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

learn more… | top users | synonyms (2)

1
vote
0answers
56 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
31 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 $a_1,.....
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
40 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
193 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
33 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
368 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
101 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 ...
118
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 $0,1,1,...
9
votes
3answers
204 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
61 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
108 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
38 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
vote
2answers
35 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
36 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 = cos(x)/(2+...
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
78 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
47 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
73 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
37 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
136 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
107 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
35 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
57 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 ...
10
votes
3answers
104 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 ...
3
votes
1answer
89 views

The angle between two clock pointers

Fun with math time The other day a friend of mine asked me for this: What is the value of the angle between two clock pointer when it's 11:50? Of course the correct answer is not $60$ degrees, ...
0
votes
2answers
123 views

Which is bigger $e^{(a+b)}$ vs $e^a + e^b$?

I understand that exponential function is a convex function so for any convex function $\theta(a+b) > \theta(a) + \theta(b)$, but can someone provide a more formal proofs ?
0
votes
1answer
57 views

How to calculate the payoff in Battleship Game Theory

Consider a 3 by 3 board and suppose that Player I hides a destroyer(length 2 squares) vertically or horizontally on this board. Then Player II shoots by calling out squares of the board, one at a time....
0
votes
0answers
29 views

Weighted Money Distribution Problem

A quantity of money must be shared between $n$ parties. Each party is assigned a weight, which determines the proportion of money they get, to a "close enough" number of pennies/cents. How would the ...
10
votes
2answers
119 views

Is it smarter to defend with one army or with two in the game Risk?

I've recently played Risk with the following rules: One player can attack another player with at most 3 armies. The defender can defend with at most 2 armies. This is independent of the number of ...
0
votes
2answers
52 views

Prove that there are only finite numbers satisfying this equation:

Coincidentally, I realized that my room number $(23)$ has the following property: $$2^3+1=3^2$$ In order to find more numbers $n$ exhibiting this property, I wrote the following equation: $$(n-1)^n+...
1
vote
2answers
97 views

If $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$

Prove that if $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$. A problem from a math competition. $A$, $B$ are 2 by 2 complex matrices. I've tried using Cayley Hamilton theorem, on $A+B$...
6
votes
1answer
93 views

$A^2$ $B=A^2-B$ then $AB=BA$

If for $2$ real $n$ by $n$ matrices we have $A^2B=A^2-B$ then prove that the two matrices commute. This is a problem from a competition. I've tried several manipulations but none of them work. Can'...
0
votes
2answers
162 views

Guess the number. Maximizing expected winnings? [closed]

A man in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from 1 dollar up to 1,000 dollars. He says that if you can ...
2
votes
1answer
47 views

The Functional Equation $f(mn)=f(m)f(n)$ where $f:\mathbb{N}\rightarrow \mathbb{R}$, $f(2)=2$, and $f(m) > fn)$ if $m>n$.

The following is exercise 3.3 from Terence Tao's "Solving Mathematical Problems." Emphasis added. Suppose $f$ is a function on the positive integers which takes real values with the following ...
5
votes
1answer
84 views

Arrangement of houses with 2 colors

From the 2016 International Mathematical and Logic Games Contest Along the coast of Maths-land, the straight beach-front road contains a line of houses, all on the same side of the road. The ...
0
votes
0answers
35 views

$n$-couples of people in a row.

For the following problem, I feel my reasoning is something wrong, so I would like if it is in the right direction or if it needs to be rephrased/corrected. The problem reads: How many ways are ...
2
votes
1answer
90 views

Maths problem: Cedric's age

We are in the year $2016$, and Cedric's age is a factor of $2016$. If Cedric adds up all the multiples of his age that are less than $365$, he arrives at the year he was born. In which year ...
0
votes
0answers
37 views

What is the name of this type of palindromic number?

A palindromic number is one that reads the same forwards as backwards,an example of which is $17071$. In the UK yesterday's date was $16/3/16$, the convention being day/month/year. Clearly $16316$ is ...
4
votes
3answers
108 views

How to calculate $10^{0.4}$ without using calculator

How to calculate $10^{0.4}$ without using calculator or if not what is the closest answer you can get just using pen and paper within say $2$ min?
2
votes
2answers
44 views

How do I find a minimal set of US states so that every state boarders one of them?

I recently played this game, in which one has to type in the names of US states. For each state one gives, all of the states which border it are removed from a list. The player continues to provide ...
3
votes
1answer
71 views

Can anybody help me with math expressions?

So , I am in $7^{th}$ grade and my teacher gave me some really hard homework. What I have to do is use math expressions that equals each number between $1$ and $100$ , only using the numbers $1,2,3,4$....
3
votes
0answers
67 views

2D walks on a square grid; The number of Paths leading to specific $(X,Y)$

Introduction Lets have a 2D plane, and place a Walker in the center $(X,Y)=(0,0)$ Lets take a example where we use all of the possible moves; Walker can make one of the 9 moves each turn: Up, Down, ...