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

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6
votes
1answer
733 views

Proof of Closed Form Formula to convert a binary number to it's Gray code

The Gray code for a binary number $x$ is given by $$ x \oplus \lfloor x/2 \rfloor $$ How can this formula be mathematically proved ?
31
votes
1answer
3k views

Minesweeper - Chance of one-click win

I'd like to know if it's possible to calculate the odds of winning a game of Minesweeper (on easy difficulty) in a single click. This page documents a bug that occurs if you do so, and they calculate ...
2
votes
2answers
855 views

Area puzzle in colored triangle [duplicate]

I have tried to figure out by calculating the area but I got same results for these, so where is gone the hole?
2
votes
1answer
142 views

kaleidoscopic effect on a triangle

Let $\triangle ABC$ and straightlines $r$, $s$, and $t$. Considering the set of all mirror images of that triangle across $r$, $s$, and $t$ and its successive images of images across the same ...
9
votes
2answers
4k views

What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
1
vote
2answers
71 views

Playing with a functional equation

I was playing with a functional equation and proved the result below: Let $f$ be such that $$f(f(z))=z$$ If $f^{-1}$ exists then $$f(z)=f^{-1}(z)$$ If $f'$ exists then as ...
2
votes
1answer
108 views

Number of days it took to climb the mountain (BdMO 2012 National Primary/Junior question)

From the Bangladesh Mathematical Olympiad 2012 National Secondary (Question 7, or ৭). When Tanvir climbed the Tajingdong mountain, on his way to the top he saw it was raining $11$ times. At ...
3
votes
1answer
242 views

Make $21$ out of $1,5,6,7$ (challenge).

Here is a small mathematical challenge. You have to use once and only once each of the four numbers $1,5,6,7$ in order to obtain, via the help of the usual operators ($+,-,/,*$ together with ...
8
votes
3answers
789 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
4
votes
2answers
175 views

Four integer numbers to express all integers from 1 to 40 [duplicate]

Let $a$, $b$, $c$, and $d$ four integers such that $0 <a <b <c <d$. We can get all integers from $1$ to $40$ by expressions containing or not only the numbers $a, b, c$ and $d$. In these ...
2
votes
5answers
610 views

Higher Dimenional Tic Tac Toe

Here we have a problem that seems very intuitive, but is hard to define mathematically. In Tic Tac Toe, we can find an equivalent of the game in any number of dimensions, it seems. The trick is to ...
0
votes
1answer
55 views

Adjust a range of given values. [duplicate]

If I have a number anywhere on the range 140 - 350 and I want to match it to the correlated range "0 - 360" what function can I run it through? i.e.:140 would go through the function and return 0.350 ...
38
votes
4answers
2k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in ...
4
votes
4answers
273 views

Find the largest number having this property.

The $13$-digit number $1200549600848$ has the property that for any $1 \le n \le 13$, the number formed by the first $n$ digits of $1200549600848$ is divisible by $n$ (e.g. 1|2, 2|12, 3|120, 4|1200, ...
2
votes
2answers
232 views

Milk and Coffee will they ever finish

Let two glasses, numbered 1 and 2, contain an equal quantity of liquid, milk in glass 1 and coffee in glass 2. One does the following: Take one spoon of mixture from glass 1 and pour it into glass ...
3
votes
3answers
362 views

Solving $\;x+y+z =8 ; \;\;\sqrt{x^2+1}+\sqrt{y^2+4}+\sqrt{z^2+9}=10 $

Solve the problem \begin{cases}x+y+z =8 \\ \\ \sqrt{x^2+1}+\sqrt{y^2+4}+\sqrt{z^2+9}=10 \end{cases} with $(x,y,z) \in \mathbb R^3$ I have already solved it, but I'd like to see others creative ...
3
votes
2answers
300 views

Interesting Problems for NonMath Majors

Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer ...
5
votes
2answers
249 views

