I have an infinite amount of massive point shaped objects, and I arrange the objects by putting one object on each point of $\mathbb{Z}^2$ within $\mathbb{R}^2$. By symmetry, the gravity ...