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

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0
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1answer
739 views

Among these figures circle, square, rectangle, isosceles triangle which has the greatest perimeter had the same area?

Among these figures circle, square, rectangle, isosceles triangle which has the greatest perimeter had the same area geometrically ?
2
votes
1answer
93 views

Throw dice, what does this mathematical expression mean in real life?

Assuming we have a dice and the event that if we throw dice for the k-th time and get a 6 is given by $A_k$, is there an actual explanation what $A:= \cap_{i=1}^{\infty} \cup_{j=i}^{\infty} A_j$ is?
3
votes
1answer
134 views

Weird system of equations

X : 2 = 7 Y : 2 = 6 X + Y = 15 Find X and Y. I think maybe this is some unpositional number system. I've tried positional, and it works for basis 21 (if we take X=D, and Y=C), but professor told me ...
1
vote
1answer
137 views

Radius ratio for four packed circles

Suppose we are given four circles $A,B,C,D$ in the Euclidean plane having radii $r_A,r_B,r_C,r_D$ such that $r_A=r_C,r_B=r_D$ and circles $A,C$ are tangent to each other and to $B,D$ but $B,D$ are ...
3
votes
1answer
138 views

Calculating the number of triangles

I am trying to calculate how many triangles that can be found in an equilateral triangle with $2n$ lines starting at the bottom angles and ending at the opposite side, such that equally many lines ...
3
votes
3answers
117 views

Don't understand this problem: There are only 2 pairs of positive integers $(x,y)$ for which…

..both $\frac{21}{x}$ and $\frac{70}{y}$ are in lowest terms and for which $\frac{21}{x} + \frac{70}{y}$ is an integer. One such pair is $(1,1)$. What is the other such pair? This is a Mathematics ...
4
votes
0answers
175 views

Can we get a 'good' approximate value of $\sqrt 2$ by an equation which uses each of $1,2,\cdots,9$ once such as $12653\div 8947\approx1.414217$?

I've been interested in representing $\sqrt 2\approx 1.414213562373095$ by an equation which uses each of $1,2,\cdots,9$ once. Suppose that the following conditions must be satisfied. Then, can we get ...
5
votes
1answer
260 views

Smullyan-To-Mock-a-Mockingbird, Find egocentric bird in L

Question (29, p. 81). Let me tell you the most surprising thing I know about larks: Suppose we are given that the forest contains a lark $L$ and we are not given any other information. From just this ...
0
votes
1answer
71 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
0
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2answers
73 views

Prove problem of Mathematical Reasoning

The gcd of two integers $a$ and $b$ (both not zero) can be described as the smallest positive integer of the form $am+bn$, where $m,n \in \Bbb Z$. Prove that every positive $x$ of the the form $x=am+...
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votes
1answer
56 views

“Convergence” of the sequence $a_k=2^{10^{\ k}}$

I've been observing final digits of each number in the sequence $$a_k=2^{10^{\ k}}$$ You get: $\ a_0=2 \\ a_1=1024 \\ a_2= ...205376 \\a_3= ...069376\\a_4=...709376\\a_5=...9883109376\\a_6=......
6
votes
1answer
273 views

Help explain a new theory on small sines

(10 Mantissa[sin(10^(-100 - r1/x))])^(r2x) The reason for the argument form .10^[-n-(1/x)] is the beautiful pattern found in sin(10^-n) for positive integer n. $$ \begin{array}{| c | r |} \hline n&...
19
votes
1answer
977 views

Why are these geometric problems so hard?

I was surprised to learn that both for the Moving Sofa Problem and Packing 11 Squares solutions have been proposed, but in either case the optimality of the proposed solution is, as of yet, only ...
3
votes
1answer
329 views

My conjecture on almost integers.

Here when I was studying almost integers , I made the following conjecture - Let $x$ be a natural number then For sufficiently large $n$ (Natural number) Let $$\Omega=(\sqrt x+\lfloor \sqrt x \...
15
votes
4answers
1k views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
10
votes
3answers
308 views

What does it mean to be $0.9-$Dimension?