Let $k \geq 3$; prove $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$

Prove: If $k \geq 3$, then $2^k$ can be written as $(2m+1)^2+7(2n+1)^2$, where $k, m, n \in \mathbb{N}$.
9
votes
4answers
2k views

Proving that none of these elements 11, 111, 1111, 11111…can be a perfect square [duplicate]

How can i prove that no number in set S S = {11, 111, 1111, 11111...} Is a perfect square. I have absolutely no idea how to tackle this problem i tried rewriting it in powers of 10 but that didn't ...
1
vote
2answers
175 views

Grand Prix Race- Differential Equations [duplicate]

Driver A has boon leading archrival B for a while by a steady 3 miles. Only 2 miles from the finish, driver A ran out of gas and decelerated thereafter at ta rate proportional to the square of his ...
5
votes
1answer
681 views

How Many Clock Hand Positions Swap to a Valid Position?

My wording will not be exactly clear, but this is what I remember. Suppose you have a clock with minute and hour hands and you switch their places to form another correct time. How many such times ...
1
vote
3answers
139 views

Five digit re-write game

In the habit of factoring numbers, a notebook I bought had a five digit item number $77076$, which factors as $2^2 3^2 2141$, which may also be $9 \cdot 8564$, and in this form the count of digits is ...
2
votes
0answers
55 views

Symmetry between differentiation and integration [duplicate]

I want to make clear, that I am interested in the question: Why does integration need a bigger spectrum of functions than differentiation and not why integration is harder!!! as experience told me, ...
6
votes
1answer
230 views

cake cutting puzzle: why do finitely many cuts suffice?

Puzzle from http://www2.maths.bris.ac.uk/~majwm/compendium/cakeslice.php A piece of angle $x$ is cut from a cake, which is purple on top and yellow underneath, and turned upside down. Then another ...
2
votes
2answers
89 views

Expected number of pieces of a chessboard

If n squares are randomly removed from a $p \ \cdot \ q$ chessboard, what will be the expected number of pieces the chessboard is divided into? Can anybody please provide how can I approach the ...
3
votes
1answer
145 views

Minimum number of coconuts

Three friends namely $A$, $B$ and $C$ collected coconuts with the help of monkey and fell asleep. At night, $A$ woke up and decided to have his share. He divided coconuts into three shares, gave the ...
2
votes
5answers
407 views

When will two cars pass each other

There was a question in my math text book the other day that stated: $2$ cars each travelling at a constant velocity around a ring , complete exactly $4$ and $7$ rounds in one hour. If they start at ...
1
vote
1answer
178 views

Raise a number to the “y” power without using exponents.

This is kind of a spinoff on my question Divide by a number without dividing. Can anyone think of some clever ways to raise any given number to any given power without using an exponent anywhere in ...
-1
votes
4answers
920 views

Divide by a number without dividing.

Can anyone come up with a way to divide any given x by any given y without actually dividing? For example to add any given x to any given y without adding you would just do: $x-(-y)$ And to ...
6
votes
1answer
2k views

Deriving the 37-percent rule for dating

I am trying to prove the theoretical "37-percent rule" for dating. The setup, if I remember correctly, is this. Suppose that you will meet exactly $N$ potential mates in your life, and you will meet ...
3
votes
0answers
117 views

A photon in expanding Universe (a snail on a tree)

I want to know how far a snail can reach in expanding universe. It has a constant speed c = 1 and tree is expanding at speed $v= H_0 D$, with Hubble constant $H_0 = 1$. Here D(T) is the distance of ...
2
votes
3answers
158 views

$2^n-3^m=1 , m,n \in \mathbb N =?$

$2^n-3^m=1 , m,n \in \mathbb N =?$ my questions are: do m,n exist? are they finitely many $m,n$? if there are infinitely many is there a way to describe them all? Same question about $3^n-2^m=1 $, ...
-2
votes
4answers
672 views

Simple Math Problem

A bat and ball cost \$1.10. The bat costs a dollar more then the ball. How much does the ball cost? If this is not the correct place to ask a question like this please tell me and I will remove it ...
1
vote
1answer
374 views

