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

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14
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0answers
527 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
12
votes
0answers
118 views

Is $2^{16} = 65536$ the only power of $2$ that has no digit which is a power of $2$ in base-$10$?

I was watching this video on YouTube where it is told (at 6:26) that $2^{16} = 65536$ has no powers of $2$ in it when represented in base-$10$. Then he - I think as a joke - says "Go on, find another ...
12
votes
0answers
1k views

This should be a piece of cake… right?

You probably know the following problem: We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only ...
11
votes
0answers
258 views

What turmite runs the longest before becoming predictable?

When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable ...
11
votes
0answers
211 views

Number of circles in configuration

Consider the $n^2$ lattice points $(i, j)$, where $1 \leq i, j \leq n$. Let the number of circles that pass through at least 3 points of this set be $C(n)$. What is a good way to count this? Is there ...
10
votes
0answers
1k views

Integer Triangle Radicals conjecture

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...
10
votes
0answers
132 views

Generalization of Gray codes

A friend of mine asked me if it was possible - physical difficulties aside - to generate all 32 combinations of raised/lowered fingers by changing status of a fixed number of fingers at every step. ...
10
votes
0answers
164 views

Take an m x n grid, and in each box pick two opposite corners at random to connect. What can be said about the resulting pattern?

Inspired by the upcoming book 10 PRINT CHR$(205.5+RND(1)); : GOTO 10 by Nick Montfort et al., whose title derives from this particular example of emergent behavior. Here's an example: (Note that ...
9
votes
0answers
219 views

Are there known pairs of simple numbers equal to huge precision, but not equal strictly?

Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they ...
8
votes
0answers
126 views

Limit approximation for $\pi$ in the four fours puzzle?

The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
7
votes
0answers
157 views

Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in $\...
7
votes
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196 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
7
votes
0answers
1k views

Solution manual to Larson's “Problem Solving through Problems”

I am working through Larson's "Problem Solving through Problems" (http://math.la.asu.edu/~ifulman/mat194/problem-solving.pdf) but many of the problems have neither solutions nor sources included. Does ...
6
votes
0answers
88 views

Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
6
votes
0answers
160 views

Separating Heavier from the Lighter Balls

This was posted Here and received a good answer, solving the general questions in either $n$ or $n+1$ moves, which is by just half a move on average "less good" than my manual solutions here. ...
6
votes
0answers
125 views

Puzzles and topology

I like problem solving. In fact, that is the reason I wanted to study mathematics; This is a field where I could learn the underlying logic of the results rather than just learning ideas even the ...
6
votes
0answers
163 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all the positive integers $m$ such that both the ratios $$ \frac{2(5^m+5)}{3^m+1}, \frac{9^m+1}{5^m+5}$$ are integers. Attempt to a solution: If the ratios are both integers, than their ...
6
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197 views

Math Behind the Dragon Illusion!

Dragon illusion has been one of the items presented in the 3rd "Gathering for Gardner". This video shows the illusion. What does it have to do with mathematics?
6
votes
0answers
193 views

Show that : $2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
5
votes
0answers
107 views

Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$, $(d=1.15096...)$ that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
5
votes
0answers
95 views

Magic square 9, Amazons, and the 2-(81,9,1) design

Consider the following order-7 magic square. The rows, columns, and diagonals all add up to the same sum: 175. Also, all the broken diagonals add up to the same sum, making this a pandiagonal ...
5
votes
0answers
90 views

Stacking circles

When I tried to stack 21 circles of radii $(30, 31, 32... 50)$ on top of each other in a tube (ID of $100$ wide), I thought they would reach the same height regardless of the order, however I was ...
5
votes
0answers
90 views

$f(x+1)=f(x)+f(\alpha\cdot x)$

I try to find an analytic increasing function $f_\alpha$ ($0\le\alpha\le1$) from $\mathbb R$ (or $\mathbb R^+$) to $\mathbb R$ such that for all $x$ $$f_\alpha(x+1)=f_\alpha(x)+f_\alpha(\alpha\cdot x)$...
5
votes
0answers
69 views

Probability of a minesweeper grid being solvable

Suppose we have an $m\times n$ minesweeper grid containing $k$ mines (for example the beginner grid is $8\times 8$ with $10$ mines). I have the following related questions: What is the probability ...
5
votes
0answers
120 views

Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the numbers $0$ through $9$, each used ...
5
votes
0answers
201 views

How many ways I can put $k$ bishops on $n\times n$ chessboard?

Is there a formula how to count in how many ways I can put $k$ bishops on $n\times n$ chessboard such that no two bishops threaten each other?
5
votes
0answers
74 views

Conway's Game OF Life maximum periods on a set x by x game board.

I have taken interest in Conway's Game of Life and want to know if you guys can help me with a mathematical problem :) That is what this website is for right? You need to be familiar with the rules ...
5
votes
0answers
114 views

Algorithm for finding “fact families”

My friend's 3rd grader encountered the following question regarding "fact families" on her math homework: I was in 3rd grade sometime in the 1980s, so I don't believe I ever encountered this term ...
5
votes
0answers
151 views

Folding sheets of paper to measure distances

You have $n\in\mathbb{N}^*$ sheets of paper with dimensions $a,b\in\mathbb{R}_+^*$ that can be folded as many times as needed. What is the set of lengths in $\left]0,\sqrt{a^2+b^2}\right]$ one can ...
5
votes
0answers
133 views

Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
5
votes
0answers
153 views

Interesting Number Game

Let's say you are given some random positive integer. Next, assume you are given the following sequence of integers: $$[1, 2, 3, 4, 5, 6, 7, 8, 9].$$ Now, assume you are allowed to use the following ...
5
votes
0answers
423 views

Paul Erdős Joke.

