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

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9
votes
3answers
268 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
1answer
64 views

Finding $a^{2014} + b^{2014} + c^{2014}$ given some conditions on $a,b,c$.

I came across this problem: "Let $a$, $b$, $c$ be nonzero real numbers that satisfy the conditions : $$a + b + c = 9,\\\mathrm{and}~ab + bc + ca = 27 $$ Calculate $$a^{2014} + b^{2014} + ...
6
votes
1answer
371 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
-1
votes
2answers
32 views

Find the missing number in the given series? [on hold]

Find the missing number in the given series? the series is $1584$, $900$, ____, $180$, $48$, $4$.
4
votes
1answer
143 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 ...
1
vote
0answers
53 views

Is it possible to solve sudoku without backtracking?

I occasionally solve sudoku puzzles on smartphone in spare time. My approach is based on the belief that at each stage in solving a sudoku puzzle there is at least one cell where there in only one ...
1
vote
1answer
26 views

Probability of 2 students being chosen the both have under 100 books at home

Suppose we select two students at random from the class of fifteen. What is the probability that both students chosen have less then 100 books at home? Data provided is the amount of books each ...
2
votes
2answers
85 views

Digit sum of $n^2$ is 44

Is there a whole number $x$ such that the sum of the digits of $x^2$ equals 44? I would like someone to tell me if my thoughts are correct. The remainder of a number a divided with 9 is the same as ...
4
votes
3answers
52 views

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 ...
0
votes
3answers
49 views

Is there a pattern present?

Jack is looking at Anne, however Anne is looking at George. Jack is married, George isn't, and Anne's status is unknown. provided this info alone determine whether: a) A married person looks at a non ...
1
vote
2answers
74 views

The product of two prime numbers

I have two expressions (both of which have a term raised to the power of $n$) and I am trying to prove that they can't be prime numbers at the same time for $n>2$. I can't post the expressions, ...
5
votes
4answers
14k views

What is the highest number that can be got from 4383 by moving exactly 2 matches?

What is the highest number that can be got from 4383 by moving exactly 2 matches? Number 1 has got 2 matches, so I thought it will be 47831 as I remove two matches from second number (3), but it ...
1
vote
2answers
58 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.
1
vote
1answer
29 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 ...
547
votes
153answers
34k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
4
votes
4answers
1k views

What is the importance of an integer sequence like the happy numbers?

I've been looking at the happy numbers which led me to the OEIS and showed me that there are many documented integer sequences. What I don't understand is the importance of these sequences. To me, the ...
0
votes
1answer
48 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 ...
3
votes
2answers
1k views

Flipping Cards Probability

You have a deck of cards, 26 red, 26 black. These are turned over, and at any point you may stop and exclaim "The next card is red.". If the next card is red you win £10. What's the ...
0
votes
0answers
27 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
votes
1answer
48 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 ...
0
votes
1answer
30 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 ...
7
votes
1answer
95 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
0answers
29 views

Time and Distance(Proof) [closed]

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
votes
2answers
50 views

Max value of Z, maybe Lagrange multipliers? [closed]

$$z = 0.0269 (x) (x-2y)^2$$ $$ x+y < 6$$ $$0 < x < 2$$ $$2 < y < 6$$ Find values of $x$ and $y$ to maximize $z$ I am stumped on how to go about this...
11
votes
0answers
96 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 ...
12
votes
2answers
7k views

Expected Ratio of Coin Flips

If you flip a coin until you decide to stop and you want to maximize the ratio of heads to total flips, what is that expected ratio? Assuming that you want to maximize the ratio, meaning ...
4
votes
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
votes
1answer
38 views

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

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
vote
2answers
54 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 ...
3
votes
1answer
67 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 ...
10
votes
1answer
71 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
votes
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 ...
1
vote
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: ...
6
votes
3answers
90 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
votes
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 ...
43
votes
2answers
928 views

Possible distinct positive real $x,y,z \neq 1$ with $x^{(y^z)} = y^{(z^x)} = z^{(x^y)}$ in cyclic permutation?

Can we have distinct positive real $x,y,z \neq 1$ with $$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutation? It does not work well if any ...
1
vote
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 ...
33
votes
8answers
20k views

What is the math behind the game Spot It?

I just purchased the game Spot It. As per this site, the structure of the game is as follows: Game has 55 round playing cards. Each card has eight randomly placed symbols. There are a total of 50 ...
5
votes
1answer
139 views

How can I show that the sequence $x_n^2$ is bounded?

Two real sequences $(x_n)$ and $(y_n)$ are defined by $$x_{n+1}=x_n-(x_ny_n+x_{n+1}y_{n+1}-2)(y_n+y_{n+1})$$ $$y_{n+1}=y_n-(x_ny_n+x_{n+1}y_{n+1}-2)(x_n+x_{n+1})$$ with initial conditions $x_0=1$ and ...
5
votes
2answers
164 views

If $abc=1$, then $\frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $

Let $a$, $b$ and $c$ be positive real numbers with $abc=1$. Prove that $$ \frac{a^{n+2}}{a^n+(n-1)b^n}+\frac{b^{n+2}}{b^n+(n-1)c^n}+\frac{c^{n+2}}{c^n+(n-1)a^n} \geq \frac{3}{n} $$ for each ...
2
votes
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
votes
0answers
49 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
votes
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 ...
5
votes
1answer
140 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 ...
9
votes
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
votes
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
37 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? ...
165
votes
5answers
6k views

Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?

My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question: I am thinking of a number... It ...
46
votes
10answers
1k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
3
votes
1answer
78 views

The Rubik Square permutation groups

This post was inspired by this webpage of mathematical challenge due to Mickaël Launay (French). Let $G_n$ be the subgroup of $S_{n^2}$ generated by the red arrow permutations as for the following ...