Math Behind the Game “Quoridor”

I'm going to write an article for middle school students to introduce them to the game "quoridor". Tha game certainly is interesting, but it will be great to add to the article some serious "math ...
7
votes
1answer
95 views

Topology of Forum Posts

Okay, so here's an interesting question regarding web forums. Let's say you have a typical forum, such as the comments section on a blog, or whatnot. Viewers can post comments in response to either ...
0
votes
1answer
586 views

total number of different mixes

Patient Age Avg Visits / Year <1 year 7.5 1-4 years 3.0 5-14 years 1.8 15-24 years 1.7 25-44 years 2.6 45-64 years ...
1
vote
1answer
169 views

You are Johnny Depp 3!

An extension of this question. As @Jared stated in his answer the solution is: we assume that the head pirate chooses between multiple possible proposals that maximize his profit by rewarding ...
3
votes
1answer
269 views

65-card deck consisting of 13 ranks and 5 suits

** I FIGURED OUT 15 out of 16 cases. I don't understand the last case of RUNT. Anyone helps? I recently went to a math event and one person presented a weird card deck, consisting of 13 ranks and 5 ...
2
votes
1answer
195 views

Heighway dragon and twindragon relation

The Heighway dragon F is defined as the limit set for the iterated function system $\begin{cases}f_1(z)=\frac{1+i}2 z\\f_2(z)=1-\frac{1-i}2z\end{cases}\quad$ starting from the two points 0 and 1. The ...
0
votes
1answer
267 views

Finding probability that a person gets $7$ when rolling a pair of dice

*I STILL DON'T GET THE ANSWERS PROVIDED. PLEASE HELP! In a game, the participant rolls a pair of dice. If the result is a $7$, he wins. If the outcome is a number $n$ different from $7$, he continues ...
3
votes
1answer
185 views

Truchet tiles on a flattened cube

We have 2 Truchet tiles and a flattened cube as shown. We randomly place copies of the tiles into faces of the flattened cube. Find the probability that the circular arcs on the Truchet tiles ...
13
votes
2answers
407 views

A card game with no decisions

A friend showed me a mindless card game he plays, in which the initial state of the deck completely determines whether he wins or loses. The game is played as follows: Shuffle a standard $52$ card ...
7
votes
2answers
358 views

You are Johnny Depp 2!

An extension of this question repeated below. A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. ...
2
votes
1answer
49 views

Number of Distinct Resistances that can be produced from n equal resistance resisters

Here is an interesting problem: The number of distince resistances that can be produced from n equal resistance resisters is given below. The Sequence Surprisingly this is also equal to the number ...
2
votes
1answer
89 views

Formula for adapting a number for cross reference

As a keen cyclist I'm trying to use the Allen Coggan Relative Power table that then relates your Relative Power 'score' to what category rider you are. My question is that given rides/segments/hill ...
13
votes
2answers
1k views

9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. Arriving on a deserted island, they now have to split up the ...
2
votes
1answer
80 views

If we were to locate another intelligent lifeform, how could we then estimate the total number of intelligent lifeforms in the galaxy?

Given the vast size of the Milky Way, it is unlikely that we are the only intelligent lifeform to be found within it. Given that we only have one data point (the Earth), we are forced to use a long ...
7
votes
3answers
211 views

Blending Colors

Three one-gallon buckets of red, blue, and yellow paint are each two-thirds full. Without the ability to measure, is it possible to equally mix all of the paint through a finite sequence of pours ...
3
votes
1answer
989 views

Cutting a cube by plane cuts

This is an extension of a 3rd grade problem. How many pieces can one get at most if one cut a unit cube with n plane cuts? 1,2,4,8, ??? And assuming cutting through an area 1 takes time t, what is ...
5
votes
1answer
248 views

Trisecting a paper using hand and without using a ruler or compass [duplicate]

This is a practical problem born while folding a paper. We can bisect a paper by using only hand. $\star$ Easy, fold it so that the two ends (of the length) coincide and press the paper to get ...