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

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11
votes
2answers
150 views

Maximum value of the smallest number of operations to obtain configuration from original configuration

Let $n$ be a positive integer. There are $n(n+1)/2$ marks, each with a black side and a white side, arranged into an equilateral triangle, with the biggest row containing $n$ marks. Initially, each ...
25
votes
3answers
762 views

Sudoku with special properties

Sudoku is a puzzle, with the objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid (also "sudoku-blocks") contains all of ...
5
votes
1answer
34 views

On the GCD of two palindromes.

I had an observation. Which I will discuss below. My question will be Is my observation correct? If so, how can one prove it? Observation: Consider the string of palindromes below: $100...01$ and $...
2
votes
0answers
42 views

What is the average maximum value of a set of random numbers? [duplicate]

Let $a_1, a_2, a_3, \ldots, a_{10}$ be ten randomly chosen real numbers in the interval [0,1]. Let $m$ be the maximal value out of these 10 numbers. What is the expected value of $m$? (i.e. If i ...
4
votes
0answers
36 views

de Bruijn sequence and sequence waiting time

this is quite a vague question, more of a puzzle than a question. I've spotted that two problems concerning word combinatorics have the same answer. I feel like there should be a connection, but I ...
9
votes
3answers
2k views

Represent $1729$ using four fours only.

I keep trying Four fours puzzle for various numbers, i.e. express a number using four fours and only four fours along with any mathematical operation. Today, I was thinking for Ramanujan number, i.e. ...
6
votes
1answer
50 views

Max size of $B \subset \{1, 2, \ldots, 3n+1\}$ for which no distinct $x, y, z \in B$ have sum in $B$

Given a set $$A=\{1,2,3,\ldots,3n,3n+1\},(n\in N^*)$$ Let $B$ be a subset of $A$, such that for any distinct $x, y, z\in B$, we have $x+y+z\not \in B$. Find the maximum number of elements $B$ ...
1
vote
1answer
23 views

Calculating the probability that the following random subsets of $\mathbb{R}^2$ are open.

For each $t \in \mathbb{R}$, select an open interval $U_t \subseteq \mathbb{R}$ containing 0. What is the probability that the set $U = \{ (t,x) | x \in U_t\}$ will be an open subset of $\mathbb{R}^2$?...
6
votes
0answers
82 views

Progressive Dice Game

You have a fair, regular 6-sided dice. The game is played for $n$ turns. Each turn you make a roll and gain that many points the rolled side is showing, then do one of the following: ...
6
votes
2answers
93 views

Mario Party 3 Mini-game Probability Question

I have a question about a mini-game in Mario Party 3. I have extracted the mathematical information from the game below. Setup: Four players $A,B,C$, and $D$ line up in some order. There are $12$ ...
-1
votes
1answer
31 views

Arc Length and Area of a Sector

A cake has a circumference of $30 \mathrm{cm}$ and a uniform height of $7\mathrm{cm}$. A slice is to be cut from the cake with two straight cuts meeting at the centre. If the slice is to contain $50\...
20
votes
5answers
536 views

Is $11^2+12^2+13^2+14^2+15^2+16^2=1111$ special?

Is this pure coincidence or is this a special case of some well-known number-theoretic result? If the latter is true, is there some notable generalization? EDIT: Thanks to the interesting answers ...
1
vote
1answer
45 views

Collecting Stickers! Evaluate occurrence of duplicates?

The Sticker Collector You need to collect $n>1$ different Stickers. Each day you get one pack with $1$ random sticker until you don't collect at least one of each kind. * Each sticker ...
6
votes
1answer
71 views

Definite integral problem of $\frac{x^n}{n!}$

I want to evaluate the following definite integral. $$\int_0^\infty\frac{x^n}{n!}dn$$ Where we have $n!=\Gamma(n+1)=\int_0^\infty t^ne^{-t}dt$ so that we can have $n\in\mathbb{R}$. I don't think ...
1
vote
0answers
30 views

Formula for (word) frequency count

I searched for a mathematical formula for the description of (word) frequency count. Its definition would be: A word frequency count is a measure of the number of times that a word w occurs in ...
6
votes
2answers
97 views

Monty Hall Problem extended

After seeing the popularity of the standard $3$ door problem, Monty thought to put a twist in the story. There are $N$ doors, $1$ car, $N-1$ goats. We need to choose any one of the doors. After we ...
0
votes
5answers
92 views

Can we prove that plumb line is vertical to ground?