We can visualize $1\mathrm D$, $2\mathrm D$, $3\mathrm D$ and we can think of a higher $M-\mathrm {Dimension}$ where $M\in \mathbb N^+$as a vector. But I recently learned that there are non integer ...
1
vote
4answers
269 views

Stuck on a basic thinking question

Stewart and Michael have arranged to meet. Michael is about to set off on his bicycle, and at the same time Stewart is going to run to meet him. Michael can cycle at a steady 20 kilometres per hour ...
0
votes
1answer
83 views

Linear Transformation Brief Question

$T:P_{3}\rightarrow P_{3}$ defined by $T(p(t))=tp'(t)+p(0)$ is a linear transformation. Determine whether $T$ is invertible. If yes, find $T^{-1}(q(t))$, where $q(t)$ is a polynomial of degree at ...
3
votes
2answers
75 views

A bishop on a grid

Suppose that we have an $n\times m$ chessboard and bishop on the square $(1,1)$. It starts to move diagonally with the following rules: If bishop is in any corner square except $(1,1)$, it stops ...
4
votes
1answer
92 views

Analysis of “Dungeon Raid”

The computer game Dungeon Raid is quite complicated, but for our purposes we can consider the following simplified version. The player has a current health $\def\hcur{h}\hcur$ and a maximum health $\...
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votes
3answers
200 views

Finding values for $x$ and $y$ given ONE equation

Ok so I'm in precalculus right now and the directions on my homework seem to make no sense to me. I'm asked to find values for $x$ and $y$ given this equation: $y=x^{1/3}$ Doesn't this mean I could ...
4
votes
3answers
282 views

Trying to work out the probabilities of a dice game I used to play

At college, my friends and I would sometimes waste time playing a game with dice. We would roll 25 dice, pick out all the dice that landed on a 6, then roll the rest. This would carry on until all the ...
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vote
3answers
7k views

How many multiples of 3 are between 10 and 100? (SAT math question)

In the figure above, circular region A represents all integers from 10 to 100, inclusive; circular region B represents all integers that are multiples of 3; and circular region C represents all ...
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2answers
11k views

What is the best strategy for Cookie-Clicker-esque games?

Today, I stumbled across the game Cookie Clicker, which I recommend you avoid until you have at least a few hours of time to waste. The basic idea behind the game is this: You have a large stash of ...
2
votes
3answers
887 views

A truth teller and liar puzzle of Ramanujan mathematical olympiad 2013

On an island each person always tells the truth or each person always tells a lie. Three people say $A$ , $B$ and $C$ have a conversation. $A$ says that $B$ is lying , $B$ says that $C$ is lying and $...
2
votes
2answers
219 views

When does the triangle have the smallest area?

The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and ...
1
vote
1answer
68 views

Guessing Game Stochastic Optimization

This is part of another post I did, but I think it has interest in its own right: Let $Y =\{X_{1},X_{2}...X_{N}\}$ be a set of $N$ random quantities with assocated set of distributions $F=\{F_{1},F_{...
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votes
2answers
245 views

Interesting problems using group/representation theory

I've been going through this representation theory lecture notes, and I've found the ''hungry knights'' problem very interesting. Do you know any interesting problems similar to that one? To define ''...
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3answers
396 views

Find all the integral solutions to $2x+3y=200$

What's the best way of going about this? $$2x+3y=200.$$
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4answers
665 views

Interesting Medieval Mathematics Lecture/Activity Ideas?

Recently I have been invited to give a talk about Medieval Mathematics or mathematics in the 500 AD - 1500 AD time frame. I have been researching the time frame for the past week and have found ...
1
vote
0answers
75 views

What are all possible numbers gotten by an digit-exchange operation?

A friend of mine taught me a number game. Supposting that $a_na_{n-1}\cdots a_1$, which satisfies $a_n\gt a_1$, is a natural $n$-digit number with decimal representation, let's consider the ...
3
votes
2answers
2k views

How to create number six using three zeroes?

How to create number 6 using only three 0, any arithmetic operation is allowed? I know it is possible, but I don't know how...
0
votes
1answer
250 views

Filling 4l, 5l bottles from two 10l bottles

There are two bottles of 10litre each filled with water. Now two persons having empty bottles of 4litre and 5litre want to take 2litres of water each from the previous 10litre bottles.. Now you ...
20
votes
1answer
490 views

Is $\sum_{k=1}^{n} k^k / \sum_{k=1}^{n} k \in \mathbb{N}$ for some $n > 1$?

