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

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Theory of mnemonics

Even for the typical most skilled (human) number theorist it is out of reach to reproduce only the first 10 digits of $\pi$ in moderate speed (without physically reading them off). On the other hand ...
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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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2answers
87 views

Reciprocal of 81 being the sequence of all natural numbers?

According to this document: http://www.answering-christianity.com/fakir60/81.htm describing the theory of scientist Peter Plichta, the reciprocal of 81 is: the ...
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43 views

Mathematics of the Ice Bucket Challenge

I've been considering the mathematics of the now global ice bucket challenge. Simple model In the simplest incarnation, there is one original seed, who then nominates 3 others, each of which take ...
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3answers
39 views

Completing the square for a quartic expression

By completing the square, find (for real $x$) the minimum value of: $$x^4 + 2x^2 + 2.$$
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2answers
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Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

I am hoping a few of you mathematicians more experienced with writing proofs might give me some guidance here and possibly give me some ideas about how to restructure the following into a more ...
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2answers
43 views

Proper way to express 0 in this case?

If 0=(x-a)(x-b)(x-c)...(x-x)..=0. So it's a product sum that we write with pi instead of sigma but how? There should be indexes but I'm not convinced that I understand what notation to use. $$\prod_{ ...
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1answer
36 views

Tiling squares with L-Trominoes

Is there a simple proof that any square besides a 3x3 square with area divisible by 3 is tileable with L-trominos?
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1answer
76 views

3D Geometry Problem

If we have 4 equal sized spheres with radius $R$ arranged surrounding another smaller sphere such as to make a triangular pyramid from the centers of the $4$ spheres with radius $R$. The radius of ...
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29 views

What is the numeral system which uses the number of digits as a signifier of value called?

Our standard notation of representing numbers has an implied infinite number of zero digits on the left of all numbers. 42, 042 and 00000000042 all represent the same number. I'm thinking of the ...
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30 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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16answers
1k views

Beautiful, simple proofs worthy of writing on this beautiful glass door [on hold]

What are some of the more beautiful proofs you know? I am measuring beauty in two dimensions -- first, how conceptually elegant is it and second, how aesthetically pleasing is it. Context: I work ...
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2answers
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Winning strategies in multidimensional tic-tac-toe

This question is a result of having too much free time years ago during military service. One of the many pastimes was playing tic-tac-toe in varying grid sizes and dimensions, and it lead me to a ...
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1answer
71 views

I'm not understanding this puzzle [closed]

At first, I was thinking, mininmum amount of sticks it takes to create the figure, and then how many draw strokes it takes to create the boxes without crossing paths...but I can't figure out what's ...
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44 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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1answer
102 views

Good Reference for Justifying (less well-known fields of) Math?

How do we as mathematicians justify the study of math to students? Or, indeed, how do we justify it to the general public? How do you justify your particular field? I'm particularly interested in ...
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2answers
54 views

4 crystal balls and a 10,000 story building

There is an analog of this question I've heard with 2 crystal balls but a higher number like 4 or more makes it much more interesting. You are given 4 crystal balls and there is a 10,000 story ...
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3answers
109 views

Blue Eyes: A Logic Puzzle, has a puzzling solution (a.k.a. What does common knowledge have to do with it?)

In Blue eyes: a logic puzzle (specifically, the follow up questions), the most common answer is that it needs to be common knowledge that someone has blue eyes for all the blue-eyed people to leave. ...
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7answers
2k views

Mathematical literature to lose yourself in

H.M. Edwards in the preface to his book on the Riemann Zeta Function, summarises his philosophy on learning Mathematics: ...I have tried to say to students of mathematics that they should read the ...
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2answers
80 views

Natural numbers verifying $P(n) = n^2 - 42n + 440$, where $P(n)$ is the product of the digits

Let $P(n)$ be the product of the digits of the number $n$, with $n \in \mathbb{N}$. What is the product of all the natural numbers $n$ that verify the equation $P(n) = n^2 - 42n + 440$? I ...
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1answer
212 views

Joke explanation: “a comathematician is a device for turning cotheorems into ffee”

Ok, so apparently there is an old joke (which I DO get) that says that in Hungary a mathematician is a device for turning coffee into theorems. I found a post by Qiaochu Yuan that has the following ...
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1answer
32 views

Csn someone produce a sudoku puzzle where guessing more than one cell's value at a time is required?

Currently I have a sudoku puzzle solver program and I've tried all the puzzles I can find that are labeled the "hardest" on various sudoku video games and puzzle books. My solver has solved them all. ...
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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 ...
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1answer
94 views

Can 23 of this polycube fit in a 5x5x5 box?

Consider the following pentacube (front and back view shown.) I have used Burr Tools to determine that 24 of these will NOT fit in a 5x5x5 box. According to my notes when working on this problem a ...
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227 views

Which is greater, $20 \uparrow\uparrow\uparrow\uparrow 20$ or $4 \uparrow\uparrow\uparrow\uparrow\uparrow 4$?

