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

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23
votes
2answers
468 views

The Plank Problem 2 Dimensions

We were trying to solve this wonderful problem, but have not succeeded to solve. It goes like this: Let $R=[0,1]^2$, and $D\subseteq R$ be a convex set which intersects each side of $R$. Define a ...
2
votes
2answers
133 views

How can I get a good estimation of the following function

The function is $$ f(n) = \sum_{i=1}^{n} \frac{1}{2i-1}$$ How can I compute for example $f(20)$ or $f(50)$ without using a calculator. I want to have an approximation
3
votes
1answer
87 views

Interesting sequence question

I saw a puzzle the other day and it was as follows: Find the next number in the sequence: 1 11 21 1211 111221 312211 ... If anyone wants to have a go at the ...
1
vote
0answers
79 views

Finding the 'best' way in a card-arrangement-game

Let $n\ge 2\in\mathbb N$. Suppose that we have a card on which $1$ is written, a card on which $2$ is written, $\cdots$ , and a card on which $n$ is written. Now these $n$ cards are arranged from left ...
1
vote
1answer
184 views

Two Genius Mathematicians

This is actually a question I find really hard to answer.any hints are appreciated. By the way feel free to edit the tags as i really do not know which category is this question is in. Two genius ...
4
votes
2answers
414 views

expected value of a game with a n sided die

Suppose we have a n-sided die. When we roll it, we can be paid the outcome or we can choose to re-roll by paying $1/n$. What is the best strategy and what is the expected value of this game? As an ...
5
votes
2answers
266 views

Rooks in 3D chess board

How many rooks are needed for a 3D chess board of size NxNxN so that every empty cube on the board can be reached by a rook in a single move?
1
vote
2answers
1k views

hitting a dart board probability

You have a dart board which is split in half. If you hit the left half, you get $2$ points, if you hit the right half, you get $3$ points. You have an 80% chance of hitting the dart board on any ...
0
votes
1answer
63 views

A Bug Crawls Along a Square

A wire of length $4$ is bent into a square. At time $t = 0$, a bug starts crawling from the corner of the square to an adjacent corner, and continues traveling along the rest of the square until it ...
1
vote
1answer
300 views

find the sales tax from a tax included price that does NOT apply tax to the portion of the total that IS tax

I, a vendor, need to find the sales tax from a tax included price that does NOT apply tax to the portion of the total that IS tax. Most answers result in overpayment of taxes. Please do not tell me ...
2
votes
1answer
165 views

Homework Question for a 15 year old

My younger brother(age: 14 years 7 months) and his classmates were given a set of eight questions by his class-teacher, which included the following two questions: (i) Find, if you can, the fallacy ...
0
votes
1answer
42 views

How to Find the Remaining Length of a Cone With Only a Part of It

I took three measurements for a certain plastic cup in my kitchen. One was of the circle on the bottom of the cup, and the other was the top(the larger opening) and the height in between the two. ...
13
votes
4answers
1k views

expected value of a sum of a 10 sided die

Suppose you have a fair die with 10 sides with numbers from 1 to 10. You roll the die and take the sum until the sum is greater than 100. What is the expected value of this sum?
1
vote
0answers
56 views

Prove that eventually Hannah and the other swimmer will settle into a pattern where they pass each other (Please refer to the context in my question)

From the 2014 Mathcamp quiz: Hannah is about to get into a swimming pool in which every lane already has one swimmer in it. Hannah wants to choose a lane in which she would have to encounter the other ...
1
vote
1answer
57 views

Graph-like problem

Each shop in a town has an odd number of customers and each pair of shops shares an even number of customers. Prove that there are at least as many customers as there are shops. Any hints are ...
1
vote
0answers
59 views

Non-symmetric polynomials, game

This is a game I thought was easy but appears to be too hard for me... I'm trying to find a polynomial in x,y,z (they commute) such that permutations of the variables only give rise to 2 different ...
1
vote
2answers
38 views

Efficient random guessing

Let $(x,y)$ be a random point on the plane with some unknown continuous distribution. Your opponent randomly chooses one of the coordinates and tells you. You shall guess whether this coordinate is ...
1
vote
1answer
53 views

counting of numbers

In a garden there are three kind of roses-red, yellow and white. No matter which 9 roses are selected at least 2 of them are white; and no matter which 10 roses are selected at least 2 of them are ...
1
vote
1answer
51 views

