# 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 ...
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 ...
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. ...
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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### 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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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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### 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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### 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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### 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 ...
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 ...
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 $X_n$. ...
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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### $5 \times 5\;$ “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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### 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 ...
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 ...
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 ...
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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### 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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### 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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### 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 ...
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 ...
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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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. ...
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, ...
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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### 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 ...
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### 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 ...
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 ...
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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### 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,...