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

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0
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0answers
23 views

Thermodynamics based proofs

What are some mathematical inequalities and theorems that follow using thermodynamics "proofs" (rigorous or just intuitive)? Any suggested books on the matter? For example, AM-GM inequality follows ...
0
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1answer
35 views

A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water $n$ times.

There is a formula in my book for questions of type, A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water. After $n$ such operations what will be the ...
1
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0answers
18 views

How to solve problems on alligation and mixture when three types are given?

Suppose there are three qualities of rice, A(1 dollar per Kg), b(2 dollar per Kg) and C(3 dollar per Kg). The salesmen want to mix these in a certain ratio a:b:c so as to make the price 2.5 dollar per ...
1
vote
2answers
41 views

Building a box from smaller boxes

John has 77 boxes each having dimensions 3x3x1. Is it possible for John to build one big box with dimensions 7x9x11? I'm leaning towards no, but I would like others opinion.
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1answer
47 views

Maximum number of chess moves

Chess has a limited number of maximum moves because of the 50-move rule (50 moves without any captures or pawn moves results in a draw). There are 30 capture-able pieces, and I've figured out that the ...
0
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1answer
29 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
-3
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0answers
29 views

Time and Distance(Proof) [on hold]

If two persons $A$ and $B$ start at the same time in opposite directions from two points and complete the journeys in $a$ and $b$ hours respectively after passing each other, then ...
-2
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2answers
50 views

Max value of Z, maybe Lagrange multipliers? [on hold]

$$z = 0.0269 (x) (x-2y)^2$$ $$ x+y < 6$$ $$0 < x < 2$$ $$2 < y < 6$$ Find values of $x$ and $y$ to maximize $z$ I am stumped on how to go about this...
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0answers
88 views

Penrose's remark on impossible figures

I'd like to think that I understand symmetry groups. I know what the elements of a symmetry group are - they are transformations that preserve an object or its relevant features - and I know what the ...
4
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0answers
33 views

How many ways I can put $k$ bishops on $n\times n$ chessboard?

Is there a formula how to count in how many ways I can put $k$ bishops on $n\times n$ chessboard such that no two bishops threaten each other?
0
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1answer
37 views

If girls average 90 and boys are double total class average is 80 what is total average of boys [on hold]

If girls in class average 90 and boys are double amount of girls total average of class is 80 what is total average of boys
1
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2answers
53 views

Ten marbles put in a box, colour of each set by toss of a fair coin. You draw (with replacement) ten white marbles. Probability all marbles are white?

The following question comes from the probability section of the Titan Test*. * I will avoid the debate around whether this test accurately measures what it aims to, nor whether such aims are ...
10
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1answer
69 views

A generalisation of Napoleon's theorem. Is this result original?

I've found a generalisation of Napoleon's theorem to general polygons. Take any regular $n$-gon inscribed in a circle and stretch it (in any direction) so that the circle becomes an ellipse and the ...
3
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1answer
56 views

The jackal, the lion, the parrot and the giraffe - logic puzzle

Here is a puzzle that appeared in a Russian magazine named Kvantik (see Tanya Khovanova's Math Blog). [The trick lies in that we don't know exactly what the hedgehog knows at each stage. The symbology ...
-1
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0answers
70 views

why is $3704554$ a magic number? [closed]

Someone told me $3704554$ is a magic number, but I am unable to see why. I've tried factoring it: $2\cdot7\cdot107\cdot2473$, binary: $1110001000011011101010_2$, but nothing seems to explain it.
3
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1answer
45 views

find a group of lowest N numbers so that no 2 pairs have the same bitwise or

I am trying to find the lowest group of N numbers (i.e. N=1000) so that no 2 pairs from the group have the same bit-wise or. more specific need to find a group $A = \{a_1,a_2,a_3,..,a_N\} $ such ...
2
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2answers
51 views

Sailor,Monkey,Coconut answer in elaborate

In Sailor, Monkey, Coconut Problem Can anyone tell me how adding 56 gives me another solution??I understand that cocount is divided into 5 piles.But how is 56 give me another solution?why wouldn't ...
1
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1answer
27 views
+100

An interesting wire-tying problem to match wire ends in as few trips as possible

You have $N$ wires that all extend from one location to a second distant location. The wire ends at both locations are unlabeled, and the goal is to label them all (on both ends) with distinct labels ...
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0answers
47 views

Proof Attempt of Brouwer (via Separating Hyperplane Theorem)

In part motivated by the discussion here, I have been playing with trying to prove Brouwer's theorem appealing as minimally as possible to topology. In the 1-dimensional case I believe one can ...
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0answers
35 views

A harder long division puzzle than the first; what should “Algebra I” solution look like?

