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

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1
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1answer
60 views

a*b = a/b = b/a (what's this symmetry called?)

I was playing around with numbers the other day, and I found an interesting symmetry, that I would like to know if it has any specific name assigned to it. Let's assume the notation n:a to refer to ...
10
votes
3answers
101 views

Mental $n-$th root of $N$

It has been a while since I started thinking about this problem: a fast method to evaluate (in an approximate way) mentally the $n-$th root of a number $N$. I'm talking about great numbers, because ...
3
votes
0answers
41 views

Chessboard four-colour theorem

Divide the infinite chess-board into countries, where squares in the same country are connected by edges. Suppose two countries are adjacent if they touch either along an edge or at a corner. What is ...
3
votes
1answer
73 views

The angle between two clock pointers

Fun with math time The other day a friend of mine asked me for this: What is the value of the angle between two clock pointer when it's 11:50? Of course the correct answer is not $60$ degrees, ...
0
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2answers
122 views

Which is bigger $e^{(a+b)}$ vs $e^a + e^b$?

I understand that exponential function is a convex function so for any convex function $\theta(a+b) > \theta(a) + \theta(b)$, but can someone provide a more formal proofs ?
0
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1answer
47 views

How to calculate the payoff in Battleship Game Theory

Consider a 3 by 3 board and suppose that Player I hides a destroyer(length 2 squares) vertically or horizontally on this board. Then Player II shoots by calling out squares of the board, one at a ...
0
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0answers
27 views

Weighted Money Distribution Problem

A quantity of money must be shared between $n$ parties. Each party is assigned a weight, which determines the proportion of money they get, to a "close enough" number of pennies/cents. How would the ...
10
votes
2answers
98 views

Is it smarter to defend with one army or with two in the game Risk?

I've recently played Risk with the following rules: One player can attack another player with at most 3 armies. The defender can defend with at most 2 armies. This is independent of the number of ...
0
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2answers
51 views

Prove that there are only finite numbers satisfying this equation:

Coincidentally, I realized that my room number $(23)$ has the following property: $$2^3+1=3^2$$ In order to find more numbers $n$ exhibiting this property, I wrote the following equation: ...
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2answers
89 views

If $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$

Prove that if $\det(A+B)=\det(A+2B)=\det(A+3B)=1$ and $AB=BA$ then $B^2=0$. A problem from a math competition. $A$, $B$ are 2 by 2 complex matrices. I've tried using Cayley Hamilton theorem, on ...
6
votes
1answer
83 views

$A^2$ $B=A^2-B$ then $AB=BA$

If for $2$ real $n$ by $n$ matrices we have $A^2B=A^2-B$ then prove that the two matrices commute. This is a problem from a competition. I've tried several manipulations but none of them work. ...
0
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2answers
157 views

Guess the number. Maximizing expected winnings? [closed]

A man in a trench coat approaches you and pulls an envelope from his pocket. He tells you that it contains a sum of money in bills, anywhere from 1 dollar up to 1,000 dollars. He says that if you can ...
2
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1answer
43 views

The Functional Equation $f(mn)=f(m)f(n)$ where $f:\mathbb{N}\rightarrow \mathbb{R}$, $f(2)=2$, and $f(m) > fn)$ if $m>n$.

The following is exercise 3.3 from Terence Tao's "Solving Mathematical Problems." Emphasis added. Suppose $f$ is a function on the positive integers which takes real values with the following ...
5
votes
1answer
84 views

Arrangement of houses with 2 colors

From the 2016 International Mathematical and Logic Games Contest Along the coast of Maths-land, the straight beach-front road contains a line of houses, all on the same side of the road. The ...
0
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0answers
34 views

$n$-couples of people in a row.

For the following problem, I feel my reasoning is something wrong, so I would like if it is in the right direction or if it needs to be rephrased/corrected. The problem reads: How many ways are ...
2
votes
1answer
88 views

Maths problem: Cedric's age

We are in the year $2016$, and Cedric's age is a factor of $2016$. If Cedric adds up all the multiples of his age that are less than $365$, he arrives at the year he was born. In which ...
0
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0answers
37 views

What is the name of this type of palindromic number?

