Questions tagged [recreational-mathematics]

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

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1answer
50 views

What is the geometry of points inside a square, that are equally distant from the square

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 ...
4
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2answers
78 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 ...
2
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9answers
2k views

Neat expressions that equal 1

I would like to see beautiful and elegant expressions involving elementary and non-elementary functions, transcendental numbers, etc. that equal 1. Be creative!
27
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0answers
465 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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2answers
813 views

Identifying 2 poisoned wines out of 2^n wines

This problem has been asked and discussed in following posts: Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints, Finding 2 poisoned bottles of wine ...
3
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1answer
11k views

Calculating equal playing time in a soccer game with minimum number of changes.

I need to produce a formula that takes the following parameters: T = time of game in minutes p = number of players on field at one time s = number of substitute players Each of these is variable on ...
2
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0answers
68 views

Smallest number not expressible using all of the first $n$ powers of $2$ (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
72 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 ...
42
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0answers
762 views
+100

Can we remove any prime number with this strange process?

This is a little algorithm I made today, which may appear to be quite complex, so I will start with an example. Questions are at the end of the post. The process goes as follows: Start with the ...
10
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1answer
173 views

Largest Numerator of Sum of Egyptian Fractions

What is the largest possible numerator when put in reduced form over all sums of the form $$\sum_{k=1}^n\frac{c(k)}{k}$$ where $c(k)\in\{-1, 0, 1\}$? An easy bound is to consider what happens when we ...
3
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1answer
50 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 $n_c$ ...
3
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6answers
204 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? ...
5
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2answers
2k views

Round Robin “King” Style Tournament

I have spent too long trying to figure this out and cannot seem to get it just right, so here I am asking the Math gods. This weekend I played in a volleyball tournament called "King of the Beach", ...
1
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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 ...
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1answer
41 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 ...
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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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0answers
269 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, ...
85
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24answers
18k views

100 blue-eyed islanders puzzle: 3 questions

I read the Blue Eyes puzzle here, and the solution which I find quite interesting. My questions: What is the quantified piece of information that the Guru provides that each person did not already ...
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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 ...
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 ...
1
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1answer
105 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
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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 ...
0
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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 ...
-1
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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 ...
2
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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{...
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2answers
51 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) ...
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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 ...
0
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1answer
151 views

How to find a $9\times 13$ matrix where one can read the squares of $1$ to $100$?

I found a programming problem containing mathematics. I was unable to find an algorithm for this one. Does anyone have any idea how to construct a given matrix? I would like to construct a $9\times ...
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 ...
3
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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
71 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 ...
21
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5answers
4k views

Shortest distance around a pyramid

Transcript: The diagram shows a square based pyramid with base PQRS and vertex O. All the edges are length 20 meters. Find the shortest distance, in meters, along the outer surface of the pyramid ...
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 ...
1
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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
109 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 ...
0
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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: ...
3
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1answer
409 views

any Math game for kids?

I need to plan a "game" for kids about 10-11 years old that involves mathematics and some physical activity or game. It must be short-time and not very difficult because it's a stage of a big game. ...
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=...
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 ...
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:
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 ...
1
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1answer
72 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 ...
4
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0answers
93 views

Does infinite mikado exist?

Let's define a mikado configuration $m$ as a countable collection $\{T_j\}_{j \in \mathbb{N}}$ of disjoint subsets of Euclidean 3-space $(\mathbb{R}^3,\cdot)$. Each $T_j$ is a "tube of radius $R>0$"...
4
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2answers
70 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
90 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 $...
2
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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