Here's another problem, significantly harder than the first, but still accessible to target audience. The statement of the problem (i.e., northwest corner only) comes from a PennyDell puzzle magazine: ...
4
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0answers
74 views
+50

Cover the grid graph with simple cycles

I have a two dimensional n x m grid graph. And I want to find in how many ways this grid can be covered with simple cycles (it can be a one cycle or it can be many ...
6
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3answers
87 views

“Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...
2
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0answers
19 views

The odyssey of spies: Kryptos

The part four $K4$ of Sanborn sculpture, a sculpture located on the grounds of the CIA in Langley remains unsolved. As you can read in [1], Sanborn released a clue for the 64th-69th letters in part ...
2
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0answers
47 views

Beal Conjecture and ($\bmod 3$) operation [closed]

When we apply a ($\bmod 3$) operation on the $A^x +B^y =C^z$ we will see some strange results. For e.g.: Let $A=6m+1$ & $B=6n+1$. Since $A$ & $B$ are odd numbers, $C$ will have to be even. ...
2
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1answer
38 views

Code Jam 2014 Cookie Clicker Alpha Proof

I was looking at the solution for the Code Jam 2014 qualification question but the proof of correctness seems to be incomplete and I was wondering if anyone could help me with it. The full question ...
9
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1answer
73 views

Branching Paths Problem

I was drawing some shapes during class, and I came across the following problem. If one takes steps of constant length, and one must deviate a constant angle $\alpha$ from one's previous step either ...
-1
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0answers
27 views

sum percentages without knowing totals

lets say that for "business choice" I have in my DB, for each item in my list, just percentage of overall sold items but not the count of them. and in another file just what i sold today so DAY1 ...
1
vote
1answer
36 views

How to get the interest rate per quarter given the semiannual interest rate?

I have this economics problem. What is the present worth of $500.00$ Rupiah deposited at the end of every three months for $6$ years if the interest rate is $12\%$ compounded semiannually? ...
5
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1answer
139 views

Stupid numbers game

Occasionally I encounter a recurring instance of stupidity when I teach mathematics, namely that someone asks if we can play a game known in Danish under the name "bum", since that person thinks that ...
7
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1answer
92 views

Smallest cylinder into which a regular tetrahedron can fit?

Given a regular tetrahedron (as shown) of edge length $b$, determine the diameter $d$ of the smallest right circular cylinder (pipe) of infinite length along which the tetrahedron can slide.
3
votes
1answer
67 views

Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
0
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1answer
31 views

Upper bound of digit sum of powers

Take $x \in \Bbb N$, $x \le9$ and $m \in \Bbb N$. Now we define a function $d_s(n): \Bbb N \to \Bbb N$ as the digit sum of $n$ in base $10$. Now let's say we have a lower bound $b_l$ and an upper ...
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0answers
25 views

recursion $t(n)=\sqrt{2} \times \frac{tn}{2} +\log{n}$

I tried substituting $m=\log{n}$ $t(2^n)=\sqrt{2} \times \frac{t2^n}{2}+m t(m)= \sqrt(2) \times \frac{tm}{2}+m$ From here I got $\log {n}$ But with induction I proofed its $\sqrt {n}$
2
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0answers
44 views

Why 6 races is not sufficient in the 25 horses, 5 tracks problem

The horse-racing puzzle has been asked on mathSE several times (1, 2, 3, 4); there is also a generalization. I restate the puzzle below: 25 horses all run at different speeds. You can race 5 ...
1
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4answers
37 views

If $n$ people are placed in a room, prove that at least $2$ of those people will have the same number of friends in the room.