I was watching the great documentary "$N$ is a Number" and in it Erdős tells a joke where he writes: PGOM LD AD LD CD Which means poor great old man, living dead, archeological discovery, legally ...
5
votes
0answers
96 views

Integer weights used to cover all numbers from 1 to N with redundancies in case of breakage

Bachet's Problem (arxiv) is a famous problem where we have to find the smallest set of positive integers such that they measure every number between 1 and 40. This can be generalized to every integer ...
4
votes
0answers
49 views

Division of a square and value of a disk

I cam across this problem and I really don't know how to solve it. So you start with a square that has value 1. You divide this square in 4 so that each new square has a new value, as given by the ...
4
votes
0answers
73 views

Formula on conference t-shirt: $\ell(\beta) = \sum_{i=1}^{N}y_i \sum_{k=0}^{K}x_{ik}\beta_{k}-\log\left(1+e^{\sum_{k=0}^{K}x_{ik}\beta_{k}}\right)$

I need some assistance figuring out what this formula signifies and what it does. It is from a shirt I got recently at a conference and am curious about it. Thanks! $$\ell(\beta) = \sum_{i=1}^{N}...
4
votes
0answers
67 views

I'm walking towards my car - when should I try the remote, in an optimal sense?

I'm interested to learn about how discrete/'event' based elements are incorporated into optimisation problems. Hopefully this is an interesting problem in its own regard, it's inspired by a daily ...
4
votes
0answers
36 views

de Bruijn sequence and sequence waiting time

this is quite a vague question, more of a puzzle than a question. I've spotted that two problems concerning word combinatorics have the same answer. I feel like there should be a connection, but I ...
4
votes
0answers
36 views

Number of iterations of an “integer-logarithm”

Let us consider the function $\sigma:\mathbb{N}\to\mathbb{N}$ defined as: If $\prod_{i=1}^{r}p_i^{\alpha_i}$ is the prime-factorization of $n$, then $$ \sigma(n)=\sum_{i=1}^{r}\alpha_ip_i $$ So in ...
4
votes
0answers
75 views

Functions with “ugly” inverses

Inspired by this post: I was amazed to see that (at least according to wolframalpha) the inverse of such a nice and simple function as $f(x)=x^3+x$ is: $$ f^{-1}(x) = \sqrt[3]{\frac{2}{3( \sqrt{81x^2+...
4
votes
0answers
308 views

Evaluating $\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x$

I need to evaluate $$\int_{0}^{\infty} \left[\left(\frac{2015}{2015+x}+\cdots +\frac{2}{2+x}+\frac{1}{1+x}-x\right)^{2016}+1 \right] ^{-1}\mathrm{d}x $$ I've been told that the way forward is ...
4
votes
0answers
37 views

Hamiltonian path on a chessboard with prescribed endpoints

On an 8 x 8 chessboard consider two squares to be adjacent if and only if they share a common side. All paths below will consist of steps which join one square to an adjacent one. Under these ...
4
votes
0answers
108 views

Two formulae for $\pi$, probably known?

I stumbled upon (in the literature) two identities for $\pi$, but they were not referenced as they are probably well-known. Hoping someone could point out who found them first. Basically, the ...
4
votes
0answers
44 views

minimum number of times to change tyres.

I saw this brain teaser. Suppose, we travel 1000 miles on a tricycle and we have 5 tyres, then how many times do we need to stop to change tyres so that each of the tyres travelled the same distance?...
4
votes
0answers
50 views

Is there a heuristic reason behind this numerical coincidence?

Write $N(m, n; c)$ for the number of $m\times n$ zero-one matrices where each zero is adjacent to precisely $c$ others, where by "adjacent" I mean up/down/left/right but not diagonally. (Notice that ...
4
votes
0answers
109 views

Are all powers of 5 Friedman numbers?

Powers of 5 seem to have a quite interesting property. Not only do the all seem to be Friedman numbers in base 10, it also seems that they don't require digit concatenation and they their 'Friedman ...
4
votes
0answers
109 views

StackEgg optimal algorithm

What is the minimum number of days that is needed to complete the StackEgg game? (It's on the right if anyone didn't notice.) There are four markers (Questions, Answers, Users, Quality) I believe each ...
4
votes
0answers
129 views

Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
4
votes
0answers
217 views

Quantifying infinitely large sums such as $\sum_{x\in\mathbb{R}^+} x$

I thought of this as a student in calculus years ago, and it may be a silly kind of question. I wondered if there were notions of different sizes of infinity a series might sum to, which then lead me ...
4
votes
0answers
97 views

Can -9 to 9 be placed in 41 lines of zero?

The cubic curve $2x^3-4x^2y+2xy^2-8x+y^3-y$ can be used to get lattice points allowing the placement of the numbers $-8$ to $8$ so that all 32 triplets that sum to 0 will be a straight line of three. ...
4
votes
0answers
143 views

$Z_n \backslash \{0\}$ splits into octets

Let $n=8m+1, m\in\mathbb{N}$. Does the set of nonzero elements of $\mathbb{Z}_n$ split into disjoint octets of the form $8_k=\{\pm a_k,\pm b_k,\pm a_k\pm b_k\}$? The computer tells me it is possible ...