A palindromic number is one that reads the same forwards as backwards,an example of which is $17071$. In the UK yesterday's date was $16/3/16$, the convention being day/month/year. Clearly $16316$ is ...
4
votes
3answers
104 views

How to calculate $10^{0.4}$ without using calculator

How to calculate $10^{0.4}$ without using calculator or if not what is the closest answer you can get just using pen and paper within say $2$ min?
2
votes
2answers
42 views

How do I find a minimal set of US states so that every state boarders one of them?

I recently played this game, in which one has to type in the names of US states. For each state one gives, all of the states which border it are removed from a list. The player continues to provide ...
3
votes
1answer
71 views

Can anybody help me with math expressions?

So , I am in $7^{th}$ grade and my teacher gave me some really hard homework. What I have to do is use math expressions that equals each number between $1$ and $100$ , only using the numbers ...
3
votes
0answers
65 views

2D walks on a square grid; The number of Paths leading to specific $(X,Y)$

Introduction Lets have a 2D plane, and place a Walker in the center $(X,Y)=(0,0)$ Lets take a example where we use all of the possible moves; Walker can make one of the 9 moves each turn: Up, Down, ...
2
votes
1answer
30 views

Generalized knight going from one corner of a box to the other

Generalize the notion of a knight from chess to a $(x, y)$-knight such that $x \leq y$ and the knight can move any of the $8$ combinations defined by $(\pm x, \pm y)$ or $(\pm y, \pm x)$. Thus a chess ...
0
votes
1answer
38 views

One Dimensional Random Path Walker

Problem The Probabilities involving 3 equally possible moves in 1D line. Imagine a one-dimensional line with a "walker" in the middle position ($x=0$) Walker can make one of the following moves ...
7
votes
2answers
107 views

What is the optimal strategy when driving to my university: Wait or take alternative route and (possibly) wait?

When bicycling to my university, I'm faced with a difficult decision. The situation is shown here: I have to get to point marked with the $\times$, but in order to do so, I must cross a big road ...
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5answers
1k views

What is the sum of the reciprocal of all of the factors of a number?

Suppose I have some operation $f(n)$ that is given as $$f(n)=\sum_{k\ge1}\frac1{a_k}$$ Where $a_k$ is the $k$th factor of $n$. For example, ...
4
votes
0answers
75 views

Functions with “ugly” inverses

Inspired by this post: I was amazed to see that (at least according to wolframalpha) the inverse of such a nice and simple function as $f(x)=x^3+x$ is: $$ f^{-1}(x) = \sqrt[3]{\frac{2}{3( ...
0
votes
1answer
19 views

pick three drawing

Our veterans club has a daily pick 3. We currently have one cage with 30 balls numbered 0 thru 9. The first number is drawn and returned to the cage and the next two the same way. I believe this gives ...
2
votes
0answers
32 views

Toroidal Split Complete Graphs

The paper On the Planar Split Thickness of Graphs shows how non-planar graphs can be split to make planar graphs. For example, they offer a split $K_{6,10}$. I would instead like to make split ...
3
votes
1answer
92 views

(Green/blue)-eye logic puzzle. Statement validation

There is a logic puzzle aiming on freeing same-color-eyed people from an island. The thing is that they must be certain of their own eye color so that they can leave. For that reason an external party ...
1
vote
0answers
26 views

Filling a grid square with 0,1,2 [duplicate]

Each of the 25 cells in a five-by-five grid square is filled with a 0, 1, or 2 in such a way that the numbers written in neighboring cells differ from the number in that cell by 1. Two cells are ...
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votes
1answer
68 views

Can this be proved using single variable calculus or there's something wrong with this problem [closed]

If $y^{1/m} + y^{-1/m} = 2x$ then prove that $(x^2 - 1)y_{n+2} + (2n + 1)x y_{n+1} + (n^2 - m^2)y_n = 0$? Where $y_n$ denotes the $n^th$ derivative of $y$. This is a question of successive ...
2
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0answers
40 views

Absolute difference and probability [closed]

Fifty tickets numbered with consecutive integers are in a jar. Two are drawn at random and without replacement. What is the probability that the absolute difference between the two numbers is 10 or ...
2
votes
3answers
67 views

The growth rate of $(\ln(x))^n$ is a lot slower than I expected

Obviously, the growth rate of $(\ln(x))^a$ is less than the growth rate of $(\ln(x))^b$ as long as $a>b$. Also, the growth rate of $(\ln(x))^n$ is apparently always less than the growth rate of ...
-2
votes
1answer
35 views

heart rate problem [closed]

The average heart rate of a shrew is 800 beats per minute, while an elephant has a heart rate of 25 beats per minute. If 1 billion heartbeats is a natural life span for each animal, on average, how ...
5
votes
2answers
70 views

Last $m$ digits of a sum

What is an efficent way (not using any computer programs and such) to find last $m$ digits of some terrible looking sum, for example I don't know ...
4
votes
3answers
88 views

How many different paths from top to bottom spell ALGEBRA?

