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Questions tagged [recreational-mathematics]

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

-2
votes
1answer
28 views

How to get nth term in a digitsum sequence

how do I get the Nth term in a sequence whose digitsums are = D and number of digits = K e.g. for D = 4, K = 3 the finite ...
3
votes
0answers
88 views

Magical Numbers

A natural number $N$ is said to be nice if we obtain a divisor of $N$ by erasing one of it's digits. A positive integer $X$ is said to be magical if: $X$ has distinct digits. $X$ is nice. The divisor ...
0
votes
2answers
48 views

AIME: I'm not sure what the question is asking for

I've encountered this problem: Let $f(x)=|x-p|+|x-15|+|x-p-15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \leq x\leq15$. I'm not sure what ...
9
votes
0answers
112 views

Strategy for board game 2

In this question the following was asked: Alice and Bob are playing the following game: They have a $4 \times 4$ empty grid and take turns coloring one square each, starting with Alice, both using ...
0
votes
0answers
29 views

Isohedron splitting into isohedra

An order 3 dipyramid can be split into two regular tetrahedra. An order $n$ dipyramid can be split into $n$ isosceles tetrahedra. A 4-scalenohedron with edges of length $(\sqrt5, \sqrt8,\sqrt9)$ ...
0
votes
0answers
14 views

Ways to express sum of $n$th-power partial derivatives?

In the context of multivariable calculus, we can express the sum of the first and second power partial derivatives of a function as follows: $$\vec 1 \cdot \nabla f = f_x + f_y + f_z$$ $$| \nabla f |...
0
votes
0answers
24 views

Is bridge a fair game: references/heuristics?

Though not knowledgeable on card games or game theory, I've just learnt the rules of bridge, and I'm starting to wonder whether it can be proven that bridge is a fair game. Of course, to give an ...
0
votes
0answers
48 views

Obtuse triangle octahedron packing

Any triangle can make an octahedron. Here are two of them for a 7-12-17 triangle. The green edges correspond to the middle length of 12. Is there any triangle $a-b-c$ where the concave octahedron'...
0
votes
1answer
28 views

Pendant Cutting dimension

Was wondering if you can help me with a calculation. I have a somewhat (not exact) square or rectangular piece of cloth. I want to cut three pendants out of the cloth. For mounting, at the wide end ...
58
votes
3answers
6k views

Optimal strategy for cutting a sausage?

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into ...
1
vote
2answers
74 views

Smallest element of cycle of length $k$ in Collatz 3x+1 map?

In studies of the Collatz conjecture, what research has asserted the existence of a $k$-length cycle and drawn conclusions about its smallest element $m$? In particular, about the behavior of $m$ as $...
0
votes
1answer
37 views

How to calculate points for f(nx) when all that's given for f(x) are arbitrary points

If you have a point list for $f(x)$, how do you determine the point list for $f(nx)$? The way I would have done it would be to plot the points for $f(x)$, identify the function and calculate the ...
0
votes
1answer
58 views

Representation as sum of irrational numbers

Let $S=\{ x\in(0,1)\setminus\mathbb{Q} \,:\, \lfloor 100x \rfloor\in \{ 0,11,22,33,44,55,66,77,88,99\}\} $. Find the smallest $k$ such that any $X\in(0,1)$ can be written as $X=x_1+x_2+\dotsb+x_k$ ...
0
votes
0answers
58 views

The Barroom Scrap interpretation: [closed]

I was given this at a tavern years ago. I called it the "Lloyd Intersection." Original What methods may be used to solve for x? Jens comment below solves this question. You also could; Convert ...
12
votes
3answers
157 views

Functional Equation $f(x+y)-f(x-y)=2f'(x)f'(y)$

I am trying to solve the equation $f(x+y)-f(x-y)=2f'(x)f'(y)$ for all $f:\mathbb{R}\to\mathbb{R}$ non-constant, differentiable functions. Here is my progress: Any solution must be an even function ...
1
vote
0answers
42 views

How many solvable configurations of the Rubik's cube have no two squares of the same color touching?

The Superflip (image below) is an example of a configuration where no two squares of the same color are adjacent. I'm interested to know how many solvable configurations (that is, those that you can ...
2
votes
3answers
110 views

What is the function $y = (1+1/x)^x$ solved for $x$?

