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

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4
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70 views

Finding every $n$ such that $n\times$ ('reverse' number of $n)=m^2$ such as $1584\times 4851={2772}^2$

Let $r(n)$ be the 'reverse' number of $n$ in the decimal system. For example, $r(1234)=4321$. Then, here is my question. Question : Can we find every $n(\in\mathbb N)$, which is not a square ...
4
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175 views

Can we get a 'good' approximate value of $\sqrt 2$ by an equation which uses each of $1,2,\cdots,9$ once such as $12653\div 8947\approx1.414217$?

I've been interested in representing $\sqrt 2\approx 1.414213562373095$ by an equation which uses each of $1,2,\cdots,9$ once. Suppose that the following conditions must be satisfied. Then, can we get ...
4
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96 views

Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)

Background: This question arose purely recreationally and doesn't really fit into any context that I know of. Let $A \sqcup B = \mathbb{R}$ be a partition of the ...
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194 views

Which chapters of Euclid's elements would be helpful for drawing a grid?

I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a ...
4
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208 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with $\...
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90 views
+50

Maximal unit lengths in 3D with $n$ points.

Given $n$ points in 3D space (V), what is the maximal number of unit distance lengths (E) between those points? Here are a few possibilities. Some of them are chromatic spindles. ...
3
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84 views

Does Graham's number have an odd or an even number of digits?

I think it is hopeless to decide whether the number of digits of Graham's number is even or odd because the only way that I can think of is determining the logarithm with accuracy $0.1$ or even better,...
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41 views

Chessboard four-colour theorem

Divide the infinite chess-board into countries, where squares in the same country are connected by edges. Suppose two countries are adjacent if they touch either along an edge or at a corner. What is ...
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66 views

2D walks on a square grid; The number of Paths leading to specific $(X,Y)$

Introduction Lets have a 2D plane, and place a Walker in the center $(X,Y)=(0,0)$ Lets take a example where we use all of the possible moves; Walker can make one of the 9 moves each turn: Up, Down, ...
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108 views

Polyhedra with identical faces

The isohedra have identical faces. They have symmetries acting transitively on their faces -- any face can be mapped to any other face to give the same figure. There are also polyhedra where all ...
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73 views

Making $66$ with $1,1,1,1,1$

How can one make $66$ with only $1,1,1,1,1$? You cannot combine these two numbers to make a new number, such as this: $66=11 \times (1+1+1)!$. This was inspired a game of dice that I used to play, ...
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65 views

Compute shooting targets for the gunmen

This is an extension of the well known "3 gunmen puzzle": N gunmen with hitting probabilities in (0,1] take turns to shoot at each other. Firing order is fixed (gunman 1 shoots first, then gunman ...
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350 views

An interesting way to visualize the Mandelbrot Set. Proofs? Simplifications? Extensions?

This is a multi-part Question. Please chime in with any interesting insights in addition to Answers. I have noticed some interesting properties of Mandelbrot series that lead to a different way to ...
3
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44 views

Fun Q2: Polygon inside a polygon. Find Test's area.

A triangle is drawn of side length $a$. Then a square is drawn inside the triangle such that the area of the square is maximum and the bottom side is shared. Then a regular pentagon is drawn inside ...
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63 views

Rifleman game with $n$ players in $D$ dimensions: what is the survivor fraction when $n,D\to\infty$?

This is a follow up to this question where the following problem is explored (for $D=2$): $n$ riflemen are distributed at random points in $[0,1]^D$. At a signal, each one shoots at and kills his ...
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224 views

The mathematics of a drinking game called Spoon

This question is about the drinking game Spoon. It was asked on reddit.com/r/math : http://www.reddit.com/r/math/comments/3i9790/drinking_game_turned_mathematical/ The question is whether the person ...
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112 views

Trajectories on a circular billiards table

This question is related to The case of Captain America's shield: a variation of Alhazen's Billard problem, but more focused. Let the unit disc in the plane be our billiards table, and let $C$ ...
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93 views

Is this proof of a mathematical olympiad problem correct?

I'm quite sure about the exactness of my proof, but I'd like to hear (constructive) criticism about my writing. This is the problem: Every non-negative integer is coloured white or red, so that: 1) ...
3
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93 views

Bitcoin math problem example

Disclaimer: I'm not a mathematician, if something is complicated, please use layman's terms. Thank you. I'm wondering about this bitcoin thing. I have heard that mining is using a computer to solve ...
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101 views

Josephus Variant

I set myself the challenge of trying to solve a variant of trying to solve a variant of the josephus problem where instead of killing every second person, every third person dies. The formula for the ...
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99 views

How much advantage would a Blackjack player gain by being able to see the underside of cards?

In the novel Spaceland by Rudy Rucker, the protagonist Joe Cube is grafted with an eyestalk that sticks vout into the fourth dimension. This lets him see under and inside three-dimensional objects ...
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209 views

Limits of infinite processes that terminate in finite time - checking my understanding?

I am a computer scientist by training, but have a fair amount of math background that I've picked up through classes, teaching, and general interest. A student of mine posed a question to me. I think ...
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220 views

Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
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237 views

Help explain why (or why not) the solution for a in $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)=0$ is 1-2$C$MRB

$C$MRB is approximately 0.1878596424620671202485179340542732. See this and this. $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)$ is formally convergent only when $a =1$. However, if you extend the ...
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507 views

Guilloché security printing — can it be cracked?

