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

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1
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0answers
15 views

Number of iterations of an “integer-logarithm”

Let us consider the function $\sigma:\mathbb{N}\to\mathbb{N}$ defined as: If $\prod_{i=1}^{r}p_i^{\alpha_i}$ is the prime-factorization of $n$, then $$ \sigma(n)=\sum_{i=1}^{r}\alpha_ip_i $$ So in ...
0
votes
4answers
42 views

BEDMAS where the order of Addition before Subtraction matters?

Here is a recent "tricky problem" that is making the rounds on FB: BEDMAS is explained here in a video, with this being the upshot: Everyone understands that ...
1
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0answers
25 views

Number of ways to get from a point to another one in the plane

I was trying to solve the following problem related to "counting cases": Consider the point $(0,0)$ in the plane and another point $(m,n)$ with $m,n>0$ integers. Suppose you want to get from the ...
4
votes
1answer
62 views

Any math competitions dedicated to calculations by hand (college level math)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
11
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0answers
966 views

Integer Triangle Radicals conjecture

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...
1
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1answer
24 views

How many different arrangements of triangles that are either red or blue around a regular heptagon are possible?

I have the following problem I have an yellow heptagon (regular $7$ sided polygon) Against every side there is a triangle. The triangle is either red or blue. How many different arrangements of ...
4
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2answers
51 views

Time-and-Work and Motorcycle Tyres

A problem about motorcycle tyres, related to Time-and-Work or rate-of-work methods. This is not a homework question, nor, as far as I know, a contest question. It is intended as a challenge for Year ...
5
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0answers
51 views

Separating Heavier from the Lighter Balls

Classic Case I think we are familiar with the classic problem where we need to find one heavier ball among the rest identical lighter $n$ amount of balls using a scale and the minimum number of ...
6
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4answers
107 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 ...
-2
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0answers
45 views

Prisoners and hats variation

Five prisoners are arrested for a crime. However, the jail is full and the jailer has nowhere to put them. He eventually comes up with the solution of giving them a puzzle so if they succeed they can ...
3
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1answer
59 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
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0answers
35 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
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0answers
45 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
52 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
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1answer
68 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
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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
21 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
31 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
22 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
59 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
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0answers
21 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 x 106 pixels ones? $${1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y ...
5
votes
2answers
51 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 ...
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2answers
91 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 ...
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2answers
38 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
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0answers
20 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
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0answers
49 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
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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
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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
34 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
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0answers
71 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
30 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
319 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
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0answers
89 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 ...
107
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
42 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
189 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 ...
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votes
2answers
40 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
101 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
33 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: $a=2$, ...
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votes
5answers
97 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
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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
31 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
47 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
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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
61 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
44 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
70 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
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2answers
31 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 ...