Starting with the A on top and only moving one letter at a time down to the left or down to the right, how many different paths from top to bottom spell ALGEBRA? ...
3
votes
3answers
42 views

How many tokens would person A have under these conditions?

Persons A and B each have a positive integer number of tokens, and the number of tokens B has is a square number less than 100. B says to A, "If you give me all of your tokens, my total number of ...
0
votes
2answers
54 views

Perimeter of Quadrilateral

The lengths of two sides of a quadrilateral are equal to 1 and 4. One of the diagonals has a lengths of 2 and divide the quadrilateral into two isosceles triangles. What is the perimeter of the ...
2
votes
1answer
50 views

How to arrange 1 to 15 such that the sum of any adjacent 3 numbers will be a perfect cube? [closed]

The numbers 1 to 15 should be arranged in a way that any 3 adjacent numbers' sum will be a perfect cube.
11
votes
1answer
163 views

On the theorem “$3$ is everywhere”

In this Numberphile video it is stated that "almost all natural numbers have the digit $3$ in their decimal representation", and a proof of this fact is proposed. A sketch of the proof follows: ...
1
vote
1answer
96 views

How many “$m$” digit numbers with digits that sum to “$N$”

How many "$m$" digit numbers can be formed whose digits sum to "$N$"? The collection of these numbers can have preceding zeros . The collection of these numbers cannot duplicate multiplicity of ...
2
votes
3answers
61 views

Formulize this sequence

There is this function defined as; $$f(x) = 10^x + 10^{x-1} + ...+10^0 $$ Which simply gives the 111.. kind of number, given the length x. What I need to do is a way to formulize this function, ...
2
votes
1answer
66 views

how to compare probability/ratios

For one location, I have: Number of lollipops selling at morning time Number of lollipops selling at afternoon time Selling periods: Every 30 minutes is a period, which sells lollies either ...
3
votes
3answers
40 views

First digits of a cube of a natural number

Can a cube of a number be of form: $2016a_1a_2a_3\dots a_n$? I have no direction, and would love to get a certain direction/proof. Thanks in advance
3
votes
2answers
51 views

I think I've found all roots to $f_k(x)=\sum_{j=1}^k x^j-x^{-j}$ for any $k$ - how to prove it?

Conjecture: The set of unique roots of $$f_k(x)=\sum_{j=1}^k x^j-x^{-j} \;,\;\; x \not=0$$ is given by $e^{i \pi \phi_k}$, where $$\frac{1}{2}\phi_k=\{0, \frac{1}{2}, ...
0
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0answers
29 views

Does this graph partitioning algorithm achieve anything interesting?

I was musing over graph clustering and partitioning, and isolating clusters, and came up with an algorithm that I think might do some interesting things. I figured I'd run it past here to get some ...
8
votes
0answers
128 views

Recreational problems in set theory?

Most areas of maths that I can think of have a number of fun, recreational problems that come under their category. Nothing deep: number theoretic stuff in olympiads, integrals, limits, products, ...
1
vote
4answers
41 views

Assumption and simple calculation

I'm having an issue with what seems to be an simple question. Here it is: Two hockey teams, team A and team B played a game, Team A beat Team B by 2 goals. The crowd was pleased as there were 8 ...
0
votes
1answer
36 views

Empirical Formula for Financial problem

I have a financial problem, which is strictly related to math of course. The problem states that on the last year the steel market price was about $450$ \$, and a company, that sells steel, used to ...
0
votes
1answer
25 views

Method to study obvious properties

Most of the time studying mathematics we come across various properties like associative, commutative,...etc. These properties are obvious and sometimes I feel why at all they are given in the text. ...