Questions tagged [recreational-mathematics]

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

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18 views

Combination of cards question

set of cards numbered 1 through 9. They shuffled their own cards and selected a card at random. If the numbers on their cards matched, they won. Wendy and Marc played this game once only. What is the ...
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1answer
114 views

Understanding solution to the secretary problem

I am trying to understand the solution to the Secretary Problem (see, e.g., Wikipedia: https://en.wikipedia.org/wiki/Secretary_problem). As I see it, it is usually solved in the following way: (1) It ...
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1answer
50 views

How to solve this problem raven matrices problem?

I am doing this free test in http://test.mensa.no/ That, as far as I know, the only problem I can't solve. Basically we shift the first row to the right. From first to second is easy transformation. ...
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63 views

Is this kind of “Gerrymandering” NP-complete?

Consider the following simplified form of "Gerrymandering": You have $n^2$ squares arranged as an $n\times n$ matrix. Each square is marked with either $0$ or $1$ which means a "voter preference" ...
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3answers
100 views

Two doors, each either trapped or safe, have signs whose truth depends on certain circumstances. Each sign reads “Both doors are safe”.

A friend of mine sent me the following puzzle: There are two doors, and behind them either a trap, or a safe passage to somewhere, and on the doors, it is written something about whether the ...
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2answers
63 views

Add the following numbers and find the answer?

Sum the $3$ numbers from the list $?+?+?=30$ Fill the boxes using $1,3,5,7,9,11,13,15$ You can also repeat the numbers
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1answer
113 views

What is the smallest number of $45^\circ$–$60^\circ$–$75^\circ$ triangles in non-trivial substitution tiling?

Let base = $45^\circ$–$60^\circ$–$75^\circ$ triangle. Over at What is the smallest number of bases that a square can be divided into? it was determined that 23 base were needed to make a $45^\circ$–$...
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0answers
46 views

An extension of $\mathbb{F}_p$ to an interval $[0,p]$ with $p \equiv 0$

I had a curious question bouncing around my head the other day. I asked myself if there were numbers with similarly nice properties to something like $e$ in a finite field, along the lines of $\mathbb{...
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1answer
47 views

Clever equalities proven similarly to Euler's Identity

From How to prove Euler's formula: $e^{i\varphi}=\cos(\varphi) +i\sin(\varphi)$?, a very elegant proof of Euler's Identity was given. Namely, observing $f(z)=g(z)h(z)=e^{-iz}(\cos(z)+i\sin(z))$, ...
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1answer
48 views

Find the odd one out

I've been trying to work this for a while and I believe the odd one is the 4th figure, but don't seem to find a satisfying justification for this. Any help? Few observations: 1) Bottom two dots ...
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1answer
41 views

Is it possible to calculate the height and distance of an object if the horizon is obstructed?

Basically, if I know $x$, $\theta_1$, and $\theta_2$ in the image below, it is possible to determine $h_1$ and $d$, or do I also need to know $h_2$ first? If the horizon is visible, then the ...
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1answer
110 views

How to win in Battleship?

Battleship explained in wiki: (also Battleships or Sea Battle1) is a guessing game for two players. It is played on ruled grids (paper or board) on which each players fleet of ships (including ...
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2answers
127 views

Distribution at First Time a Sum Reaches a Threshold

Consider the following problem. Roll a die many times, and stop when the total exceeds $M$, for some prescribed threshold $M$. Call this time $\tau$, and call the running score after $n$ rolls $X_n$. ...
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2answers
55 views

Proof or disproof of the existence of a way to flip all bits on a (undirected) graph

Suppose there is an undirected graph $G=(V,E)$. Each node $v\in V$ is allowed to be labelled either "$0$" or "$1$". Define an "operation" $f_v$ for each $v\in V$ which toggles the label for $v$ and ...
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403 views

$5 \times 5\;$ “square additive set”

Problem: IBM Research - Ponder This - January 2019 monthly contest (which was closed few days ago) leads to the problem: Find sets $A = \{a_1,a_2,\ldots,a_n\}$, $B = \{b_1,b_2,\ldots, b_m\}$ such ...
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1answer
49 views

Proof for a question regarding order of $n$-tuples of real numbers

Question as an image Let $x = (x_1, x_2, x_3,..., x_n)$ and $y= (y_1, y_2, y_3,..., y_n),$ where $x_1, x_2, x_3,..., x_n ,y_1, y_2, y_3,..., y_n$ are real numbers. We write $x > y$ if, for some $...
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2answers
56 views

