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

learn more… | top users | synonyms (2)

6
votes
4answers
144 views

Minimum number of marked squares on $n × n$ board

Came across this question: Consider an $n × n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two different squares on the board ...
3
votes
1answer
66 views

How to prove a regular pentagon is formed by knotting a rectangular strip of paper?

I found this interesting problem from a friend (From Arthur Engel's-Problem Solving Strategies book). The method to begin the problem is as follows- Step 1.Take a rectangular strip of paper ...
1
vote
0answers
37 views

Can we evaluate the alternating sum of the digits of an irrational number?

Suppose you had a summation $\sum(-1)^na_n$, where $a_n$ is the $n$th digit of $e$ and $a_0=2$. I know it diverges, but I want to know if its possible to evaluate anyways. Since it is alternating, ...
1
vote
0answers
47 views

Defining Logic Algebraically, Math Functions & Integers

Introduction I wanted to define some functions algebraically to be used as "logical conditions" that would be assigned to a term $t$ to "control" its value. Or in some other words, I wanted to ...
3
votes
0answers
66 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 ...
0
votes
1answer
19 views

Algebraic manipulation from $a^2+b^2 = abm$ with all variables in Z to a|b?

I know that $a^2+b^2=a\,b\,m$, with $a,b,m$ integers ($a,b$ positive integers) How can I show that a|b from this? I know it's true intuitively. I can recall the definition of divides to be $ak=b$ ...
2
votes
1answer
73 views

Division Without Division

Ok. I'm having a bit of a problem with a mathematical task my friend challenged me with, find the answer to two divided numbers without any division and the method used has to work for all whole ...
1
vote
1answer
20 views

Index set of dyadic partition

Imagine if you have a line from 0 to 1, and you begin partitioning it dyadically. The first point will be at 0, the second at 1 and the third at 0.5, the fourth at 0.25, the fifth at 0.75 etc. Let's ...
0
votes
1answer
22 views

Recurrence relations, trouble understanding the statement

I have been struggling with the English in some recurrence relations problems, since I am studying it on my own and I'm not in a combinatorial environment. Here is one in which I can't grasp what it ...
3
votes
1answer
33 views

Collecting integers puzzle.

Given a set of $n$ integers and a starting point, one has to collect all $n$ numbers moving at most a distance 1 from any number previous picked. For example if $n=5$ one solution would be: ...
0
votes
1answer
30 views

Recurrence relation. Application to ternary sequences

The question is: How many ternary sequences have no double zero? For this I understand that our $n$-digit sequence either have $0,1,\dots,n$ zeroes, is this ok? If the answer of above is positive, ...
0
votes
1answer
61 views

What is the first absolutely normal number to be discovered?

What is the first absolutely normal number to be discovered? Is it the Chaitin's constant? $$\Omega_F = \sum_{p \in P_F} 2^{-|p|}$$
0
votes
0answers
27 views

How to generalize Tupper's self-referential formula?

How do I generalilze Tupper's self-referential formula so that it can graph arbitrarily big images, and not just $17 \times 106$ pixels ones? $${1\over 2} < \left\lfloor ...
5
votes
2answers
55 views

Where is a good source for serious math (wall-size) posters?

Where is a good source for math wall posters that give glimpses of serious and beautiful mathematics? I'm a faculty member looking to find some wall posters (e.g. 2 ft x 3 ft) to hang in a handful of ...
-1
votes
2answers
92 views

Infinite product of negative numbers? $-1\times -1\times-1\times -1\dots=$ [closed]

Edited: Making the question as brief as possible to avoid future confusion and misunderstanding. Note This was moved as a separate question from: Product of all real numbers in a given interval ...
-1
votes
2answers
39 views

How Many Possible Triples $(x,y,z)$ are there? [closed]

Given that $xyz= 2015$, and $x,y,z$ are positive integer, how many possible triples $(x,y,z)$ are there ?
0
votes
0answers
24 views

Travelling salesman - organising a tour of any European destination based on the cheapest flights available.