Characterization of nowhere differentiable functions

Let $N:=\{f\in C([0,1])\vert \text{ f is nowhere differentiable } \}$ and $A_n = \{f\in C([0,1]) \vert \exists x\in [0,1]s.t. \forall y\in[0,1]: |f(x)-f(y)|\leq n |x-y|\}$. Now I have already ...
4
votes
3answers
1k views

Unbounded sequence with convergent subsequence

I'm just wondering if anyone knows any nice sequences that are unbounded themselves, but have one or more convergent sub-sequences?
0
votes
1answer
65 views

How often does $p^k$ divide the Fibonacci numbers?

I would like to know about the Fibonacci numbers $F_n = 1,1,2,3,5,8, \dots$ in $\mathbb{Z}/p^k\mathbb{Z}$. $$ \mathbb{P}[p^k \text{ divides } F_n ] = \frac{\#\{1 \leq n\leq N: F_n \equiv 0 \mod ...
5
votes
1answer
307 views

Geometrical question just for fun

Was puzzling with the following (home made) puzzle: Given the square $ABCD$ with $A = (1,1)$, $B = (1,-1)$, $C = (-1,-1)$ and $D = (-1,1)$ And given point $E = (0,2)$ What is the smallest (by area)...
1
vote
2answers
96 views

Arrangement of Numbers to Get a Common Sum

I'm having trouble with a math problem. I need to arrange 6 numbers on a certain diagram: At every intersection of two circles, I have to put one of these six numbers: 4, 5, 5, 6, 6, or 7. The sum ...
6
votes
6answers
508 views

Do the differences of perfect squares apply to perfect cubes and more?

I'm curious about a special property of squares. The difference between perfect squares starting from 0 are 1,3,5,7,9..., where each difference goes up by 2. I want to know if there are any patterns ...
4
votes
2answers
101 views

Mandelbrot Set Equation

Is there a graphable equation that graphs the Mandelbrot Set? It seems like an interesting design, but I want to find a way to show all of the details via a graphing calculator.
2
votes
2answers
430 views

Puzzle with pirates

That one I'm pretty low on ideas of how to approach it. Five pirates of different ages have a treasure of 50 gold coins. On their ship, they decide to split the coins using this scheme: The oldest ...
3
votes
1answer
171 views

Marriage puzzle

I need some insight for the following problem: There are 100 girls and 100 guys. The girls each have their individual preference list for which guy they’d like to marry. Each list contains all 100 ...
1
vote
2answers
77 views

Continue the sequence

I'm stack at that question as I'm not sure which law does the sequence obey What number comes next in this series: 1 11 21 1211 111221 312211 13112221: 12113331 1113213211 13221113 ...
3
votes
1answer
148 views

Paint a cube with 6 colors

I have a unit cube and $6$ colors to paint its sides in. How many different cubes to I get if I use one color per side? I think I know the answer, but just want to be sure regarding my solution. Let ...
5
votes
4answers
761 views

How to solve a puzzle with numerals

Any insights are welcome for this puzzle. The following equation is wrong: $103 - 102 = 3$. Move one numeral to make it correct. The numeral moved is: $0,1,2$ or $3$?
1
vote
1answer
153 views

Question About the Solutions to the Eight Queens Problem [closed]

How is $a_{15}n_8e_9k_5f_{10}d_7b_4m_6$ a solution to the Eight Queens problem? J. W. L. Glaisher, On the Problem of the Eight Queens, Philosophical Magazine, 1847 says that each one of these terms (...
4
votes
0answers
86 views

Derangements and the “other” secretary problem

I just found out that the name "Secretary problem" is given to two different problems. The first one talks about a secretary who mixes letters and envelopes, and ask for the probability that no letter ...
1
vote
4answers
11k views

Probability that a leap year has 52 Sundays

For the Question "Find the probability that a leap year has 53 Sundays". The Solution goes : For 53 Sundays, we proceed as: $\frac{366}{7} = 52.28$; So we can be sure that there are 52 Sundays, ...
1
vote
3answers
505 views

Find Differences between Ages of A and B.

Question: A says to B, I am twice as old as you were, when I was as old as you are. If the sum of ages is 63 years. Find the difference between their ages. My Question: I understand that we need to ...
0
votes
2answers
152 views

Liar - Truth-Sayer - Tourist Problem. Construct the answer with the given 2 sub-statements.