Money uses Security printing, and often uses Guilloché patterns. These curves are inscribed by wheels on wheels on wheels, ten wheels deep in some cases. For example, the back of the US $1 bill has ...
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118 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
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47 views

How to find the point in a closed geometrical figure which maximizes the “direct-line-of-sight function”

To expand upon the title, and put it in clear terms, I phrase the problem thusly: Consider the interior of any continuous, closed, non-self-intersecting curve in the plane. (I'm not sure if I'm ...
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952 views

Turning radius of a vehicle

What's the minimum turning radius of a vehicle, rectangular in shape, with length l units and width w units? One key point to consider, would be that, the inclination of the front wheels can be ...
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170 views

Is there a way to determine how many solution does “ The hundred Fowls problems” have looking at the coefficients?

I was looking at the different versions of the hundred fowls problems. I came across the problem posed by Chinese mathematicians posed in 5th century,and then Indians in 9th, 10th centuries. Alcuin of ...
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158 views

For which positive integers n does there exist a prime whose digits sum to n?

Motivated by this earlier question, I thought of this problem: Question: For which positive integers $n$ does there exist a prime whose decimal digits sum to $n$? We can make two "easy" ...
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440 views

“Green Globs” question

When I was in high school geometry, we had a fun little game on the computer called Green Globs (the website for the software is http://www.greenglobs.net/index.html). A number of targets (globs) are ...
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695 views

Four candle problem: Using candles as timers

The candles each take one hour to burn completely. Cutting off bits of the candles is forbidden, but the candles are placed on a raft of fork handles so they may be burnt at both ends (e.g. to time $1/...
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108 views

Least characters in a numerical representation of integers

I was wondering what the shortest way to represent any given number is. For example, $387420489=9^9$. So, for this case, the smallest representation is of order 2 (2 numbers). Alternatively, $10=2\...
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39 views

Trying to understand the properties of a combinatorial game

Consider the following game for $n \geqslant 3$, which I will demonstrate with $n=4$: draw an $n$-gon and place the value 0 at each of the vertices, except one vertex which we circle and place the ...
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43 views

Decimal Multiplication Without Multiplication

My friend has recently been challenging me to solve some maths problems, the latest challange is to find a method of finding the answer of $2.5 \cdot 2.5$ without ever using multiplication. Now with ...
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0answers
33 views

Toroidal Split Complete Graphs

The paper On the Planar Split Thickness of Graphs shows how non-planar graphs can be split to make planar graphs. For example, they offer a split $K_{6,10}$. I would instead like to make split ...
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0answers
145 views

How to find Misiurewicz Points without solving huge polynomials? (Mandelbrot Set)

Here is a plot of 17,723 Misiurewicz Points. The code below generates a set of polynomials u[m,n], the roots of which have periodicity (m-n) starting at iteration n. I stopped at 17,723 points ...
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0answers
45 views

The hardest game of mahjongg

I was playing Mahjongg solitaire the other day. It got me thinking... The board has $2n$ pieces at the beginning and assuming that the game is winnable. The game would be trivial if there would be ...
2
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0answers
151 views

Number of ways to color a grid?

I have a $N \times M $ grid and I am trying to calculate the number of ways I can color this grid in maximum $k$ colors (I can use only $2$ colors or all $k$ colors) with the exception that two ...
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0answers
238 views

How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which ...
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0answers
59 views

Could you suggest books, papers or problems that could be used as good “general” motivating examples of calculus application?

I would like to stress the kind of reference I am looking for... In statistics there are lots of motivating (and sometimes unexpected) examples that are interested for everyone such as Birthday ...
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0answers
100 views

When solving a big Rubik cube (100x100x100), do you reduce the solution to like 50x50x50, and then 25x25x25, and then like 10x10x10 and then 3x3x3?

My question is about Rubiks cube. Say you're solving a 100x100x100 cube (you can see examples in youtube by computer program - https://www.youtube.com/watch?v=0cedyW6JdsQ) When solving a big Rubik ...
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0answers
50 views

Queen moves — The Squared Chain Puzzle

Karl Scherer made the interesting Squared Chain Puzzle. Start with a $7\times7$ board, with a queen somewhere. Make a legal move with the queen, placing coins over all squares visited. For subsequent ...
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34 views

Alternative Arithmetics

In Anderson et. al 2010, "Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns", the experimenters taught people how to solve a novel system of ...
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69 views

Is there a contiguous locus of the equality $a = b$ in $\zeta(\rho + \varepsilon)=a + î b$ in the near of a root?

This is just by an accidental couriosity: In the near of a root $\rho$ of the Riemann's zeta - can there be a continuous line starting from $\zeta(\rho)=0$ to $\zeta(\rho + \varepsilon_j)=a_j + î b_j ...
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67 views

Start with a circular cake…

This is a random math question I saw and it piqued my interest. I like cake. Start with a circular cake and cut it with five straight slices. What is the largest number of pieces that you can create ...
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0answers
28 views

Help with understanding Rating Systems

Hey guys so I'm trying to come with a rating system from 1-10 that will be determined based on how close a user is to an average value that I have. I don't have any data(other than the average value) ...
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56 views

Puzzle about decomposing numbers with $+, -,\times, \div$

In the book The Moscow Puzzles by Kordemsky I came across the following puzzle (no. 47): Express 100 three ways with five 5s. You can use brackets, parentheses, and these signs: $+, -,\times, \...
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109 views

A fashion victim puzzle

Consider $n \in \mathbb{N}$ fashion-sensitive kids, each wearing a T-shirt; for simplicity, kid $i \in \{1, \ldots, n\}$ initially wears shirt $i$. Tastes over the shirts are summarized in an $n \...
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56 views

The odyssey of spies: Kryptos

The part four $K4$ of Sanborn sculpture, a sculpture located on the grounds of the CIA in Langley remains unsolved. As you can read in [1], Sanborn released a clue for the 64th-69th letters in part ...