Using a plumb line to make sure a wall is vertical for instance, is as far as I know one of the most primary tools in the sense that the very-very ancient builders used it as an instrument. I was ...
0
votes
0answers
47 views

Minimizing the effort after toilet visit

We live together with 5 people (4 men and 1 woman) and the woman wants everyone to close the toilet after every turn (i.e. bring the seat and cover down, for smell reasons). To me this seems unfair. ...
1
vote
0answers
55 views

Find all natural numbers of the form $2^n$ whose all digits are even

Find all natural numbers of the form $2^n$ whose all digits are even. For example: $2, 4, 8, 64, 2048$ (I believe they are the only such numbers). For $n \geq 11$, so far, I can prove that the last ...
3
votes
2answers
53 views

Factoring the factorials

Just for the fun of it, I've started factoring $n!$ into its prime divisors, and this is what I got for $2\leq n\leq20$: $$\begin{align} 2! &= 2^\color{red}{1} &S_e=1\\ 3! &= 2^\color{red}...
1
vote
2answers
88 views

Scheduling gym class

My cousin came to me with this problem yesterday: She has 8 students in her gym class. In tomorrows class she has planned 4 different activities to rotate them through, each of which requires ...
0
votes
1answer
29 views

What counts as a “Neighbor” in Conways' game of life?

I have looked everywhere but I cannot find an answer for this. Since I am bored, I am trying to create this game, but I can't seem to figure out what is considered a "Neighbor". Is it only directly ...
0
votes
0answers
49 views

An Interesting Variation to the “Pebbling a Checkerboard” Puzzle

Pebbling a Checkerboard (or chess board) was a puzzle proposed by Maxim Kontsevich in 1985, which was very interesting and fun to try, and you can find a great video on it at: https://www.youtube.com/...
4
votes
4answers
644 views

Curious inequality: $(1+a)(1+a+b)\geq\sqrt{27ab}$

I was recently trying to play with mean inequalities and Jensen inequality. My question is, if the following relation holds for any positive real numbers $a$ and $b$ $$(1+a)(1+a+b)\geq\sqrt{27ab}$$ ...
1
vote
0answers
37 views

How much information is missing?

If we know the value of $\frac{(a-b)}{(c-d)}$, can we calculate the value of $\frac{(a-d)}{(c-b)}$ That is : Let $\frac{(a-b)}{(c-d)}=k$ , can we calculate $\frac{(a-d)}{(c-b)}$ in terms of $k$ And if ...
0
votes
1answer
41 views

Does the first player have a winning strategy?

Two players play a game where they alternatively cross out a number from the numbers written on the board ($1-21$). They stop when two numbers are remaining. If thie sum of these two numbers is ...
0
votes
0answers
4 views

Sequence of nested sets in [0,1] with gaps at most $(1+\epsilon)/N$

What is the best possible $\epsilon$ and sequence $(a_n)_{n=1}^{\infty}\subset [0,1]$ you can find such that $$ d_{N}:=\sup_{x\in [0,1]}\inf_{n=1}^N |x-a_n|\leq \frac{1+\epsilon}{N} $$ for all $N\in ...
2
votes
0answers
43 views

Decimal Multiplication Without Multiplication

My friend has recently been challenging me to solve some maths problems, the latest challange is to find a method of finding the answer of $2.5 \cdot 2.5$ without ever using multiplication. Now with ...
1
vote
1answer
56 views

What is the area of the shaded region of the rectangle? [closed]

what will be the area of the shaded region of the following rectangle. Where 2m*2 and 3m*2 are the areas of the enclosed triangles. I'm trying to figure out any help. thanks
0
votes
3answers
52 views

Round to the nearest tenth position

I'm new to this forum, but I wanted to post this question hoping to know if anyone has come across it. Is there a formula in math to round down to the closest power of ten? For example, $n$ : the ...
5
votes
0answers
103 views

Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$, $(d=1.15096...)$ that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
-4
votes
3answers
59 views

What is the next number in this sequence of five? [closed]

I have the following sequence of numbers: $100, 121, 144, 202, 244 ...$ What will be the next number in the sequence?
1
vote
2answers
14 views

Can the state of a system after applying the operation “absolute value” be got back using elementary operations or transformations?

