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Questions tagged [recreational-mathematics]

Mathematics done just for fun, often disjoint from typical school mathematics curriculum. Also see the [puzzle] and [contest-math] tags.

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10 votes
2 answers
317 views

If we are drawing red and black cards out of an infinite deck, and we draw red with probability 4/5, what is E(num draws to draw 3 consecutive blacks)

This is the full question: John is drawing red and black cards out of an infinite deck. The probability of drawing a red card is 4/5. Calculate the expected number of draws to draw three black cards ...
Claire's user avatar
  • 101
-5 votes
0 answers
37 views

Question about balls [closed]

You have 8 identical balls and one is slightly heavier. Using a balance whats the minimum number of times you need to weigh yo find the heavier ball
Lojaine Ahmed's user avatar
1 vote
0 answers
77 views

Prove $\sqrt{\frac{a+6bc}{a+6}}+\sqrt{\frac{b+6ac}{b+6}}+\sqrt{\frac{c+6ab}{c+6}}\ge 3.$

I found the inequality here (#25) : Let $a,b,c$ be nonnegative real numbers such that $ab+bc+ca+abc=4$ Prove that $$\sqrt{\frac{a+6bc}{a+6}}+\sqrt{\frac{b+6ac}{b+6}}+\sqrt{\frac{c+6ab}{c+6}}\ge 3$$ I ...
30 Anh Ti 711's user avatar
6 votes
1 answer
125 views

Conjecture: The sum of the coefficients of terms in the $n$-th derivative of $\sec (x)$ is $n!$ (and another pattern)

Let $f^{(n)}$ stands for the $n$th derivative. The conjectured pattern is: The summation of coefficients of the terms of $(\sec {x})^{(n)}$ equals $n!$ For example: $(\sec {x})^{(0)}=\sec (x)$, the ...
Math Admiral's user avatar
  • 1,554
0 votes
0 answers
62 views

Chessboard Domination by too few queens?

I am looking for patterns of Domination by Independent queens - one source says there is a pattern for 7x7 with only 4 queens with no queens attacking another ie independent. I cannot find it so far. ...
J King's user avatar
  • 29
4 votes
1 answer
112 views

Solutions to $(f(x)-f(y))^3=f\left(x^3\right)-f\left(y^3\right)$

I was wondering, if there are more solutions to the functional equations, than $f(x) = const$. Maybe someone has an idea of how to find all solutions (or all continuous solutions)? Find all the ...
Vlad Boiko's user avatar
0 votes
1 answer
27 views

Tetrahedron analogue of a triangle Cevians property

It's a cute Olympaid geometry problem to prove that given a triangle $\triangle ABC$ with three Cevians $AA_1,BB_1, CC_1$ intersecting at an interior point $M$: $$ab+bc+ca+2abc = 1 \quad (1)$$ where $$...
dezdichado's user avatar
  • 14.1k
4 votes
0 answers
33 views

The Binary Counter Turmite

For 2D Turing Machines, or Turmites (MathWorld, Wikipedia) the best known is Langton's Ant. An ant is on a grid of white/gray 0/1 squares. If white, turn right, otherwise turn left, change the color ...
Ed Pegg's user avatar
  • 21.4k
0 votes
1 answer
155 views

Seeking "900 Geometry Problems" Book – Any Leads on Its Whereabouts?

I have been on a quest to find a book titled "900 Geometry Problems" that I've heard a lot about. The authors are Titu Andreescu and Vladimir Crisan. Geometry is a subject I am deeply ...
MathsGuy's user avatar
4 votes
1 answer
182 views

Creative Algebra Net Problem Solving Question

I came across a problem that I found pretty tedious and difficult to answer and I would appreciate any views or solutions for this question. The diagram shows the net of a cube. On each face there is ...
Jonathan Xu's user avatar
0 votes
1 answer
90 views

Are there infinitely many numbers for which there is no power of 3 with the hamming weight of its binary representation equal to that number?