I came across this function in algebra ($e$ being its limit as $x$ goes to infinity) while studying compounded interest. Since this function is a little modified from the real interest formula $y=(1+1/...
2
votes
2answers
143 views

Why does $11! - 10!$ equal $10!$ plus an extra digit in “ones”?

Pardon my English as I'm not a native speaker of the language and I'm not a big math guy, so please bear with me and my ignorance for a bit. I've unconsciously stumbled upon something that's most ...
1
vote
3answers
49 views

Find the number of trailing zeros in 50! [duplicate]

My attempt: 50! = 50 * 49 *48 .... Even * even = even number Even * odd = even number odd * odd = odd number 25 evens and 25 odds Atleast 26 of the numbers will lead to an even ...
1
vote
1answer
29 views

Add component to equation so a value turns into a negative?

This is a variation of another question I posted. The difference is that here we have an equation that mostly works. We just need help finding how to push a Case. Honestly, we're just trying out ...
0
votes
4answers
68 views

Rules of Roman Numerals

I've been working on a problem that involves discovering valid methods of expressing natural numbers as Roman Numerals, and I came across a few oddities in the numbering system. For example, the ...
1
vote
0answers
52 views

What planar concave polyforms bound 3D space?

A square can bound a cube or a heptacube if a cube is glued to each face. An equilateral triangle can bound a tetrahedron and other shapes. The diamond (2 equilateral triangles) can make a ...
4
votes
0answers
55 views

Does infinite mikado exist?

Let's define a mikado configuration $m$ as a countable collection $\{T_j\}_{j \in \mathbb{N}}$ of disjoint subsets of Euclidean 3-space $(\mathbb{R}^3,\cdot)$. Each $T_j$ is a "tube of radius $R>0$"...
0
votes
0answers
30 views

Circular track - finding distances between the bus stops.

Tried different combinations with simultaneous equations, but could not get the answer. Any easier way to get the answer. this is my workout.
23
votes
7answers
2k views

What problems have been frequently computationally verified for large values?

Although any theorem (or true conjecture) can be computationally checked, many long-standing open problems have been computational verified for very large values. For example, the Collatz Conjecture ...
4
votes
1answer
91 views

Can a $10\times 10$ square be entirely covered by 25 $T$-shape bricks?

Let $ABCD$ be a square in which length of a side is $10$ meters. Suppose that we have $T$-shape brick which consists of $4$ smaller squares in which a side of each smaller square has length of $1$ ...
9
votes
3answers
206 views

Getting an airplane around the world

I'm interested in a generalization to the following riddle: You're the director at an airport on the equator, and each plane at your airport can hold enough fuel to fly half way around the world. ...
3
votes
0answers
105 views

Explaining MENSA's IQ Problem on Alien Fingers

There is one probably quite well-known IQ puzzle below from Mensa: There are a number of aliens in a room. Each alien has more than one finger on each hand. All aliens have the same number of ...
0
votes
2answers
62 views

Lissajous Curve dense in a Volume

The Lissajous-type curve $\left(\sin(\omega_1 t), \sin(\omega_2 t)\right)$, with $t \in \mathrm{R}$, is dense in a certain region of the plane. This can be seen for instance from excellent answers ...
2
votes
0answers
31 views

Symmetric 3-pire map

Awhile ago, Martin Gardner introduced Scott Kim's symmetric 2-pire map. There are 12 empires, each with two regions. All twelve empires share a border. This is a 2-pire solution for the Empire ...
16
votes
5answers
265 views

Multiples of $999$ have digit sum $\geq 27$

How could we prove the following claim? The sum of the digits of $k\cdot 999$ is $\ge 27$ I checked $k = 1$ up to $9$. And I found that if it's true of $d$ it's also true of $10\cdot d$. I also ...
25
votes
3answers
651 views

Hat 'trick': Can one of them guess right?

There are $n$ boys and $n$ girls. Each of them is given a hat of only 4 possible (known) colors and doesn't know its color. Now each can only see all the colors of hats of those of the other gender ...
7
votes
1answer
92 views

Expected area “sweeped” on an infinite Minesweeper board

Given an average density (x/y) of x mines in y squares, is it possible to calculate the expected number of squares you can "sweep" (i.e. identify whether there is a mine or not) on an infinitely sized ...
9
votes
0answers
137 views

What would you see inside a spherical mirror?