This past Wednesday's What-If had this image at the bottom: In particular, I am interested in $20 \uparrow\uparrow\uparrow\uparrow 20$. I immediately thought of Graham's Number, but clearly that ...
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1answer
41 views

Why does the filling up of odd order magic square with numbers follow the knight movement?

Why does the filling up of odd order magic square with numbers follow the knight movement? I was reading about magic square, where I came up with the knight movement filling up of the magic square ...
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1answer
67 views

How can “Lucky Numbers” be approached rigorously?

To begin with, "Lucky Numbers" are a sequence of numbers generated by a sieve similar to the Sieve of Eratosthenes for finding primes. It starts with the set of natural numbers. Begin by selecting ...
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2answers
79 views

Prove withoui calculus: the integral of 1/x is logarithmic

It was known in the 17th century that the function $$ t \mapsto \int_{1}^{t} \frac{dx}{x} $$ is logarithmic: a geometric sequence in the domain produces an arithmetic sequence in the codomain. This ...
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1answer
84 views

2048 algorithm for merging

Ok, here's a question my friend just sent me, ive mastered it to some extent, but am failing, so, please help a little: Your target is to merge these blocks in such a way that one bigger number is ...
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2answers
26 views

Suppose T(k) denotes the smallest number of steps needed to move from k to 100.Find y & z such that T(9)= 1+ min (T(y),T(z)).

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre-specified pair of integers i,j ...
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1answer
38 views

what is the formula for determining the next year in which a given month/day will occur on a specific weekday

So, I was trying to express the formula for determining the next year on which a given date (month/day) will fall on a given weekday. The internet has plenty of sites that explain how to determine ...
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2answers
54 views

maximum number of independent bishops on a nxn chessboard

So I came across this problem where we have to find the maximum number of independent bishops on a nxn chessboard such that no two bishops attack each other . So after drawing the cases for $3$x$3$ , ...
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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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1answer
29 views

Proof involving an isosceles triangle

I came across this problem in some (maybe) high school book: Let $ABC$ be an isosceles triangle s.t. $AB=AC$. Also, $\alpha>\beta$. It is known/given: ...
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1answer
61 views

State whether the following claim is true, and then prove or disprove it. Give a detailed structured proof, justifying every step. [closed]

State whether the following claim is true, and then prove or disprove it. Give a detailed structured proof, justifying every step. $$\forall n \in \mathbb{N}, [(\exists k \in \mathbb{N}, n = ...
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1answer
53 views

How many bit strings of length 15 have exactly three 0s?

I need help with this question: How many bit strings of length 15 have exactly three 0s?
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1answer
87 views

An equation of the form A + B + C = ABC

So I was on a SPOJ spree until I came across this question . The question says $$\tan(\frac{1}{A}) = \tan(\frac{1}{B}) + \tan(\frac{1}{C})$$ where we have to find the $\min(B+C)$ for a fix $A$ where ...
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2answers
91 views

What is the Smallest Integer $N$ Where Reversing the Digits Makes $3N$?

What is the smallest positive integer N such that the integer formed by reversing the digits of N is triple N? (Does such an integer even exist? If not, then for what multiplier for $N$ will such an ...
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2answers
2k views

Interview puzzle with a deck of cards, some cards upside-down

You are sitting in a dark room. It is completely dark. You can't see anything and there is no way that you can make light. Basically, just assume that you are blind for this task. There is a table in ...
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1answer
69 views

How much distance did messenger cover? [closed]

A column of troops $80$m long is moving along a straight road at a uniform pace. A messenger is sent from the head of the column, delivers a message at the rear of the column and returns. He also ...
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51 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'' ...
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2answers
75 views

how to find taxicab numbers but for squares?

Natural numbers that can be written as the sum of squares in two or more ways. The first ten numbers are 50, 65, 85, 125, 130, 145, 170, 185, 200, 205. $$ n = a^2 + b^2 = c^2 + d^2\\ a^2 − c^2 = d^2 ...
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0answers
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Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
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1answer
1k views

Mathematical Intuition Behind Schizophrenic Numbers?

Schizophrenic numbers (A014824) are numbers whose square roots "look" like rational numbers. They were first discussed in 2004 by Darling in the Universal Book of Mathematics (page 282), and I ...
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422 views

Breaking chocolate bars game

About two weeks ago, a friend of mine taught me the following game without his knowing the answer. It may be famous, but I haven't known it. There are $N\ (\in\mathbb N)$ chocolate bars composed of ...
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1answer
70 views

total number of combinations?

Patient Age ---> Avg Visits / Year <1 year ---> 7.5 1-4 years ---> 3.0 5-14 years ---> 1.8 15-24 years ---> 1.7 25-44 years ...
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117 views

$\pi$, $e$, $\phi$, and sunflowers

While reading some internet materials on design, I came across this picture and comment: I found it a little bit surprising. I knew that the real sunflower follows golden ratio in some way (but I ...
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Evenly space holes in circle

A picture is worth a thousand words: This gear is part of an interactive SVG Spirograph I'm creating. I'm dynamically generating the gear based on a number of parameters (gear radius, number of ...
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105 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
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0answers
122 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 ...