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

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7
votes
0answers
193 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
34
votes
3answers
4k views

Can a Rubik's cube be mapped knowing only two sides?

Is it possible to know the entire configuration of a Rubik's cube looking at only two sides and not rotating the cube? In other words: what is the minimum information required to create a ...
3
votes
1answer
181 views

For which chess boards do solutions exist for this generalised Knight's Tour problem?

We know from a theorem by Schwenk that for any (m x n) chess board with $m \leq n$ it is always possible to create a knight's tour unless one or more of these three conditions are met: m and n are ...
2
votes
2answers
412 views

Putnam 2001 - Problem A-1 (On a binary operation)

Let $*$ be a binary operation acting on a non-empty set $S$, such that $$(a*b)*a=b,$$ for all $a,b\in S$. Prove that $$a*(b*a)=b,$$ for all $a,b \in S$.
1
vote
2answers
59 views

polynomial series and root multiplicity

Excuse me, because I know this is a double post but I can't for the life of me find the original post. Given a sequence $(a_n)$, one can construct a polynomial of the form ...
0
votes
3answers
114 views

How many persons do you think are liar?

There are 10 person. First person says: At least one of the person is liar. Second person says: At least two of the person is liar. Third person says: At least three of the person is liar. Fourth ...
0
votes
1answer
272 views

Base 12 Versus Base 16

I'm not good when it comes to math, so forgive me. I'm doing a personal study of is there a better base number for our culture to use? I have to consider factors like: the number of digits to write, ...
0
votes
2answers
89 views

Is There An Alternative To Using 0 As A Placeholder?

I'm no math wiz here, but I have a question that I can't wrap my head around. In fact, I don't even know how I would even go about asking the question properly. Is there an alternative to using 0 as a ...
7
votes
2answers
176 views

Mental Arithmetic

This is very possibly not the best place to ask this, however it's the best I could find but please suggest anywhere else that might be better suited. I'm building a sort of challenge revolving ...
14
votes
3answers
8k views

The Best Strategy and Highest Possible Score for the “Threes!” Game.

[There's still the strategy to go. A suitably robust argument that establishes what is statistically the best strategy will be accepted.] Here's my description of the game: There's a $4\times 4$ ...
1
vote
3answers
174 views

What is the next number? [closed]

What is the next number in the following set ? $$1,11,21,1211,111221, \ldots$$
4
votes
1answer
92 views

Is there any way to analyze an absurdly large exponent?

On a recent Giant Bombcast, someone wrote in and asked an absurd question (as is usual for this podcast). In short, the question was: Given a 1080p TV, how long would it take to view every ...
0
votes
0answers
53 views

Integral of a function and its inverse

This comes from the comments section of this question. The original question was to show the following identity for some increasing invertible function $f$ ...
46
votes
1answer
1k views

How likely is it not to be anyone's best friend?

A teenage acquaintance of mine lamented: Every one of my friends is better friends with somebody else. Thanks to my knowledge of mathematics I could inform her that she's not alone and ...
0
votes
1answer
72 views

Degree sequence in $O(n)$

How can we determine the whether a sequence of non negative integers is a valid degree sequence in $O(n)$. I have determined an $O(n\log n)$ algorithm using erdos-gallai theorem.
1
vote
1answer
140 views

NCAA bracket and binomial coefficients

Given that March Madness is almost here I was trying to figure out the probability of constructing a perfect bracket if you just flipped a coin for every game. I came up with two possible solutions. ...
-1
votes
1answer
153 views

Question on Speed and Distance

X, Y and Z move along a circular path of length 1.2 km with speeds of 6 km/h, 8 km/h and 9 km/h respectively. X and Y move in the same direction but Z moves in opposite direction. If they all start at ...
3
votes
1answer
880 views

How can I use math to fill out my NCAA tournament bracket?

With the NCAA basketball tournament right around the corner and the conference tournaments just beginning, it's the perfect time to consider strategies to fill out an NCAA tournament bracket. How can ...
1
vote
0answers
81 views

Describing the sequence A224239.