Take the operation or transformation "addition". You can get back the original state of the system by doing the opposite operation, i.e., "subtraction". But, if the operation is "absolute value", you ...
4
votes
0answers
35 views

Number of iterations of an “integer-logarithm”

Let us consider the function $\sigma:\mathbb{N}\to\mathbb{N}$ defined as: If $\prod_{i=1}^{r}p_i^{\alpha_i}$ is the prime-factorization of $n$, then $$ \sigma(n)=\sum_{i=1}^{r}\alpha_ip_i $$ So in ...
0
votes
4answers
60 views

BEDMAS where the order of Addition before Subtraction matters?

Here is a recent "tricky problem" that is making the rounds on FB: BEDMAS is explained here in a video, with this being the upshot: Everyone understands that ...
1
vote
0answers
30 views

Number of ways to get from a point to another one in the plane

I was trying to solve the following problem related to "counting cases": Consider the point $(0,0)$ in the plane and another point $(m,n)$ with $m,n>0$ integers. Suppose you want to get from the ...
7
votes
1answer
162 views

Any math competitions dedicated to calculations by hand (on a college level)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
10
votes
0answers
1k views

Integer Triangle Radicals conjecture

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...
1
vote
1answer
27 views

How many different arrangements of triangles that are either red or blue around a regular heptagon are possible?

I have the following problem I have an yellow heptagon (regular $7$ sided polygon) Against every side there is a triangle. The triangle is either red or blue. How many different arrangements of ...
4
votes
3answers
64 views

Time-and-Work and Motorcycle Tyres

A problem about motorcycle tyres, related to Time-and-Work or rate-of-work methods. This is not a homework question, nor, as far as I know, a contest question. It is intended as a challenge for Year ...
6
votes
0answers
156 views

Separating Heavier from the Lighter Balls

This was posted Here and received a good answer, solving the general questions in either $n$ or $n+1$ moves, which is by just half a move on average "less good" than my manual solutions here. ...
6
votes
4answers
146 views

Minimum number of marked squares on $n × n$ board

Came across this question: Consider an $n × n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two different squares on the board ...
3
votes
1answer
71 views

How to prove a regular pentagon is formed by knotting a rectangular strip of paper?

I found this interesting problem from a friend (From Arthur Engel's-Problem Solving Strategies book). The method to begin the problem is as follows- Step 1.Take a rectangular strip of paper ...
1
vote
0answers
37 views

Can we evaluate the alternating sum of the digits of an irrational number?

Suppose you had a summation $\sum(-1)^na_n$, where $a_n$ is the $n$th digit of $e$ and $a_0=2$. I know it diverges, but I want to know if its possible to evaluate anyways. Since it is alternating, ...
1
vote
0answers
47 views

Defining Logic Algebraically, Math Functions & Integers

Introduction I wanted to define some functions algebraically to be used as "logical conditions" that would be assigned to a term $t$ to "control" its value. Or in some other words, I wanted to ...
3
votes
0answers
84 views

Does Graham's number have an odd or an even number of digits?

I think it is hopeless to decide whether the number of digits of Graham's number is even or odd because the only way that I can think of is determining the logarithm with accuracy $0.1$ or even better,...
0
votes
1answer
19 views

Algebraic manipulation from $a^2+b^2 = abm$ with all variables in Z to a|b?

I know that $a^2+b^2=a\,b\,m$, with $a,b,m$ integers ($a,b$ positive integers) How can I show that a|b from this? I know it's true intuitively. I can recall the definition of divides to be $ak=b$ ...
2
votes
1answer
74 views

Division Without Division

Ok. I'm having a bit of a problem with a mathematical task my friend challenged me with, find the answer to two divided numbers without any division and the method used has to work for all whole ...
1
vote
1answer
21 views

Index set of dyadic partition

Imagine if you have a line from 0 to 1, and you begin partitioning it dyadically. The first point will be at 0, the second at 1 and the third at 0.5, the fourth at 0.25, the fifth at 0.75 etc. Let's ...
0
votes
1answer
22 views

Recurrence relations, trouble understanding the statement

I have been struggling with the English in some recurrence relations problems, since I am studying it on my own and I'm not in a combinatorial environment. Here is one in which I can't grasp what it ...