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

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0
votes
1answer
23 views

Adding the intersections of circles. [on hold]

Given the grid attached, how can you place the numbers $1-20$ at the intersections so that each circle adds to the same sum. I haven't been able to figure this out.
0
votes
0answers
10 views

Making ease-out-bounce formula have a linear start

I'm using a bounce ease out formula, the code for it: https://github.com/jesusgollonet/processing-penner-easing/blob/master/src/Bounce.java#L9. The function is copied here: ...
720
votes
26answers
114k views

How long will it take Marie to saw another board into 3 pieces?

So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long ...
10
votes
0answers
303 views

Prime factor of $2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4$

I checked the prime factors of $$2 \uparrow \uparrow 4 + 3\uparrow \uparrow 4 = 2^{2^{2^2}} + 3^{3^{3^3}}$$ with trial division and found non below $8*10^9$ Nevertheless, the given number has still ...
4
votes
1answer
35 views
+100

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 ...
6
votes
1answer
43 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 $B$ ...
2
votes
2answers
214 views

63% chance of event happening over repeated attempts

I saw this online: If there is a $1 / x$ chance of something happening, in $x$ attempts, for large numbers over $50$ or so, the likelihood of it happening is about $63\%$. If there's a $1$ in ...
1
vote
0answers
58 views

If two parallel lines meet at infinity, then what is their angle? [duplicate]

Since lines that meet at some point have an angle. And if parallel lines meet at infinity, then that what is the angle of two parallel lines that meet at infinity?
-1
votes
0answers
17 views

Playing with spheres

Suppose I have $N$ spheres, each of which has radius $R_i$ and cost $C_i$. How do I place them in Euclidean Space such that the total cost is maximized. The total cost is calulated as follows: for ...
21
votes
3answers
701 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 ...
4
votes
4answers
636 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 $b$ $$(1+a)(1+a+b)\geq\sqrt{27ab}$$ ...
-2
votes
0answers
45 views

Number of Holes in a Number [closed]

In a recent puzzle I was working on, it asked to find the number of holes in a given number as a string. I was wondering if there was a mathematical solution to this rather than creating a list of ...
4
votes
1answer
29 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 ...
6
votes
0answers
67 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: ...
1
vote
0answers
38 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 ...
0
votes
0answers
10 views

finding the optimal representative subset of a group of $n$ members

I am interested in comparing different teams that have the same number of members. For example, assume for a team activity, $n=22$, and I have come up with a skill rating for each member of the ...
4
votes
0answers
31 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. ...
16
votes
3answers
223 views

Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
-5
votes
0answers
42 views

math riddles with connections to pigeon hole principle and binary representation [closed]

1) Let an infinite sequence of numbers be : $a_1=1,a_2=1,a_n=a_{n-1}+a_{n-2} \ (mod \ 10)$ so the sequence goes like this : $1, 1, 2, 3, 5, 8, 3, 1,...$ and so on. Does this sequence periodic? (i.e. ...
1
vote
1answer
21 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 ...
6
votes
1answer
167 views

Card game probability

Suppose the following solitaire with a standard deck. I turn four cards visible on the board and on each turn, I remove those suits that appears more than once in the board. Then I fill the board such ...
5
votes
2answers
83 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
vote
2answers
626 views

Determining Formula (Game Mechanics)

WARNING I believe that the data below has errors in the defense strength, so is therefore not solvable. I will update it when I have more information. Thank you. I play a game (Empire: Four ...
6
votes
0answers
140 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. ...
19
votes
5answers
500 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 ...
2
votes
1answer
432 views

Counting the number of cubes in an isometric view

I had seen questions in a sample aptitude test, where an isometric view of an object made up of cubes was given, with some of the cubes removed. We were supposed to count the number of cubes present ...
-1
votes
1answer
20 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 ...
1
vote
1answer
37 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 ...
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 ...
6
votes
1answer
68 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 ...
11
votes
1answer
259 views

Mathematics of the Ice Bucket Challenge

I've been considering the mathematics of the now global ice bucket challenge. Simple model In the simplest incarnation, there is one original seed, who then nominates 3 others, each of which take ...
1
vote
2answers
82 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
5answers
81 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 ...
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
66 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
0answers
45 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
50 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 ...
1
vote
2answers
399 views

Arithmetic progressions of perfect powers

Find the largest positive integer $n<100$, such that there exists an arithmetic progression of positive integers $a_1,a_2,...,a_n$ with the following properties. $1)$ All numbers ...
2
votes
2answers
48 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! &= ...
0
votes
1answer
25 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 ...
37
votes
2answers
3k views

Minesweeper - Chance of one-click win

I'd like to know if it's possible to calculate the odds of winning a game of Minesweeper (on easy difficulty) in a single click. This page documents a bug that occurs if you do so, and they calculate ...
5
votes
0answers
189 views

Show that : $2, 5, 13, 17, 29, 421, 401, 53, 281,…,\rightarrow \infty$? $a_{n+1}=\operatorname{ GPF}(qa_n+p)$

I denote by $\operatorname{ GPF}(n)$ the greatest prime factor of $n$, eg. $\operatorname{ GPF}(17)=17$, $\operatorname{ GPF}(18)=3$. Is there a way to prove that the sequence $a_{n+1}=\operatorname{ ...
0
votes
0answers
43 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: ...
1
vote
0answers
34 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 ...
-2
votes
1answer
90 views

Puzzle: Players $A,B,C,D$ are in a line

Players $A,B,C,D$ stands in a line. Players $A, D$ do not move. round $1:$ player $B$ moves one distance closer to the midpoint of $A$ and $C$ round $2:$ player $C$ moves one distance closer to ...
4
votes
1answer
155 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. $C$ ...
0
votes
1answer
40 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
3 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 ...
4
votes
2answers
86 views

How to find the formula for the sequence $1, 3, 6, 10, 15…$?

Before I say anything, I have to say that this isn't an advanced mathematics question; I'm just a $15$ year-old student, who came across a mathematical problem. I saw a picture displaying a ...