# Tagged Questions

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

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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### “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 ...
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### “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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### 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 $A+B$...
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### $A^2$ $B=A^2-B$ 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'...
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### 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 ...
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### 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 ...
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### 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 ...
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### $n$-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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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 $1,2,3,4$....
### 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, ...