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

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114 views

Inserting +/- into 123456789…

I'm looking at a generalization of the problem of inserting + and/or - into the blocks $123456789$ and $987654321$ to create a formula for $100$, like this: $$123 - 45 - 67 + 89 = 100$$ $$9 - 8 + 7 ...
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82 views

Megaminx parity

I have an old 12-colored Megaminx that I put all new stickers on because the old ones were falling off. This Megaminx was in more of a state of disrepair than I originally thought, though, and when I ...
2
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639 views

Sum of cubes of the digits of a number equal to to the number

I have a number, I don't know how large or small, but if I cube the digits of the number and sum them, the sum is equal to the number itself. In other words, $$\sum_{k=1}^n{a_k^3}=\sum_{k=1}^{n}{a_k ...
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59 views

How to prove $\sum_{n=1}^\infty (-1)^n(x n^{1/n}+y n)=(c-1/2) x-1/4 y$?

Could someone help me prove this theorem where regularization is used so that the sum that formally diverges returns a result that can be interpreted as evaluation of the analytic extension of the ...
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36 views

For which minimal $k$ true is that ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$, ${a}_{i}\in {1,2,3,4,5,6}$?

I've got the following inequality, which bounds Minkowski distance. ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$ and values of ${a}_{i}\in {1,2,3,4,5,6}$ We know all ...
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118 views

Dates and times with no repeated digits?

I have a digital clock that shows the date and time like this: $$ \mathsf{YYYY-(M)M-(D)D\qquad (H)H:MM \; [:SS]} $$ That is, the seconds display is optional, and if the month or day or hour is ...
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121 views

A system of linear equations in integer squares - solvable?

I consider this more a "recreational math" problem, possibly lacking a solution (because it stems from the question of "magic square of squares") or simply intractable with reasonable effort. ...
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73 views

Tilting sealed drums for fun

A sealed cylindrical drum of radius r is filled with 9% of water. Now if the drum is tilted to rest on its side, show that the fraction of the curved surface area (not counting the flat sides) ...
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133 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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278 views

Does this approach to the Collatz Conjecture make any sense.

I was playing around with the Collatz Conjecture and came up with the following: Take any positive integer. If it's odd, multiply by three and add one. If it's even, divide by two. Repeat ...
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530 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 ...
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499 views

Social Golfer Problem - Quintets

I wrote an article on the Social Golfer Problem, which has questions like: Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with ...
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158 views

$n$ by $n$ Primally Magic Squares

(Again copied verbatim from a September 2009 thread I made.) A Primally Magic Square (PMS) is exactly like a traditional magic square with a change of criteria. Where a traditional magic square is ...
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36 views

Game idea “square or not”

I have an idea of a quadrilateral / square game, and am looking for help. For the moment lets call it the "Square or Not " game. Imagine we have a big stack of cards with on each card some property ...
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31 views

Shannon number upper and lower bounds

What are the best proved upper and lower bounds for the Shannon number, i.e. number of possible positions of chess? Is the upper bound 7728772977965919677164873487685453137329736522 given in ...
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37 views

Which numbers have the sum of their digits equal to the sum of the digits of their inverse?

$n$ is a number such as $n \in \mathbb{N}$ and $n >0$.(Eg. $n = 8$) $p$ is the sum of the digits of $n$ in base $10$.(Eg. $n=80$, $a = 8+0 = 8$) $q$ is the sum of the digits of $1/n$ in base ...
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23 views

Finding out a person's age in days given their birthday dd/mm/yyyy?

It has to be somebody alive today. Assume that the day is today - September 15, 2014. This is convenient because the leap years will be regular (once every for years; the weird rule applies to $1900$ ...
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68 views

Folding sheets of paper

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 ...
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34 views

Are (odd) perfect numbers divisible by a repdigit (in another base)? How about by a repunit?

[This has been cross-posted to MO.] A positive integer $N$ is said to be a perfect number if $$\sigma(N) = 2N,$$ where $\sigma(x)$ is the sum of the divisors of $x$. For example, $6$ is perfect ...
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48 views

Primes made from sequential digits

While messing around, I noticed that across some prime numbers contain only sequentially increasing digits, e.g. $23, 67, 89,23456789$. If we adopt a convention of returning to $1$ after a $9$, we ...
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45 views

Fusible numbers — can we prove fuse(4) is finite?

Fusible numbers have been discussed here before. Other links: fusible numbers. OEIS A188545. You have an unlimited number of irregularly burning fuses that will nevertheless burn for exactly 1 minute. ...
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20 views

Generalized-knight's tour

Define an $(a,b)$-knight as a chess piece that moves $a$ squares horizontally and $b$ squares vertically, or vice versa, in either order. Thus a normal chess knight is a $(1,2)$-knight. For which ...
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49 views

How do you evaluate $a^b$ where b is irrational using only basic operators.