Inspired by https://mathoverflow.net/questions/22629/are-there-primes-of-every-hamming-weight#:~:text=Standard%20heuristics%20on%20the%20distribution,Hamming%20weight%20h%E2%89%A53, I decided to look ...
codebender's user avatar
1 vote
1 answer
70 views

Describing Frequentist vs Bayesian probability using casual dialogue

I am trying to presented Frequentist and Bayesian probability in casual dialogue format (i.e. a discussion between friends) for my younger sister, who is unfamiliar. The goal of this exercise is to ...
Abhishek Divekar's user avatar
3 votes
0 answers
70 views

Is there a $b$ for every $a$, such that the decimal representation of $b^b$ contains $a$?

Is there a $b$ for every $a$, such that the decimal representation of $b^b$ contains $a$? ($a$ and $b$ are positive integers) This is a question my friend asked recently. I've checked that there is ...
codebender's user avatar
5 votes
0 answers
51 views

Existence of non-intersecting semicircular arcs connecting every points in a given set

This is a problem that I came up with a few months ago. Let $S$ be a set of $n$ points on a plane ($n \ge 3$), and let $e(c)$ denote the set of the two end points of the semicircle $c$. A ...
Kevin Cheng's user avatar
0 votes
2 answers
87 views

Optimal Strategy : Dice Game

I was asked this question in an interview: You are given a fair dice. You can roll the dice any number of times. Your reward will be the sum of the face value of ...
Md Kaif Faiyaz's user avatar
1 vote
1 answer
110 views

Number of ways to place $4$ kings on an $n \times n$ chessboard

I have an $n \times n$ chessboard and $4$ kings on it. My goal is to count the number of arrangements where some of them are non-attacking or mutually attacking, for example: In the case where the $4$...
Cardstdani's user avatar
2 votes
4 answers
1k views

Paradox in Prisoner's dilemma

I have come across a curious paradox concerning The Prisoner's Dilemma Suppose 4 things : prisoners A and B are rather stupid people and decide to use an artificial intelligence program to decide ...
Arnaud's user avatar
  • 872
0 votes
1 answer
71 views

Linearity of expected value and what it implies

In some problem I was doing, it involves a trial where given some value $x$, $50\%$ of the time it will go to $0$ at the next step, and $50\%$ of the time it will go to $2x$. Notation-wise, is it fine ...
Aiden Chow's user avatar
  • 2,821
1 vote
1 answer
59 views

A Problem with two programmers working on a program, one after another - Solution Verification

Below is a recreational math problem that I did with a friend. Problem: Greg and Bob are two computer programmers. They need to write a program. Either one could complete the program in $10$ hours ...
Bob's user avatar
  • 4,064
12 votes
3 answers
1k views

Combinatoric Problem in Stardew Valley about Keg Layout

I will first give the mathmatical description of the problem, which I think is a good problem for high school MOers. Given positive integers $m, n \geq 3$, where $m$ is an odd number, consider a ...
EggTart's user avatar
  • 507
3 votes
1 answer
93 views

A Problem with two programmers working on a program, one after another

Problem: A and B are two computer programmers. They need to write a program. Either one could complete the program in 10 hours working alone. Unfortunately, they only have 1 computer available, so ...
Bob's user avatar
  • 4,064
28 votes
8 answers
4k views

Where is the pentagon in the Fibonacci sequence?

It is common wisdom that "When you see $\pi$, there is a circle close at hand". For example: The periods of sine and cosine equal $2\pi$? Properly constructed, the right triangles that ...
No Name's user avatar
  • 1,055
0 votes
1 answer
39 views

Expression for dividing by 100, based on the number of factors, minus one factor.

I'm trying to convey a way to calculate the aggregate device downtime in a proposal that I'm writing, and I'm struggling, since I don't have a formal higher-math background. My premise is that, if x ...
user501798's user avatar
3 votes
0 answers
74 views

What are the order-3 non-fractal irreptiles?

An irreptile is a shape that can be dissected into smaller copies of the same shape. An order-3 irreptile would divide into three similar copies of the original shape. What are the order-3 non-fractal ...
Ed Pegg's user avatar
  • 21.4k
3 votes
1 answer
200 views

Smallest "diamond-number" above some power of ten?