I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, ...
1
vote
1answer
626 views

Cyclic tower of hanoi problem [duplicate]

If I have 3 rods in a circle and it is allowed to move the disks only in the clockwise direction. How many moves is necessary to move n disks from first rod to the third rod?
1
vote
1answer
67 views

$[x-\frac{1}{n}, (n-1)x+\frac{1}{n}]$ contains an integer $\forall x\in \mathbb{R}$ and $\forall n\in \mathbb{N}$

For any real number x: Prove that among the numbers x,2x,...,(n-1)x ,there is one that differs from an integer by at most $\frac{1}{n}$. any hints for a pigeon solution. Non-pigeon solution ...
1
vote
1answer
119 views

shorter proof of generalized mediant inequality?

Show $\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\frac{a_{i}}{b_{i}}$, where $b_{i}>0$. Attempt Assume the largest is $\frac{a_{n}}{b_{n}}\Rightarrow$ ...
4
votes
2answers
183 views

Evaluate $\sum_0^\infty \frac{1}{n^n}$

Courtesy of this xkcd comic I now know that $$ \sum_{n=1}^\infty \frac{1}{n^n} \approx \ln^e(3) $$ Echoing the views of the comic itself, if I ever find myself taking $\ln^e(x)$ then something has ...
2
votes
1answer
150 views

Guests at a table

Fifteen chairs are evenly placed around a circular table. On the table are the name cards of fifteen guests. After the guests sit down, it turns out that none of them is sitting in front of his own ...
2
votes
1answer
84 views

Ping Pong players

A and B play ping pong game multiple times. The person serving first has a probability p of winning that game. A serves the first game and thereafter the loser serves first. If p(n) = pbt that A ...
10
votes
2answers
219 views

Prove $\sum_{n=1}^\infty(e-\sum_{k=0}^n\frac1{k!})=1$

This comes from the comments section of this question here, credits Lucian. The statement is $$\sum_{n=1}^\infty\left(e-\sum_{k=0}^n\frac1{k!}\right)=1$$ This looks really interesting, so I was ...
2
votes
1answer
257 views

Sequences, sets and element position in the set.

I have a sequence Q with the length of N. This is the fragment of this sequence: 68 70 72 74 76 78 80 The sequence has been divided into the sets of 4 elements ...
-4
votes
3answers
871 views

Fun math riddle

In his will , a farmer left 17 horses to his 3 sons with the following instructions. 1) The eldest son is to get half of the total horses. 2) The middle son is to get one third of the total horses. ...
1
vote
4answers
239 views

What is the most awe-inspiring math equation you have come across [closed]

What is the favorite equation of your life? I know this might be a subjective question, and may be not-so-on-topic here, so if anyone decides to close this, could you link me somewhere I can ask this? ...
10
votes
2answers
941 views

Is it possible to shuffle a 3x3 Rubik's cube so that there's no more than 2 pieces of the same color in every face?

I'm not sure if this question belongs here but I see lots of Rubik Cube's questions around so here it goes: Can I take a standard 3x3 Rubik's Cube and shuffle it so that, for every face, there are no ...
4
votes
1answer
152 views

General approach to puzzles such as the “6 books puzzle”

Six different books (A,B,C,D,E,F) of identical size are stacked as in the figure. We know A and D are not touching. E is between two books which are both vertical or both horizontal. C touches ...
8
votes
2answers
176 views

Mathematicly Untangeling Untangle.

I have a new addiction, I play Untangle to often, and i am wondering what is the mathematics behind it. some free games: (but be warned highly addictive) Javascript: ...
-1
votes
2answers
146 views

Proving $1 + 1 = 2$ [duplicate]

How do you break down the theory of $1 + 1 = 2$? How do you provide a proof, please be precise. This is for one of my discrete math courses and I don't know how this is relevant to the course. And ...
0
votes
1answer
143 views

Proving using squeeze principle

This problem sounds very confusing. Please help me solve this problem.
1
vote
1answer
59 views

Prove the derivative

Let $f(x) = (x^2-1)^{\frac{1}{2}}, x>1$. How do I prove that the $n$th derivative of $f(x) > 0$ for odd $n$, and the $n$th derivative of $f(x) < 0$ for even $n$?
1
vote
0answers
40 views

Convex quadrilateral

In a convex quadrilateral (the two diagonals are interior to the quadrilateral) prove that the sum lengths of the diagonals is less than the perimeter but great than one-half the perimeter.
0
votes
2answers
65 views

A question of divisibility.