I apologise if this has only a tenuous link to a mathematics forum I'm sure everyone is familiar with the £10 one-way flights by Ryanair and similar airlines in Europe. I was wondering whether there ...
1
vote
0answers
54 views

Scaling factor closest to 1 in an infinite sequential rectangle packing

The Ammann Chair can be used in an infinite dissection of a rectangle, where the pieces have a scaling factor of $ k = 1/\sqrt{\phi} = 0.786151...$. The largest piece has area $\sqrt{5}$ and longest ...
1
vote
2answers
29 views

The number with minimum sum of differences

Let $a_1,a_2,...,a_n\in\mathbb{R}$. I wonder how to find the number $x$ with $$|x-a_1|+...+|x-a_n|=\mbox{min}\{|a-a_1|+...+|a-a_n|\mid a\in\mathbb{R}\},$$ namely the sum of the differences with ...
0
votes
0answers
22 views

What is the topology within a circle in order to map hypotenuses at the correct length? (see image)

Each slice of the triangle has a hypotenuse with a corresponding curve of equal length within the circle. What is the topology of the inside of the circle so that the curved lines equal the lengths ...
0
votes
1answer
35 views

How to divide 6.4 miles per hour into integer blocks

I promise this is not a homework problem, but my brain cannot figure out the math to solve this problem that is relatable to a similar situation to my own: You want to run on a treadmill at an ...
1
vote
0answers
123 views

Is this translation into symbols correct?

Me and my friend came up with a cool game - we take turns in taking some mathematical theorem stated in English and turn it into a symbolic statement. The rules are this: you are only allowed to use ...
0
votes
1answer
32 views

Is there a metric that is zero for translations?

First define a relation $\sim$ on $\mathbb{Z}^k$ such that for any $a,b\in \mathbb{Z}^k$ where $a=(a_1,\dotsc,a_k)$, and $b=(b_1,\dotsc,b_k)$ we write $a\sim b$ if and only if $a-b=(n,n,\dotsc,n)$ for ...
10
votes
3answers
331 views

Product of all real numbers in a given interval $[n,m]$

READ-ME I have now what I can call for myself answers to all my problems and subquestions proposed in this post, thus I accepted Strings answer as the answer to this question since it was of most ...
1
vote
0answers
99 views

Progressive packings in a convex shape

Take a shape, and scale it by 1 to $n$. For a tiny set of tightly related shapes, such as isosceles right triangles with shortest sides 1 and sqrt(2), scale the set of shapes by 1 to $n$. What is ...
115
votes
9answers
15k views

There are apparently $3072$ ways to draw this flower. But why?

This picture was in my friend's math book: Below the picture it says: There are $3072$ ways to draw this flower, starting from the center of the petals, without lifting the pen. I know ...
2
votes
1answer
43 views

Interesting Property of Primes involving Modulo?

Primes & Modulo What I have observed is that for the following expression, choose a positive integer $m$, and if it is prime then for positive integers $n=1,2,3,\ldots$ the results will be ...
9
votes
3answers
200 views

How to win Matt Parker's jackpot - finding the median of the following distribution

In a recent video the legendary Matt Parker claimed he kept flipping a two-sided (fair) coin untill he scored a sequence of ten consecutive 'switch flips', i.e. letting $T$ denote a tail and $H$ a ...
-1
votes
2answers
46 views

The hands of a clock are observed continuously from 12:45 pm onwards. They will be observed to point in the same direction some time between [closed]

The hands of a clock are observed continuously from 12:45 pm onwards. They will be observed to point in the same direction some time between A).1:03 pm and 1:04 pm B).1:04 pm and 1:05 pm C).1:05 pm to ...
17
votes
9answers
1k views

Function that maps the “pureness” of a rational number?

By pureness I mean a number that shows how much the numerator and denominator are small. E.g. $\frac{1}{1}$ is purest, $\frac{1}{2}$ is less pure (but the same as $\frac{2}{1}$), $\frac{2}{3}$ is ...
0
votes
2answers
104 views

“Canceling Out The Zeroes” In A Mathematically Sane Way $\frac{0\times x}{0\times 1}$

Introduction Lets look at the product sequence: $$(n-1)(n-2)(n-3)...(n-k)$$ Where $n,k\in \mathbb N$ and $n\le k$ ; the expression will always have value $0$ But what if we remove the $n$th term in ...
1
vote
1answer
36 views

“Binary-Like” Function?; In Consecutive Products as Multi-Factorials…

Summary Is there a function $Z(a,b)$ or how would one find such a function so that for $a,b\in \mathbb N$, it would produce $0$'s on for each $a$th step for each $b$th value? For example: $a=2$, ...
-1
votes
5answers
100 views

How To Combine 1,2,3,4,5 into 333? [closed]

