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

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1
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2answers
83 views

Swap two integers in different ways

This is a famous rudimentary problem : how to use mathematical operations (not any other temporary variable or storage) to swap two integers A and B. The most well-known way is the following: ...
-1
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1answer
78 views

Computation of integral $\int_{0}^{1}\ln(p)\ln(1-p)p^{2}\,dp$

I want to compute this integral: \begin{equation*} J=\int_{0}^{1}\ln(p)\ln(1-p)p^{2}dp \end{equation*} It will be great if you can detail the proof. I tried to do change of variable it does not ...
1
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1answer
59 views

For how many seconds do I need to turn the pedal?

For how many seconds do I need to turn the pedal of a bike, so that the number of turns is equal to the value of my velocity in the given moment measured in km/h.
0
votes
1answer
119 views

How to better compute this eta formula

In the question $C$ MRB proofs wanted , I gave the following excerpt from http://www.perfscipress.com/papers/UniversalTOC25.pdf . . I accurately worked formula 44 in Matheamatica in the following ...
4
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0answers
67 views

Finding every $n$ such that $n\times$ ('reverse' number of $n)=m^2$ such as $1584\times 4851={2772}^2$

Let $r(n)$ be the 'reverse' number of $n$ in the decimal system. For example, $r(1234)=4321$. Then, here is my question. Question : Can we find every $n(\in\mathbb N)$, which is not a square ...
2
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1answer
158 views

Monotonically increasing path in a complete graph

Given a complete graph with n vertices such that all edge weights are distinct. Prove that we can find a monotonically increasing path of length n-1. I tried finding such a path by sorting the edges ...
1
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0answers
30 views

Converting dot producto to set of arithmetic mean differences?

Ok so I am reading a book on linear algebra ( Gilbert Strang to be specific) and I am on second problem set, challenge problem, problem 29. In solutions it appears that the author states that: ...
2
votes
1answer
441 views

roulette wheel sequence

Is the sequence of numbers around a European roulette wheel (the integers from 0 to 36 inclusive) random or is there a pattern to it? It is said to have been devised by Pascal, which might be thought ...
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0answers
63 views

Net for both cube and regular tetrahedron

At how to fold it by Joseph O'Rourke, there is a net given that can be folded into a cube or irregular tetrahedron. Is there a net that can be folded into either a cube or regular tetrahedron?
3
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0answers
445 views

Guilloché security printing — can it be cracked?

Money uses Security printing, and often uses Guilloché patterns. These curves are inscribed by wheels on wheels on wheels, ten wheels deep in some cases. For example, the back of the US $1 bill has ...
4
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2answers
616 views

Expected number of points on circle to form an acute angled triangle

This problem was asked to me in an interview. We keep on adding points on a circle uniformly until there exist three points on the circle which form an acute angled triangle. What is the expected ...
5
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2answers
2k views

Can you divide a square into 5 equal area regions

Given this shape: Is it possible to divide the cyan area into 5 equal area shapes such that: Each shape is the same Each shape has an edge touching the red square Each shape has an edge touching ...
2
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0answers
571 views

K non-intersecting diagonals in a polygon

Given a regular N-sided polygon, how many ways can you draw K non-intersecting diagonals? Any pair of diagonals must not intersect strictly inside the polygon. For e.g. N = 4 and K = 2 -> 2 ways ...
2
votes
1answer
56 views

What points do have three points with minimal distance on $x^2$?

Suppose you have a function $f: \mathbb{R} \rightarrow \mathbb{R}$ and a Point $P = (x,y) \in \mathbb{R}^2$. Now you want to find all $x_1, \dots, x_n$ such that $$\forall \tilde x \in \mathbb{R} ...
4
votes
3answers
680 views

What is the most mathematical flag?

I know it sounds stupid. But my professor asked us which is the most mathematical flag. As I know most of the flags of the countries are rectangle then what does he mean by most mathematical flag?
5
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1answer
100 views

Investigating the density of special integers

I was working on a problem earlier and came up with a solution using the following type of integers: Call an integer $n \geq 3$ convenient if the following hold: $n$ is a squarefree odd integer if ...
0
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0answers
116 views

Where to find more of these puzzles?

Examples: A packing company supplies storage boxes in three different sizes: small, medium, and large. All three types of box have the same ratio of width:length and height:length. It is noted ...
4
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0answers
389 views

MRB constant proofs wanted

This article has been edited for a bounty. $C$ MRB, the MRB constant, is defined at http://mathworld.wolfram.com/MRBConstant.html . There is an excellent 56 page paper whose author has passed away. ...
2
votes
1answer
113 views

Why these exact values?

In Mathematica I have In[181]:= FullSimplify[ArcSin[10^(1/2)] == (Pi/2 - ArcSinh[3] I)] Out[181]= True In[206]:= FullSimplify[ArcSin[100^(1/3)] == (Pi/2 - ArcCosh[10^(2/3)] I)] Out[206]= True ...
0
votes
3answers
192 views

Coded language puzzle!! [closed]

Here is a puzzle I can't crack. It goes like this: In a certain coded language MANGO=3/5 ORANGE=2/6 APPLE=1/5 Then, POTATO=?? The answer is 5/6. I would like to know to arrive at the answer.
7
votes
1answer
335 views

Why do my professors ignore my work? [closed]

I'm a high-school student finishing in December and about to pursue a career in mathematics. In my free time, I like to ''research'' hard problems and come up with unique proofs or combine ...
2
votes
0answers
72 views

How many Hamiltonian loop are there in a big rectangle?

Suppose I have some big rectangle made of $n \times m$ squares, and I want to place tiles on it in a manner that makes a picture of a hamiltonian loop. I can transform this problem into a problem ...
5
votes
1answer
304 views

Tricky Puzzle!! Please help.

I stumbled upon a puzzle I can't crack. It goes like this: In a certain Code language: 7321=6 5342=3 8645=15 Then 9312=? The Answer is 9. But I can't seem to find the logic behind it??
0
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1answer
28 views

What can be the value of $m$ in following equation

During calculations I got this step $$(e^m/((m+1)^{m+1}) )^{3n/4} = 1/2^n$$ I want the value of m here??
6
votes
2answers
237 views

How small is the smallest circle a car can drive?

Lets say we have a model of a car with two fixed back wheels and two wheels in front that steer in the same angle: The wheelbase $w$ is the fixed distance of the two wheel axes. $\alpha_m$ is the ...
5
votes
0answers
338 views

Paul Erdős Joke.

I was watching the great documentary "$N$ is a Number" and in it Erdős tells a joke where he writes: PGOM LD AD LD CD Which means poor great old man, living dead, archeological discovery, legally ...
1
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1answer
64 views

Finding the time taken by a flight

I came across a question today which is as follows: A Flight takes off at 2 A.M from northeast direction and travels for 11 hours to reach the destination, which is in northwest direction. Given ...
0
votes
1answer
540 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
88 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?
2
votes
1answer
125 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
110 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
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1answer
120 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
114 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
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0answers
168 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
197 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 ...
0
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1answer
69 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
68 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 ...
6
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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= ...
6
votes
1answer
261 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 ...
18
votes
1answer
847 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
288 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 ...
14
votes
4answers
835 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 ...
9
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3answers
271 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
207 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
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1answer
80 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
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2answers
72 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
86 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 ...
0
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3answers
199 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 ...
3
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3answers
209 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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3answers
4k 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 ...