Let $p$ and $q$ are relatively prime integers. Consider $S = \{\frac{p}{q}\} + \{\frac{2p}{q}\} + \{\frac{(q-1)p}{q}\}$, where $\{x\}$ is the fractional part of the real number $x$. Prove that $2S$ ...
0
votes
1answer
117 views

25 coins are arranged in a 5 by 5 array.

25 coins are arranged in a 5 by 5 array. A fly lands on one and tries to hop on to every coin exactly once, at each stage moving only to an adjacent coin in the same row or column. Is this possible?
0
votes
2answers
528 views

How much the shopkeeper loses? [duplicate]

I struck with this tricky math question A girl went to a shop and bought a Rs.$200$ show piece. She gave a Rs.$1000$ note to shopkeeper. Shopkeeper didn't have any change so he went outside and ...
6
votes
4answers
182 views

A puzzle that came when I am half awake

When I am about to wake up in the morning, a puzzle crept into my mind.It is when $\sqrt{a}$ and $\sqrt{b}$ are both non-integers where $a,b$ are positive integers is it possible for $\sqrt{ab}$ to ...
1
vote
2answers
104 views

Ball bouncing in a box, will it meet a vertex.

I have no idea upon how to solve this: A box 5cm by 3cm with a ball projected from a vertex at 45 degree angle, it reflexes at a 45 degree angle and keeps reflecting at a 45 degree angle. Will it ...
0
votes
3answers
482 views

How to find the planar embedding of a graph in general?

I need to find the planar embedding of a graph in general if one exists and specifically want to solve the problem for the graph in the figure below. I am acquainted with the graph algorithms but have ...
1
vote
1answer
80 views

matrix row/col mapping

Many square matrices are symmetric. i.e. $a_{i,j}=a_{j,i}$ For such matrices, we can only store the upper triangle elements, i.e. any $a_{i,j}$ for which $i<=j$. Assume a 10x10 matrix. Using this ...
5
votes
1answer
89 views

Hockey Classics at Matheletics '13

I'm trying to solve a challenge from Matheletics '13: Micheal Nobbs is organizing a training camp for identifying new talents in Indian Hockey. The camp witnessed a total of ($3K+1$) players. Each of ...
4
votes
1answer
2k views

BINGO Probability: Controlling average game duration

I wandered over here from StackOverflow and my understanding of advanced mathematics is limited, so bear with me... A standard, BINGO game card has 24 numbers arranged in a 5x5 format. The center of ...
10
votes
5answers
371 views

What went wrong? [One-dimensional-inverse-square-law]

Intrigued by this question, one-dimensional inverse square laws, I started to try to find an answer and came up with what follows. However, I calculated the derivatives to double check myself, and ...
6
votes
1answer
222 views

Integer coefficient polynomial - values as powers of 2

Does there exist a polynomial f with integer coefficients such that $f(0) , f(1) ... f(n) $ are all distinct powers of 2 ? I have no clue about how should i start thinking about this problem but ...
2
votes
1answer
78 views

Functions which satisfy $\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w)$

Let $\mathrm{f}$ be a complex-valued function with the following property: $$\mathrm{f}(wz) =w\,\mathrm{f}(z)+z\,\mathrm{f}(w) $$ for all $w,z \in \mathbb C$. Necessary conditions are that ...
88
votes
12answers
10k views

Logic puzzle: Which octopus is telling the truth?

King Octopus has servants with six, seven, or eight legs. The servants with seven legs always lie, but the servants with either six or eight legs always tell the truth. One day, four servants met. ...
0
votes
3answers
10k views

How do you find the altitude in a pyramid? (SAT math question)

The pyramid shown above has altitude h and a square base of side m. The four edges that meet at V, the vertex of the pyramid, each have length e. If e = m, what is the value of h in terms of m? A) ...