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

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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
33 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
101 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 ...
-1
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 ...
1
vote
0answers
42 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 $x$...
-2
votes
1answer
37 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 $$1^{1000}+2^{1000}+3^{1000}+\ldots+(10^{1000})^{1000}?...
4
votes
3answers
105 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
43 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
53 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
166 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
99 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
64 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
44 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}, \underbrace{\frac{\...
0
votes
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 ...
10
votes
1answer
155 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
26 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. ...
1
vote
1answer
28 views

Bingo-like Game

In one board game, each player has a unique 4 x 4 grid with squares randomly labeled with each integer from 1 to 16. As the integers 1 to 16 are randomly called, each player puts an "X" in the square ...
3
votes
0answers
108 views

Polyhedra with identical faces

The isohedra have identical faces. They have symmetries acting transitively on their faces -- any face can be mapped to any other face to give the same figure. There are also polyhedra where all ...
1
vote
2answers
50 views

Mapping two integers to one: deriving formula

I have an interesting puzzle: Given two non-negative integers, let's call them $x$ and $y$, work out a formula for $z$ as shown in the table below: ...
0
votes
0answers
15 views

Limiting points

For a system of coaxial circles why are there only 2 limiting points? Shouldn't there be infinite limiting points? After all system of coaxial circles are pairs of circles which have same radical axis,...
1
vote
1answer
72 views

Expected Value for Number of Consecutive Cards of the Same Suit

Here is the setup. Shuffle a deck of 52 cards so their order is random (i.e., determined by a uniformly distributed random variable). Now flip through the cards and find the maximum number of ...
0
votes
1answer
30 views

Using the Fibonacci sequence and deduction to prove… [duplicate]

Using the Fibonacci sequence and induction prove that $$F_{n-1}F_{n+1}-F_{n}^2 = (-1)^n, \space \space n=1,2,3...$$ My efforts so far: The basis holds for $n=1$ Induction step: $$F_{n-1}F_{n+1}-...
5
votes
0answers
95 views

Magic square 9, Amazons, and the 2-(81,9,1) design

Consider the following order-7 magic square. The rows, columns, and diagonals all add up to the same sum: 175. Also, all the broken diagonals add up to the same sum, making this a pandiagonal ...
8
votes
0answers
126 views

Limit approximation for $\pi$ in the four fours puzzle?

The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
3
votes
1answer
126 views

Golf Problem Math [closed]

Hey guys i cant seem to draw the diagram for this. I dont understand this question at all. I got this triangle but i dont know how to solve it. I only have 2 sides on it and i cant use the sin/cos law ...
0
votes
1answer
34 views

Trignometry Building Problem

Ok guys this is one of the trig recrational problems i was doing and i cant seem to draw the problem right... Please help.. A surveryor standing 69 meters from the base of the bulding measures the ...
2
votes
2answers
60 views

Does probability depend on knowledge?

There is at least $2/3$ probability that this question is rather silly, but being an almost absolute beginner in Probability, I will ask it anyway. Consider the following problem, proposed at AIME ...
0
votes
1answer
78 views

Connection between 7 and 13 [closed]

While there have been many numbers that have been deemed 'lucky' or 'unlucky', 7 and 13 are two of the most prominently known. So, this had led me to wonder if there were any connections between 7 ...
1
vote
1answer
79 views

How many 10-letter words do not contain all the vowels

I can't find where I am overcounting in the problem How many 10-letter words do not contain all the vowels. What I do is to count all the words that have at ...
2
votes
4answers
106 views

Suppose a function is expressed by: $f(x)=f(x+1) - f(x-1)$ and $f(16)=20 , f(20)=16$ What is $f(20162016)$?

Math quiz bee question Suppose a function is expressed by: $$f(x)=f(x+1) + f(x-1)$$ and $$f(16)=20 , f(20)=16.$$ What is $f(20162016)$? Attempt at solution: $f(17)+f(15)=20$ $f(21)+f(...
0
votes
0answers
39 views

Does this resilience/resource scheduling analogy make sense?

A firend has recently presented an analogy for the rescheduling of Doctors (big topic in the UK atm) across a 7 day week as opposed to a 5 day week with a skelton staff at weekends - I'm ignoring A&...
7
votes
2answers
199 views

Chinese New Year Equation 2016

In the spirit of Chinese New Year, here's a problem to commemorate the year. $\color{black}{\text{Solve the following equation for positive integers $a$ and $b$:}}$ $$\color{red}{a^2+b^2+(a+8)^...
0
votes
0answers
26 views

Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors function?

(Note: This post is a bit related to this earlier MSE question.) The title says it all. Is $\sigma(2^r)$ a palindrome (in base $10$) for some $r > 2$, where $\sigma$ is the sum-of-divisors ...
25
votes
1answer
358 views

Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which ...
0
votes
1answer
13 views

Write bounds for a quantity in terms of bounds on another quantity

I have an expression that gives me the bounds for a certain variable X in terms of another variable Y. How can I write these ...
1
vote
1answer
59 views

Taking away pocket money [closed]

This is my fist post on Mathematics, so I hope my question isn't too trivial - you won't need a degree to answer this one! - but it's messing with my head.... Allow me to explain..... I'm helping my ...
5
votes
1answer
52 views

Factor the matrix (scalar $\times A$) into permutations of $A$

Here's an example of $A . B = scalar \times C$, done with magic squares. The last square does not have a consecutive range of digits. Drop the magic square requirement. In $2\times2$ matrices we ...
1
vote
1answer
81 views

Puzzle. Transfer maximal coal using a train [duplicate]

We need to transfer coal from point $A$ to point $B$ using a train. There is $9000$ tonne at point $A$. The distance between point $A$ and $B$ is $3000$ km. Train can carry only $3000$ tonne included ...
1
vote
2answers
43 views

Area of piece of paper folded around straight line of orientation $\theta$

Imagine drawing a straight line $l$ through the center of a square piece of paper with area $1$. Now fold the paper along that line. Q: What is the function for the area covered by the folded piece ...
1
vote
4answers
86 views

Finding the height of a Building at Night

EDIT: Method $1$ is false, as pointed out by Hetebrij. If it is night, how would one find the height of the building? By assuming I am trying to find the height of a building at night, I am ...
0
votes
0answers
23 views

Property of all k-dimesional shapes

A close friend showed me an intuitive pattern that given a k-dimensional convex polytope P, if one doubles every sidelength to generate a $P'$ it is possible to fit $2^k$ copies of $P$ within the ...
-3
votes
1answer
62 views

Counting and Abstract Problem Solving [closed]

Suppose that you have a bucket holds fiv-sev c, and one holds tw-one c. How could you use them to measure out thre c of water?