Image to build a huge spherical shell made of semitransparent glass, and to cover the internal part with a reflecting material. In such structure some light can enter, and an observer inside it (e....
4
votes
1answer
124 views

Is there a simpler single polygon toroid?

In B.M. Stewart's book Adventures Among the Toroids, toroidal shapes of many sorts are made. One of them is the ring of 8 octahedra, with 48 faces. The toroid is made with a single polygon -- the ...
3
votes
0answers
88 views

Generalized Pythagorean integer solutions

We know that $a^2 + b^2 = c^2$ has infinitely integer solutions which can be written as 3-tuples for example (3,4,5). Can we conjecture that $$x_{1}^n +x_{2}^n + \cdots +x_{n}^n = y^n$$ has ...
6
votes
1answer
686 views

A bug's journey

I encountered this question here (question 6) http://sections.maa.org/iowa/Activities/Contest/Problems/Probs98.htm The Question: A bug is crawling on the coordinate plane from (7,11) to (-17, -3). ...
7
votes
1answer
78 views

Packing problem - how to fit things nicely with just the aspect ratio of the objects

Suppose you have a rectangle that is w units wide and h units tall (bounding rectangle). You also have an even ...
5
votes
1answer
107 views

How many colors are needed to color an infinite grid so that no sqaure have the same color in all 4 vertexes? [duplicate]

Suppose on a 2 dimension infinite grid, all nodes, or equivalently, all $(p,q)$ points, where $p, q \in \mathbb Z$, need to be painted with a color. Is there a way that, given enough kinds of paiting ...
3
votes
1answer
67 views

Volume in cone segment bent from rectangle.

A flexible rectangle sheet size $(a,b),a>b $ is folded half along side $a$ and glued to make a circular cone cut segment of vertex angle $60^{\circ}$ as shown with three edges $(b,a,b).$ ( $60^{\...
206
votes
5answers
10k views

What is the maximum volume that can be contained by a sheet of paper?

I was writing some exercises about the AM-GM inequality and I got carried away by the following (pretty non-trivial, I believe) question: Q: By properly folding a common $210mm\times 297mm$ sheet ...
8
votes
1answer
182 views

Area of a mushroom-shaped curve

Inspired by a discussion on this question, I discovered the following hybrid function: The curves in red are defined as $$f(x)=\exp\left((\sin x)^{(\sin x)^{\sin x}}\right)$$ and the curves in blue ...
3
votes
2answers
65 views

What is the probability that the stock trader gains money if the price of a share goes up or down with the probability of 50% each day?

Preface My brother was applying for a university. In his entrance exam there was the following question (translation done by me): John buys 100 shares of a company. He estimates that during one ...
6
votes
1answer
70 views

Edge set tetrahedra

For a set of 6 edge values, up to 60 distinct tetrahedra can be made in 30 mirrored pairs. Can a tetrahedron with a given edge set exist entirely inside a second tetrahedron with the same edge ...
2
votes
0answers
74 views

Largest Nontrivial Integer Definition

The largest Fermat number with a known factor is $F_{3329780}$ with prime factor $193\times2^{3329782} + 1$ At googology are many examples of huge integers. In general, for any integer $a$, it's ...
13
votes
2answers
1k views

Can the sum of two distinct factorizations of a number be equal?

Given two distinct factorizations of a positive integer with the same number of factors (not necessarily prime or all distinct), must the sums of the respective sets of factors also be distinct? This ...
0
votes
1answer
41 views

Finding largest prime that divides an integer that is equal to sum of the squares of its four smallest positive divisors

Sorry about the title... wasn't sure what to put?? My question is: A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides ...
4
votes
0answers
48 views

Maximum path crossings in a square grid

Make a path from one corner of a square grid to the diagonally opposite corner, using horizontal and vertical edges between the grid points. What is the maximum number of times the path can cross ...
4
votes
1answer
89 views

Closed form of multi-variable integral with Dirac delta function

I want to give you this exercise to see if there are simpler solutions to the one that I've chosen! I found it a very interesting exercise. Question Find the closed form for all N of following ...
1
vote
0answers
48 views

If I have a number $N$, what operation can I use to make a number $X^N$

I'm trying to create a method to a way to take a number and make it a part of another “family” of numbers. For example, given a number $X$, to make it a part of the family of even numbers, just ...