How would you evaluate $a^b$ where b is irrational and you can only use +,-, multiplication, division, and rational powers. For example $2^\pi.$ We know $2^2$ = $2\times2$ etc... but when the power ...
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69 views

Finding the 'best' way in a card-arrangement-game

Let $n\ge 2\in\mathbb N$. Suppose that we have a card on which $1$ is written, a card on which $2$ is written, $\cdots$ , and a card on which $n$ is written. Now these $n$ cards are arranged from left ...
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45 views

Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
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40 views

Non-symmetric polynomials, game

This is a game I thought was easy but appears to be too hard for me... I'm trying to find a polynomial in x,y,z (they commute) such that permutations of the variables only give rise to 2 different ...
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37 views

Tools (electronic notebook and ontology viewer) of mathematical formulas (definitions and facts)

I am reading math books and articles (for applications in other disciplines, mostly about logics for computer science and AI) and the hardest part is to memorize formulas, to look up some ...
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22 views

Convex quadrilateral

In a convex quadrilateral (the two diagonals are interior to the quadrilateral) prove that the sum lengths of the diagonals is less than the perimeter but great than one-half the perimeter.
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38 views

On a certain type of card game

Suppose two players are playing a card game, which is described as follow. Each player is allowed to construct their own decks of exactly $n$ cards with an additional repeatable card, where each ...
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58 views

A Problem for the year with prime decomposition

I have noticed (and hope there are no errors) that: $$2013=3\times 11\times 61$$ $$2014=2\times 19\times 53$$ $$2015=5\times 13\times 31$$$ while 2012 and 2016 are not the product of exactly 3 ...
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49 views

Finding code in 5-guesses only

Suppose you want to crack a code composed of 4 digits (each between 0 to 5 when repetitions allowed) and you get feedback like in mastermind, how can you find it in less than 5 guesses in an ...
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30 views

Converting dot producto to set of arithmetic mean differences?

Ok so I am reading a book on linear algebra ( Gilbert Strang to be specific) and I am on second problem set, challenge problem, problem 29. In solutions it appears that the author states that: ...
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45 views

Net for both cube and regular tetrahedron

At how to fold it by Joseph O'Rourke, there is a net given that can be folded into a cube or irregular tetrahedron. Is there a net that can be folded into either a cube or regular tetrahedron?
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347 views

K non-intersecting diagonals in a polygon

Given a regular N-sided polygon, how many ways can you draw K non-intersecting diagonals? Any pair of diagonals must not intersect strictly inside the polygon. For e.g. N = 4 and K = 2 -> 2 ways ...
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54 views

How many Hamiltonian loop are there in a big rectangle?

Suppose I have some big rectangle made of $n \times m$ squares, and I want to place tiles on it in a manner that makes a picture of a hamiltonian loop. I can transform this problem into a problem ...
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68 views

What are all possible numbers gotten by an digit-exchange operation?

A friend of mine taught me a number game. Supposting that $a_na_{n-1}\cdots a_1$, which satisfies $a_n\gt a_1$, is a natural $n$-digit number with decimal representation, let's consider the ...
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66 views

Existence of a Vampire number on the form $v = xy = a^bb^a$?

A number $v = xy$ with an even number $n$ of digits formed by multiplying a pair of $n/2$-digit numbers (where the digits are taken from the original number in any order) $x$ and $y$ together. ...
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49 views

Visually apealing holologous transformation of a given contour

There is this problem which roughly says: You want to put a framed picture onto the wall with a cord to the picture frame. The cord is a single one, and both ends are attached to the frame. ...
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57 views

Linear algebra function that creates decreasing product vector of original vector

For vector $y=[y_1,y_2,\dots y_n]$ , let $\gamma = \sum_{i=1}^n \gamma_i$ , and $\gamma_i(n-i+1)=y_n*y_{n-1}*\dots y_i$ so that $\gamma$ looks like $[y_1*y_2*\dots y_n, y_2*\dots*y_{n}, \dots y_n]$ ...
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79 views

Probability of occurrence of games in a football league

This question just came to me as I was watching a football game. There is a football league with 20 teams. Each team has to play every other team at home and away, which means each team will play a ...
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88 views

gcd finding method

An integer $d$ is a $\gcd$ of two non-zero integers $a$ and $b$, if $d$ divides $a$ & $d$ divides $b$ '$c$ divides $a$ & $c$ divides $b$' implies '$c$ divides $d$' for any integer $c$. If ...
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39 views

maximising the frequency of mode.

I have 4 numbers 5,5,3,1. Now I have the number 5, which I can distribute in any manner to ...
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78 views

Zome construction

I'm preparing a presentation on Penrose tiles, and I want to talk about how we can get non-periodic tilings of the plane by taking projections of slices of higher-dimensional periodic tilings (in ...
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862 views

A recreational math problem, integers in a grid

I was thinking of the following recreational math problem: We have a $4\times 4$ square filled with integers $a_{1,1},...,a_{4,4}$. It has $30$ sub-squares $A_{i,j,k}$, corners of the form ...
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12 views

Has the mathematics of 4d-tetris, or any other 4-dimensional polyforms been studied?

There are a few variations of 4d tetris games floating around the internet, but I'm more interested to know if there's been mathematical research done in the area of 4d polyforms. I assume that the ...
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120 views

Prime number distribution theory for dummies

For the distribution of prime numbers there is a hypothesis which predicts the possible positions of prime numbers called Riemann hypothesis ...
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24 views

Limiting behaviour of a system

A friend of mine offered me the following problem. Suppose we have a rabbit and a fox in $\Bbb R^2$. The rabbit starts at time $t=0$ at the point $(0,0)$ and runs with constant speed $(1,0)$. The fox ...
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42 views

Is it possible to calculate how many people pay full price from the following numbers?

I'm currently analysing the Activision Blizzard earnings call for Q2 2014 and 2014 to see if I can figure out how many North American and EU subscriptions there are of the 6.8 million World of ...
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58 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...