Let us call a positive integer $N$ a "diamond-number" if it has the form $p^2q$ with distinct primes $p,q$ with the same number of decimal digits. An example is $N=10^{19}+93815391$. Its ...
Peter's user avatar
  • 85.1k
0 votes
1 answer
57 views

Division based recurrences instead of subtraction based: $F(x)=F(x/2)+F(x/3)$

The most famous (and simplest non-trivial) recurrence is the Fibonacci recurrence $F(n)=F(n-1)+F(n-2)$ with $F(0)=0, F(1)=1$. What if we consider instead division based recurrences, the simplest non-...
D.R.'s user avatar
  • 8,945
3 votes
2 answers
67 views

Search for centers of generic sets of points (other then the centroid)

I proposed to a group of friends o'mine the following problem for discussion: Given two finite sets of points $A$ and $B$ on the plane, how to efficiently determine if one is a rotation (not ...
Alma Arjuna's user avatar
  • 3,901
0 votes
1 answer
38 views

Proving Symmedian intersects intersection of tangents

I'm going through Evan Chen's "Euclidean Geometry in Math Olympiads" and I've come to Chapter 4's section on Symmedians. Proposition 4.24 says: Let $X$ be the intersection of the tangents to ...
PabloGamerX's user avatar
0 votes
0 answers
9 views

modifing the word vectors for Cosine Similarity based on associated words

I am trying to use cosine similarity for summing a group of words that will output a tag. think of "knight" + "queen" + "king" = "chess" however due to my tag ...
Hojo.Timberwolf's user avatar
1 vote
1 answer
69 views

Probability That Your Age is Equal to the Current Year Minus Your Birth Year

This problem arose as a thought I had: When someone tells me when they were born, for example, in 2001, I often will say they are 2024-2001=23 years old. Of course, this is not always correct and is ...
PotusOtis's user avatar
  • 135
0 votes
1 answer
166 views

How to Model Stake's : "Mines Gambling Game"

Mines is a gambling game on stake that I came across after watching a streamer play it. There are 25 tiles, and the player has the option to choose between 1-24 mines to be randomly placed under the ...
PotusOtis's user avatar
  • 135
1 vote
0 answers
51 views

Sequences formed by picking cards from 52 face down in a row

Place 52 cards from a deck face down in a line. Turn the first card, and then move number of cards further equivalent to the face value of the opened card. Now open the card that you land on and ...
Abhinav Sood's user avatar
12 votes
2 answers
876 views

Numbers that are four times their decimal reverse

Given an integer $n \geq 4$, find all $n$-digit numbers which are 4 times the reverse of their decimal digits e.g. $8712 = 4 \times 2178$. For each value of $n$, there is only one solution. An ...
1123581321's user avatar
  • 1,102
3 votes
0 answers
76 views

A very interesting function equation $f(a,b)=f(a,c)+f(c,b)$ implies $f(a,b)=g(a)-g(b)$.

Function equation: $f(a,b)=f(a,c)+f(c,b)$ for all positive reals $a>c>b\geq 0$. My solution: $f(a,c)=f(a,b)-f(c,b)$, Let $b=0$, $f(a,c)=f(a,0)-f(c,0)$. Define $g(a)=f(a,0)$. We get the answer. ...
dodo's user avatar
  • 828
0 votes
2 answers
115 views

What did I miss while solving $f(x+f(x)) = x+f(x)$?

What is the number of linear functions $f$ satisfying $$f(x+f(x)) = x+f(x)$$ $\forall x \in R$ ? I began by setting $g(x) = x+f(x)$. Then I applied $f$ on both sides, which gave me $$f \circ g(x) = f(...
AryanSonwatikar's user avatar
7 votes
3 answers
1k views

Prove that the numbers 2008 and 2106 are not terms of this sequence.

The sequence $(x_n)$ is given recursively with $x_1 = 188$, and $x_{n+1}$ is obtained from $x_n$ by adding twice the sum of the digits of the number $x_n$. Prove that the numbers 2008 and 2106 are not ...
user avatar
0 votes
1 answer
65 views

Logic Puzzle with truths & lies to questions

Puzzle : We are at a crossroad of 2 paths. One path leads to a swamp and one path leads to the treasure. At the crossroad we meet 2 people , of whom we know that 1 is always telling the truth and 1 is ...
Root Groves's user avatar
4 votes
1 answer
161 views

