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

learn more… | top users | synonyms (2)

0
votes
1answer
36 views

Integer solutions of an equation that is set to a number

How many integer solutions for $a$ and $b$ in $(ab)/(a+b)=3600$? My attempt: $(ab)/(a+b)=3600$ = $ab=3600(a+b)$ = $ab=3600a+3600b$ =$ab=3600a=3600b$ Dividing $3600b$ on both sides ...
3
votes
3answers
128 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
2
votes
1answer
109 views

How are Sudoku puzzles created?

I recently read about the connection between solving Sudoku puzzles (and other graph coloring problems) and Groebner bases. This doesn't lead to an efficient solution technique, but it does link a ...
0
votes
0answers
170 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 ...
2
votes
3answers
159 views

Fun proofs for layperson?

I'm not quite sure whether this question belongs here, because it has no definite answer. But I'll give it a shot. If any of the mods objects, then I will, of course, respectfully delete this ...
2
votes
1answer
114 views

Mathematics of paper fold-cutting

Take a square of paper... ... and fold it any number of times using consecutive straight folds... ... then cut off any number of pieces using consecutive straight cuts... ... and unfold the ...
0
votes
0answers
34 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 ...
5
votes
2answers
518 views

What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
-2
votes
1answer
34 views

Equation that outputs digit in 1's 10's 100's slot [duplicate]

I need an equation that outputs the digit in the slot of my choosing EX1: I want the 10's slot in 1837 EX2: I want the 10's slot in 123456789 EX3: I want the 1000's slot in 93037352 I also need it ...
-2
votes
1answer
37 views

Challenge - “Highscore” output equation

I need an equation capable of processing 2 inputs to make one output that is either = to input 1 or 2. This is how it works. Since it is working with scores and such, Input1 will be "Last Score", and ...
20
votes
10answers
3k views

Get $5$ by doing any operations with four $7$s

How can one combine four sevens with elementary operations to get $5$? For example $$\dfrac{(7+7)\times7}{7}$$ (though that does not equal $5$). I am not able to do this. Can you solve it or prove ...
0
votes
1answer
38 views

Form $4$ new symbols with the most common symbols

Suppose we have $6$ symbols, say $A,B,C,D,E,F$. We are asked to form $4$ new symbols using the $6$ symbols with the addition operation. For example, the $4$ new symbols can be $A+C+E, F+E+A, ...
4
votes
0answers
120 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 can ...
0
votes
2answers
138 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 ...
7
votes
0answers
130 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 ...
0
votes
3answers
69 views

Completing the square for a quartic expression

By completing the square, find (for real $x$) the minimum value of: $$x^4 + 2x^2 + 2.$$
1
vote
2answers
231 views

Proving that an equilateral triangle in the plane cannot have vertices on integer lattice points

Thanks for the help! I've written a more detailed proof. The hints were great.
1
vote
2answers
50 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_{ ...
3
votes
1answer
66 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?
0
votes
2answers
95 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 ...
1
vote
1answer
56 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 ...
1
vote
0answers
38 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 ...
15
votes
16answers
2k views

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

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 ...
7
votes
2answers
139 views

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 ...
1
vote
0answers
60 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 ...
9
votes
2answers
186 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 ...
1
vote
2answers
67 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 ...
0
votes
3answers
233 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. ...
26
votes
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 ...
4
votes
2answers
85 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 ...
13
votes
1answer
489 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 ...
2
votes
1answer
43 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. ...
4
votes
0answers
89 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 ...
9
votes
1answer
117 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 ...
23
votes
1answer
247 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 ...
1
vote
1answer
60 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 ...
3
votes
1answer
79 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 ...
3
votes
2answers
106 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 ...
0
votes
1answer
104 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 ...
0
votes
2answers
43 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 ...
1
vote
1answer
57 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 ...
1
vote
2answers
123 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$3$ , ...
0
votes
0answers
83 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 ...
3
votes
1answer
71 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: ...
1
vote
1answer
818 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?
0
votes
1answer
107 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 ...
2
votes
2answers
96 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 ...
16
votes
2answers
3k 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 ...
2
votes
1answer
70 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 ...
0
votes
0answers
107 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'' ...