A tourist A comes to a country where people are divided into two categories: Liars (L) and Truth Sayers (T). Ls always lie and Ts always speak the truth. Intending to walk to the capital, the tourist ...
3
votes
2answers
368 views

2 Trains and Fly Problem. Find the number of trips made by the fly back and forth.

Question: A Train A is approaching at a speed of 10m/sec, another Train B moving in the opposite direction at a speed of 20m/sec. A fly whose absolute speed is 50m/sec goes repeatedly from A to B and ...
0
votes
1answer
470 views

What is the subset of the letters in the word 'numbers'

What is the subset of the letters in 'numbers'? I thought it meant to make a word out of the letters in 'numbers' so I tried to make a word out of it, but I couldn't make a word and so I don't ...
0
votes
1answer
436 views

The famous Portia's casket problem

Gold casket: the portrait isn't in the silver casket. Silver: the portrait isn't in this casket. Lead: the portrait is in this casket. At least one of the statements was true and at least one of them ...
9
votes
2answers
644 views

Euler's identity in matrix form

I assume everyone is familiar with the famous mathematical identity due to L. Euler: $$ e^{i \, \pi} + 1 = 0,$$ where $i^2 = -1$ and $e$ is the base of natural logarithms. I was wondering if this ...
1
vote
1answer
3k views

Solve Spherical Shell for Radius given Thickness and Volume

I'm looking to calculate the outer radius of a spherical shell of a desired volume and thickness. I don't know if the years have knocked some obvious obstacle out of my perception, but here's what i'...
1
vote
1answer
51 views

Find a polynomial mod $n$ injective on a given set

This question is inspired by this challenge on CodeGolf.SE, in which the goal is to create a hash function with specified collisions. I thought a polynomial over the integers mod $n$ might be a nice ...
0
votes
1answer
96 views

Repeating cycles in the $3n-1$ problem

While tracking sequences beginning with 1-to-3 digit integers, I have found 3 different repeating cycles in the $3n-1$ problem (similar to the Collatz Conjecture). They are 1, 2, 1..., 5, 14, 7, 20, ...
1
vote
0answers
59 views

Tools (electronic notebook and ontology viewer) of mathematical formulas (definitions and facts)

I am reading math books and articles (for applications in other disciplines, mostly about logics for computer science and AI) and the hardest part is to memorize formulas, to look up some definitions/...
18
votes
10answers
785 views

hand evaluate $\sqrt{e}$

I have seen this question many times as a example of provoking creativity. I wonder how many ways are there to evaluate $\sqrt{e}$ as accurately as possible. The obvious way I can think of is to use ...
1
vote
1answer
81 views

Finding the minimum value of a 6th degree polynomial algebraically

Is it possible to answer this question using methods of basic algebra? Find the least value of the expression $a^6 + a^4 - a^3 - a + 1$ for real value of $a$. This question is from the 2013 Philippine ...
1
vote
2answers
363 views

Is it possible to reach 50k in the game 2048 with only having a highest tile of 512?

Is it possible to reach a score of 50,000 in the game 2048 with only having a highest tile of 512(Only one 512 was present) and one 256 tile and other small numbered tiles such as 2,4,8,16,...,64. ...
2
votes
1answer
189 views

Getting multiple of 11

Take any $2$ digit number having different digits. Now add the bigger digit in that number. By continuing this process, you will get a multiple of $11$ i.e. both of the digits will be equal of a ...
6
votes
4answers
4k views

What does “twice as likely” mean?

Once in a while I hear people say something like X is twice as likely as Y. What they usually mean is: $$p(X) = 2 \cdot p(Y)$$ and - in the context they refer to - they usually have $p(Y) < ...
8
votes
2answers
164 views

What is the integral $\int x^t/\Gamma(1+t) \, dt$? (In general: relation between series and integrals)

(The question arises from playing with translating series into integrals) I wanted to see, what it means to have a "continuous" relative for powerseries and other series; the most simple one ...
0
votes
2answers
513 views

Google Code Jam's Cookie Clicker Program…

Today, the Google Code Jam's cookie clicker problem was something like this. Problem In this problem, you start with 0 cookies. You gain cookies at a rate of 2 cookies per second, by ...