Geometrical proof for length of chord passing through vertex of parabola

In a parabola $y^2=4ax$ , the length of focal chord making an angle $\theta$ with the x - axis is $4acosec^2\theta$ . If a chord is drawn parallel to that focal chord which passes through vertex of ...
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4answers
81 views

Linear equations applications

I'm reading Jane Eyre in class and we have to do questions. However, there are page numbers in the questions that relate to a different copy of the novel that I have. I was wondering if I could create ...
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1answer
26 views

Distributing a number over a course of time

Hi I want to know how to evenly distribute a cost evenly over a course of time between three people. the total cost of the fee is $160 dollars over 34 days. however, 2 out of the 3 people have been ...
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3answers
89 views

Math puzzles suitable for printing on a mug

I need to design a cup for a reception for first-year college students and i'm searching for some challenging and entertaining math puzzle or game to use. Previous years it has been used the "Three ...
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1answer
234 views

Five Similar Triangles from 4-5-6

A 4-5-6 triangle can be divided into 5 similar scalene triangles. I have the solution, but I figured out it was a 4-5-6 after putting together the component triangles. Can anyone else find it? The ...
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1answer
265 views

Why does momentum appear to be not conserved in this elastic collision?

Here's a bit of a fun physics paradox, which I will pose, and then answer below. (These ideas were inspired by Grant Sanderson's fascinating video on how the digits of $\pi$ are hidden within elastic ...
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2answers
58 views

Solving complicated system of equations:

I'm trying to solve this system of equations, and was wondering if there is any possible algebraic manipulation I can do to solve this question. Here are the equations: $x+y+z=338$ $xy+yz+zx=335$ ...
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1answer
229 views

Prove that $2019^{2018}+2020$ is divisible by at least three primes.

Prove that $2019^{2018}+2020$ is divisible by at least three primes. I try to use modular arithmetic, but I believe the only prime I can find is 11. This means I have to find one more factor, but I'...
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2answers
97 views

Suggestions of long and complex formulas/equations, for practicing memorization (It's my hobby.)

First, I'm not a mathematician; my hobby is mainly memorization. I want to practice math formulas and/or equations memorization. In that way, I'm looking for large and complex formulas or equations ...
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2answers
55 views

Problemsolving with weights and their labels

The problem is stated as follows: With a balance scale and six given weights; 1g, 2g, 3g, 4g, 5g and 6g, is it possible to make sure the labels on the weights are put on correctly only using the ...
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0answers
73 views

I'm a celeb get me outta here: optimal strategy to open locks

Suppose there are two 4 digit locks and a given and finite set $A$ of 4 digit tuples. Two elements of $A$ open the locks. What shall I do? Stick to the number or stick to the lock? I. Choose the ...
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1answer
43 views

Generating rotating groups for a seminar

One of my teachers is planning a seminar for his English class and he asked me if there was a way to generate the groups for the days other than brute-force random generating. I really think there ...
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1answer
67 views

Maximum number of ufo that can visit any planet

Consider an infinite alien 2d world consisting of infinite planet, so that distance between any two planets is not same. Now at some point of time, a ufo leaves each planet and goes to planet nearest ...
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1answer
20 views

Expected number of well addressed parcels

Consider the following mad postman scenario. The mad postman has $n$ parcels which should go to $n$ different destinations. However, the mad postman assigns destinations to the parcels randomly. What ...
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1answer
65 views

Candy machines and optimal strategy in terms of expected value

Problem We have three candy machines: call them G (good), B (bad) and M (mixed) . G always gives you a candy when you put 1\$. B never gives you a candy when you put 1\$. M gives you a candy with ...
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1answer
54 views

On which place should you stand in a line, to get a bonus.

Customers are going inside a store, the first customer whose birthday matches the birthday of someone that has already entered the store will get a bonus discount. Where on the line to stand to get ...
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1answer
94 views

Existence of 2006 distinct natural numbers

Does there exists $2006$ distinct natural numbers such that the sum of any two divides the sum of all the given numbers?
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1answer
179 views

Classic Stats Problem, New Twist: Romeo and Juliet meet for a date. Solve Without drawings.