Let $ A = \sum_{k=1}^{n} k^k $ and $ B = \sum_{k=1}^{n} k$, where $n >1 $ is a positive integer. Is $A/B$ ever an integer?
10
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3answers
193 views

Conway's Game of Life

Is there a mathematical way to directly calculate iteration n from the first iteration skipping calculating the iterations in between in Conway's Game of Life? I would assume, if it is possible, it ...
4
votes
1answer
83 views

A persistent difference

Here's a fun math problem. I wasn't able to get it - am curious what you guys have to say. Pick a four-digit number whose digits are not all the same. From its digits form the smallest four-digit ...
3
votes
2answers
93 views

Simple question that I can't solve [duplicate]

Here is a relatively simple question that I'm unable to solve :/ There are $10000$ closed lockers in a hallway. A man begins by opening all $10000$ lockers. Next, he closes every $2^{nd}$ locker. ...
7
votes
3answers
312 views

Can we identify the time if we know every angle between three hands of a watch?

Let $M, H, S$ be the minute hand, the hour hand, the second hand of a watch respectively. Also, let $A_{MH}, A_{MS}, A_{HS}$ be the angle between $M$ and $H$, $M$ and $S$, $H$ and $S$ respectively. ...
3
votes
1answer
75 views

A neat application of Chebyshev's inequality

An interesting little fact I noticed today while problem-solving: Show that, for any positive reals $x_1, x_2, ... x_k$ and any positive integers $m, n$ with $n >m$, $$ x_1 x_2 \cdots x_k \ge 1 \...
22
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2answers
396 views

The next number that has this property?

I noticed that $1/8 = 0.125$ and the sum of the digits of the number $0.125$ is $0+1+2+5=8$. It's lovely. I searched other numbers who have that propriety : I only found $1$, $3$ and $8$. Is there ...
6
votes
7answers
3k views

answer to iq test with colored squares

What is the best (whatever this means) answer and why?
1
vote
2answers
138 views

How to calculate x in formula $ y =\left( \frac{\frac{17000}{x+400}+8.5}{100}+ 1\right) x $?

I am not even sure how it's officially called (so not sure with tag to give it). As an example if you have a math problem $y = x + 1$. You have a $y$ value, but not $x$. To you revese the problem $y - ...
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1answer
79 views

given a positive integer $n\geq 2$, we have a positive integer $m$ such that $m+2,m+3,\dots m+n$ are composite. (TIFR exam $2012$)

Question is to prove that : given a positive integer $n\geq 2$, we have a positive integer $m$ such that $m+2,m+3,\dots m+n$ are composite. I tried checking for small numbers to see if there is any ...
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vote
0answers
99 views

Existence of a Vampire number on the form $v = xy = a^bb^a$?

A number $v = xy$ with an even number $n$ of digits formed by multiplying a pair of $n/2$-digit numbers (where the digits are taken from the original number in any order) $x$ and $y$ together. ...
40
votes
19answers
2k views

Literary statements that are false as mathematics [closed]

I recently wanted to use the title of the famous short story "Everything that Rises must Converge" in a poem of mine. However, the mathematician in me insisted on changing it to "Everything that Rises,...
5
votes
1answer
179 views

Surprising limit (probability of no two coinciding pairs)

I stumbled upon this question by random chance. The motivation is kind of long, the question is pretty short; if you're just here for the limits, feel free to skip to the break. I'm taking five ...
2
votes
1answer
423 views

How many $n$-disk legal configurations are there for the Tower of Hanoi?

This question comes from this homework assignment from ECS20 at UC Davis. How many $n$-disk legal configurations are there for the Tower of Hanoi? A "legal configuration" means that no disk is ...
2
votes
1answer
170 views

A puzzle related to three cars which leave a town and reach another.

I'm trying to solve the following puzzle: $C_1, C_2$ and $C_3$ are three cars that leave town $T_1$ and reach town $T_2$. For a car, say $C_k, k$ is considered to be the car number. The car number ...
0
votes
1answer
185 views

Lattice Squares; Basic Interesting Facts and Problems

I'm going to write an article in an educational magazine for middle school students, about the game Square It. The purpose of the game is to make lattice squares: I want to introduce the game, and ...
3
votes
1answer
119 views

Find the values of $(a,b,c)$ such that $a^{2013}+b^{2013}=c^{2013}$ and $a^2+b^2=c^2$.

My professor likes to give our class some questions for fun every once in a while. He posed the following problem in class yesterday, and I've been stuck. Find the values of $(a,b,c)$ such that $a^{...