Counterexample for a proof

Let $n$ and $k$ be positive integers and $$T = \{ (x,y,z) \in \mathbb{N}^3 \mid 1 \leq x,y,z \leq n \}$$ be a lattice cube of length $n$. Suppose that $3n^2 - 3n + 1 + k$ points of $T$ are colored red ...
Charles Green's user avatar
-1 votes
1 answer
100 views

Example of functions $f$ and $g$ where $f\circ g(x) = x^2$ and $g \circ f(x) = x^3$ for range $(1, \infty)$ [closed]

I'm just looking for examples of real functions $f$ and $g$ where $f \circ g(x) = x^2$ and $g \circ f(x) = x^3$ and the domains and codomains of $f$ and $g$ is $(1, \infty)$. No complex functions but ...
Coach Jonathan Ramachandran's user avatar
1 vote
1 answer
51 views

Determine the minimal tiling, allowing for both overhang and overlap, from a small shape to a larger one.

Context of Concrete Problem: While I have run into similar problems in other games, this specific one is for the game Stardew Valley. You would like to make a farm involving a scarecrow and the lowest ...
Noaline's user avatar
  • 39
-3 votes
1 answer
66 views

nth derivative recursive formula

I'm trying to find a recursive series representation of the $n$-th derivative of the following function. $D^{(1)}_{a, b}(x) = b\sqrt{\frac{p}{q}}D_{a + p - 1, b - 1}(x) - a\sqrt{\frac{q}{p}}D_{a - 1, ...
Ghull's user avatar
  • 87
0 votes
0 answers
76 views

Equivocal numbers : A base system question

Caltech Harvey Mudd Mathematics Competition ( CHMMC ) $2012$ Team Round Problem $7$ : A positive integer $x$ is $k$-equivocal if there exists two positive integers $b$, $b’$ such that when $x$ is ...
Aashita's user avatar
  • 69
19 votes
0 answers
1k views

Placing triangles around a central triangle: Optimal Strategy?

Now cross-posted to MathOverflow (link). Question: There is an equilateral triangle. Two players alternate turns placing non-overlapping equilateral triangles of the same size that touch the original ...
Benjamin Dickman's user avatar
2 votes
0 answers
139 views

Is it just a coincidence that 2187 and 2197 are so close?

I noticed that $3^7 = 2187$ and $13^3 = 2197$. So, a fun way to combine these is: $$\left(10+3\right)^{3} = 10+3^{10-3}$$ If Pillai's conjecture holds, then such a difference of $10$ between perfect ...
Zubin Mukerjee's user avatar
2 votes
2 answers
92 views

Positive integers $n$ such that $ n= 37\times$ times the digit sum of $n$

ARML 2021 I-5 , Sum of digits : For a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Compute the sum of all positive integers $n$ for which $n = 37s(n)$. It's easy to derive ...
Aashita's user avatar
  • 69
2 votes
1 answer
105 views

Calculating deflection on a beam

This is for a hobby project, and to learn a little about elasticity along the way. I have a triangle wedge comb piece of decreasing width and angle for which the cross section is shown here: For each ...
vallev's user avatar
  • 408
0 votes
0 answers
36 views

Concrete mathematics page 11 (The Josephus Problem - Proof)

Can someone check my proof for the odd case of the recurrence relation for the Josephus problem from Concrete Mathematics, Chapter 1, page 11 Our recursive equation: \begin{equation} \begin{split} ...
Emoanimelovr's user avatar
1 vote
0 answers
72 views

Are integrals just approximations of area? [duplicate]

When we try to solve some problems with integration, we face some problems and a bit of confusion. Now, if we want to determine the area of an object, we have some specific formula to determine the ...
shanto khan's user avatar
1 vote
1 answer
70 views

Game - two players take turn moving a marker to an adjacent square in a 9x9 grid

A marker is placed in the centre of a $9$x$9$ grid. Ann and Beth take turns moving the marker to one of the adjacent squares (one sharing a side) provided that this square has never been occupied by a ...
Abhinav Sood's user avatar
1 vote
0 answers
62 views

Number of sets that can be built using length-$n$ combinations of commas and braces

Let $a(n)$ be the number of sets that can be built using length-$n$ combination of either a commas and braces. Here's a manual calculation of $a(n)$ for $0<n<13$ (duplicate sets have been ...
Bryle Morga's user avatar
  • 1,029

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