If $n$ people are placed in a room, then at least $2$ of those people will have the same number of friends in the room. I want to prove this statement. Here are some of my thoughts: If all the ...
2
votes
1answer
93 views

Most Number Of Lowest Numbers

I'm looking for a formula for the following problem. Hopefully I can explain this clearly and it all makes sense. No, it's not my homework, it's part of a competition I'm involved with managing and ...
5
votes
2answers
342 views

Impossible numbers drawn from tricky function

The function is this: $f(\frac{a}{b},\frac{c}{d})=\frac{a+c}{b+d}$ where $0\lt \frac{a}{b} \lt 1$ $0\lt \frac{c}{d}\lt 1$ $a,b,c,d$ are all integers $a/b$ and $c/d$ are in lowest terms Are there ...
17
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4answers
405 views

Why do siamese magic squares have real eigenvalues, symmetric around zero?

There is a standard method to construct magic squares of odd size, known as the Siamese construction. I'll write $S_m$ for the $m \times m$ Siamese square. For example, here is $S_5$. ...
0
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1answer
73 views

Evaluating function

Here is the function: $$f(a)=\sqrt{f(a)+\sqrt{f(f(a))+\sqrt{f(f(f(a)))+\cdots}}}$$ Is there another way to represent this function so that it only has $f(a)$ on one side and no $f(a)$'s on the other ...
0
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2answers
34 views

Eliminating non-integer solutions to $ab / (2\sqrt{ab} + a + b)$

I am writing a program to output all $a,b \in \mathbb{N}$, where $a \le b \le n$ (for a given $n \in \mathbb{N}$), such that $$ \frac{ab}{2\sqrt{ab}+a+b}=c\in \mathbb{N} $$ For example, $a=9$, ...
0
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0answers
16 views

Lines of intersection of four similar planes are coplaner.

If lines joining four corresponding vertices of two tetrahedrons are concurrent, then the lines of intersection of four corresponding planes are coplanar, and the converse also holds true. What would ...
-4
votes
1answer
54 views

How many minimum weights do you need to measure all weights from $1$kg to $1000$kg [closed]

You can place weights on both side of weighing balance and you need to measure all weights between $1$ and $1000$. For example if you have weights $1$ and $3$, now you can measure $1,3$ and $4$ like ...
-2
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1answer
245 views

Nice combinatorics puzzle [closed]

INTRODUCTION Around 1980 my father told my a simple yet interesting mathematics puzzle. It was similar to the "two professors puzzle". I don't remember all the details, but it had three "I don't ...
33
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3answers
915 views

Guessing the length of a playlist on “shuffle random?”

The other night I was hanging out with some friends and someone put on a playlist on shuffle random, where the songs are drawn uniformly at random from a fixed playlist. The person who put the ...
1
vote
1answer
44 views

Does this game have infinite expected payout?

Consider the following game: Suppose the initial value of the pot is $ S $. Our player Josephine then rolls a fair $n$-sided die. If the roll is not $1$, then the pot is multiplied by that roll, and ...
2
votes
1answer
42 views

Puzzle: Determining the structure of a bipartite graph

Consider the bipartite graph $G = (X, Y, E)$, with $|X| = |Y| = n$. We can think of $X$ and $Y$ as clusters of $n$ switches on either end of a long hallway. Each switch on one end of the hallway has ...
0
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1answer
34 views

$f(x)=y$ while $g(y)=x$; Is it possible to find two not reverse functions that behave such at least for a given set of inputs and outputs?

I want to know if it is possible to program such a code that could determine two distinguish, not inverse, functions, say $f$ and $g$, that is true for the below statements at a given input and output ...
1
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3answers
37 views

understanding how multiplying a number by itself and than using the result for division gives you a consistent result

hey guys i am basically a programmer and just came across a peice of mathematical calculation that i was curious to understand , have a look below :: ...
12
votes
2answers
371 views

Sum numbers game

$2n-1$ numbers are lined up as follows: $n$ , $n-1$ , $n-2$ , $\cdots$ , $2$ , $1$ , $2$ , $3$ , $\cdots$ , $n-1$ , $n$ At each step, one can choose any number in the line and add it to each of ...
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0answers
19 views

Iterated digit product

A very interesting calculator at http://www.micmaths.com/defis/defi_01.html repeatedly calculates the product of the digits of a number and stops when it reaches a single digit. It asks what is the ...