I am trying to figure out how it is possible to combine 1,2,3,4,5 into 333. Apparently there exists some way that makes this work, but I am not sure how. 1,2,3,4,5 can only be used once, and I am ...
1
vote
2answers
34 views

For what value of constant a is function continuous

I know there is a similar question. I had a read through it and it didn't help me so I'm posting this one. The question is For what value(s) of the constant $a\in \mathbb R$ is $$f_a(x) = \left\{ ...
1
vote
1answer
33 views

Find points on curve where tangent is horizontal

I've looked for a similar question on here but couldn't find any. I have found a similar question on Google but it still didn't help me. My question is Find the points on the curve y = ...
2
votes
1answer
49 views

Given more than $3$ dimensions, would I be able to slice my apple more than one time and still being able to place it in a table in a particular way?

My english is okay, but not good enough to describe this, so I made a picture. This is what happens in our real life (boring) $3$D world, Note that if we slice the apple one more time (unless you ...
0
votes
0answers
28 views

calculus book recommendations [duplicate]

i want to learn single variable calculus i completed schooling and i love calculus for now i am focusing on single variable calculus i tried many books like Calculus - "A Complete Course 7th ed - R. ...
2
votes
1answer
67 views

What's the probability to win (or lose) this solitaire? [duplicate]

Me and my friends used to play a "solitaire" and always asked ourselves which are the odds to win, or lose. I studied Maths and many of them did as well, but nobody could find a good answer to this ...
0
votes
0answers
46 views

Pairs of Numbers such that the sum of their digits is Equal

How many pairs of numbers $(n,m)$ whose digits add up to the same sum, where $n\ne m$ and $(n,m)=(m,n)$ such that $m,n\le k$ , are there for a given $k$? Observing this in base 10 we are looking at ...
0
votes
1answer
72 views

Count number of ways that people can ride a chairlift

I've come across a fun problem that I couldn't generalize. Description 3 students arrive at a chairlift. They are free to use up to 3 consecutive chairlifts (no empty chairlifts between them). So ...
1
vote
2answers
32 views

Pink Kangaroo Maths Challenge: Ria Bakes Six Raspberry Pies

I have been doing some practice papers for an upcoming UKMT Maths Challenge. There's one question I can't seem to grasp. I'm not sure entirely sure where to start. I'm open to any ideas. Thank you ...
7
votes
1answer
1k views

A curious property of $\operatorname{frac}(e\cdot k)$

Let $\alpha > 0$ be a real number and let us consider the set $S(\alpha)$ of those natural numbers $n$ such that the fractional part of $\alpha \cdot n$ "begins" with the representation of $n$ (in ...
2
votes
2answers
132 views

A unit square contains 1 million rectangles without any common points. Show that the total area of rectangles is less than 1.

"A unit square contains 1,000,000 rectangles without common points. Show that the total area of rectangles is less than 1." This statement is somewhat imprecise. Let's say that these are closed ...
4
votes
1answer
93 views

What are some PDE applications in recreational mathematics?

I have to do a final project for my PDE subject and last year I did one about Game Theory (specifically, Prisonner's Dilemma and Snowdrift game) for my ODE subject, which the rest of the students ...
1
vote
0answers
34 views

Need help to visualise Topological Puzzle

I am curious about this puzzle on unlinking the fingers of a rubber man. (https://www.youtube.com/watch?v=olHV4qvSDg8) However, despite the illustrations in the above video, I can't visualize the ...
0
votes
1answer
19 views

balancing stats for equality

Not sure if this is the right forum, please comment on the correct one. OK: I am creating a game where the user inputs stats and attack values, and I want the ai to ...
-1
votes
1answer
54 views

Fill in operators (7 7 7 7 7 7 7 7 = 820)

My kid's git the following as his homework - the problem is to fill in arithmetic operators between eight digits 7 to get 820, that is: 7_7_7_7_7_7_7_7=820 This drives me mad, but I myself cannot ...
1
vote
1answer
60 views

a*b = a/b = b/a (what's this symmetry called?)

I was playing around with numbers the other day, and I found an interesting symmetry, that I would like to know if it has any specific name assigned to it. Let's assume the notation n:a to refer to ...
9
votes
3answers
99 views

Mental $n-$th root of $N$

It has been a while since I started thinking about this problem: a fast method to evaluate (in an approximate way) mentally the $n-$th root of a number $N$. I'm talking about great numbers, because ...
3
votes
0answers
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 ...