This question has been asked before here. It usually goes like: Romeo and Juliet have a date at a given time, and each will arrive at the meeting place with a delay between 0 and 1 hour, with all ...
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0answers
116 views

References about the mathematics of mazes or labyrinths

I am looking for references on mazes or labyrinths. I prefer books, but research articles are welcome, too. I am looking for the mathematical point of view of mazes, not their history or development. ...
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1answer
174 views

Using a time machine that can jump to a permutation of the current year

This problem is easy, I provide it just for fun of the community. Like many other mad scientists, I believe that time travel is possible. One of such possibilities was described by a Soviet, ...
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1answer
51 views

Incircle of a triangle

In the above image, it says $$AE = \frac{bc}{c+a}$$ and $$AF = \frac{bc}{a+b}$$ But $AE$ and $AF$ are tangents from $A$ to the incircle. As tangents on a circle from a given point are equal, $AE=AF$ ...
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2answers
40 views

Solve for x when another variable is present

I am wanting to reverse-engineer an equation due to a previously unnoticed rounding error and I am trying to determine if it's possible to solve for x. In theory, the values of z and y would always be ...
3
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1answer
128 views

How to solve this puzzle by using Axiom of Choice? [duplicate]

In this article, at the end of page 6, it is given the following puzzle, An evil wizard has threatened a village where an infinite number of gnomes reside. The wizard will cast a spell that will ...
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1answer
256 views

How can I mathematically model and analyze an incremental game like Cookie Clicker?

Recently, I've been interested in the optimization of the infamous incremental game Cookie Clicker. From Wikipedia: The user initially clicks on a big cookie on the screen, earning one cookie per ...
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4answers
257 views

If $f(x)=\int_{x-1}^x f(s)ds$, is $f$ constant? Periodic?

I was thinking of periodic functions, and in particular the following type of condition: If a function $f:\mathbb{R}\to\mathbb{R}$ always "tends to its average", then it should be periodic. To ...
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1answer
51 views

The function $f(x)=\arcsin(a\sinh(\sin(x)))$

While playing around with Desmos, I came across the function $$f(x)=\arcsin(a\sinh(\sin(x)))$$ where $a\in\mathbb{R}$ is a constant (whose value changes the function drastically; if you see the graph,...
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2answers
172 views

What is the probability of rolling two 5s or better with 5 dice?

Preamble My question is in the context of a dice game called 31. You have 6 dice and the goal is to have the highest score. If you roll 36, you score 6 points. If you roll 35 you score 5 points and ...
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0answers
41 views

Given two formulas $A$ and $B$, if $A$ follows from $B$ and $B$ follows from $A$ then is it true that $A$ and $B$ are equivalent?

This is true if $A$ and $B$ are statements, but formulas are statements too, so I expect the answer to be yes. But let's consider this simple example: Formula A: $$\sum_{k=0}^{n-1}x^k=\dfrac{1-x^n}{...
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4answers
452 views

Selecting a menu

While perusing old unanswered puzzle questions, I came across this one. I found it quite interesting, but a bit vague, so I've decided to recast it. A party is to be held at a restaurant. The ...
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1answer
37 views

Permutation of boxes with two colors

Given a row with $n$ black boxes and $n$ white boxes, how many permutations exist where each box can be adjacent to at most one other box of the same color? In other words, three or more consecutive ...
5
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1answer
277 views

Egg drop problem - minimize AVERAGE case

In the egg drop problem, you have two eggs and you want to determine from which floors in a $n$ - floor building you can drop an egg such that is doesn't break. You are to determine the minimum ...
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2answers
48 views

Equation to locate a square in a square

Good evening, I have been experimenting with different Sudoku checker and have come across a problem: For a nxn Sudoku where n is a square number (4,6,19,25 etcc.), there would be an n number of ...
4
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2answers
145 views

Problem of rooms

A rectangle is divided into some smaller rectangles.Each two adjacent rectangles share a door which connects them.Prove that we can start from one of the small rectangles and pass them all without ...
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6answers
1k views

Integral for the New Year $2019$!

Does the following transition between $2018$ and $2019$ hold true?$$\large\bbox[10pt,#000,border:5px solid green]{\color{#58A}{\color{#A0A}\int_{\color{#0F5}{-\infty}}^{\color{#0F5}{+\infty}} \frac{\...