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

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1
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1answer
26 views

Technical meaning of two alike combinatorial problems

I am confused in how to interpret two alike combinatorial problems, because to me they both look the same. These are the problems: How many ways are there to put $24$ distinguishable flags on $18$ ...
17
votes
3answers
427 views

How, if at all, does pure mathematics benefit from $2^{74207281}-1$ being prime?

So a couple of days ago the $17$ million digit number $2^{57885161}-1$ was beaten by the $22$ million digit number $2^{74207281}-1$ at being the largest known prime number. Are there any specific ...
0
votes
1answer
67 views

There are two buckets A and B. [closed]

There are two buckets A and B. Initially A has 2 litres of water and B is empty. At every hour 1 litre of water is transferred from A to B followed by returning litre back to A from B half an hour ...
0
votes
0answers
66 views

Mr.Smith commute word problem. Solved through logic, where is the argument unsound?

Mr. Smith commutes to the city regularly and invariably takes the same train home which arrives at the his home station at 5 pm. At this time, his chauffeur always just arrives, promptly picks him up ...
1
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1answer
98 views

How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? [closed]

Consider the function $x^2 + bx + c$ How do the (real or complex) roots of the equation change if $b$ is held constant and $c$ is changed? I.e. Which patterns are evident? What would it look like if ...
2
votes
2answers
30 views

Measurement Question Related to a Race Car

I recently got this peculiar interview question, and I wanted some help figuring out how to reach an appropriate solution. Imagine that we have a race car that is driving on a $50$-mile-long race ...
2
votes
1answer
59 views

Trivia Crack Probability

In the game Trivia Crack, you answer a question with 4 possible answers. A "life line" you can use allows you to guess a second time if you were wrong the first time. What is the probability you will ...
4
votes
1answer
247 views

How to draw a Mandelbrot Set with the connecting filaments visible?

The M-Set is connected. But the M-Set viewers I’ve found create cool pictures that don’t really show the connecting filaments. This mini-Mandel beetle should be connected to a larger min-Mandel by a ...
13
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4answers
1k views

Volume of 1/2 using hull of finite point set with diameter 1

It's easy to bound a volume of a half. For example, the points $(0,0,0),(0,0,1),(0,1,0),(3,0,0)$ can do it. The problem is harder if no two points can be further than 1 apart. Bound a volume of 1/2 ...
3
votes
1answer
57 views

Find Three Mutual Friends in a Mathematical Society

I am having trouble with the following combinatorics/graph theory problem: A mathematical society has three divisions (Pure, Applied, and Statistics), and exactly $n$ mathematicians in each ...
0
votes
1answer
164 views

Maths Puzzle - Logic

Somebody asked me this puzzle, but they don't have answer to it. 1+2+3+4 = 61 2+3+4+5 = 52 3+4+5+6 = 51 4+5+6+7 = 50 7+8+9+10 = ? I want to know whether my reasoning and ...
13
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0answers
113 views

Every natural number in binary can be cut and added so that it is a power of $2$? [duplicate]

I was watching a google techtalk with Donald Knuth and he mentions for every binary number $\overline{a_1a_2a_3\dots a_n}$ there exists $c_1<c_2<\dots <c_r=n$ so that: ...
1
vote
1answer
49 views

Counting duplicates

I have been doodling around and have stumbled across the following problem: Say I have a set $p = \{p_1, p_2, p_3, ..., p_n\}$ where $p_i \in \mathbb{N}$ $p_i$ could represent the amount of a ...
0
votes
1answer
13 views

Getting X based on table values

I have a table of values; $$3.00 \mapsto 12.15$$ $$3.10 \mapsto 12.82$$ And I'd like to know an equation for getting the inbetween values. For example: $$3.05 \mapsto \frac{12.15 + 12.82}2$$ But ...
3
votes
3answers
84 views

Computing $2016$ using basic operations on the fewest integers, in sequence

Using the operators $$+,-,\div,\times,\exp,(,),!$$ what is the least $n$ to come up with the number $2016$ using the sequence of numbers $1,2,3,\ldots,n$ in that order. You cannot combine numbers, so ...
5
votes
1answer
117 views

Possible all-Pentagon Polyhedra

If a polyhedron is made only of pentagons and hexagons, how many pentagons can it contain? With the assumption of three polygons per vertex, one can prove there are 12 pentagons. Let's not make that ...
1
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0answers
40 views

$\exists\ n \gt 34131$ with more than $7$ odd divisors $d_i \gt 1$ such as when $d_i+1$ are accumulated in increasing order to $1$ the sums are prime?

In the same style as a previous test, I did a little test today looking for all the numbers such as the odd divisors, ordered in increasing order excluding $1$, when they are accumulated one by one to ...
1
vote
2answers
44 views

Ratio of even and odd divisors

I've been given of this problem: Let $r$ be an integer which has $k$ even divisors and $k-3$ odd divisors. Furthemore let $x$ denote sum of all even divisors and $y$ sum of all odd divisors. What are ...
0
votes
1answer
26 views

If a particular constant contains every single possible combination of numbers in it's decimal expansion, does it also contain itself?

If a particular constant $a$ contains every single combination of numbers in it's decimal expansion, does it then imply that at one point the series should also contain itself?
15
votes
1answer
208 views

Show there is an uncut square lying in a larger square cut by lines

I found this problem on Keith Ball's blog sometime ago but I've never really worked it out. Show that if a square is cut by two lines (shown above in green) then there is an uncut square at ...
0
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2answers
26 views

Cuboid room, hooks and strings proof

I'm trying to do the following problem: In a cuboid shaped room a hook is placed in the centre of each wall, the floor, and the ceiling. Every pair of hooks has either a piece of red or blue ...
0
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2answers
25 views

What is the length of one turn along the axis in strip winding?

In strip winding of a cylindrical surface like this What is the length of one turn along the axis? Or what is the distance between two similar points on consecutive turns along the axis of ...
1
vote
2answers
39 views

LIM without x->a

I was working through the first few pages of Problem-Solving Strategies by Arthur Engel (Which may or may not be a little above my level), and I came upon an interesting form of notation I haven't ...
0
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1answer
54 views

Ramsey's Theory and Tic-Tac-Toe analogy.

I read, briefly, about a connection between Ramsey's Theory and Tic-Tac-Toe. From my understanding, it went like this: Imagine playing Tic-Tac-Toe in a k-dimensional hyper-cube. There is such a ...
0
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1answer
283 views

How to solve simple programming problem with strange math question?

Here is the question: A cookie recipe calls for the following ingredients: 1.5 cups of sugar 1 cup of butter 2.75 cups of flour The recipe produces 48 cookies with this amount ...
1
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1answer
71 views

What is the shortest LOOP program that outputs 2016? [closed]

Use a minor restriction of the LOOP language described under Wikipedia's "LOOP (Programming Language)". The restriction is to eliminate constants. So, the language contains increment: $x_i++$, ...
0
votes
0answers
43 views

Solving Trigon Puzzles

The Trigon puzzle consists of a grid of triangles arranged in shapes with both outside borders and "holes" in the center. Each triangle has a sum between 0 and 18. The goal is to assign values to ...
0
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1answer
393 views

A function f(n) satisfies the recurrence f(n)=4f(n/2)+n for real numbers. Give an upper bound for f(n)?

A function f(n) satisfies the recurrence f(n)=4*f(n/2)+n for real numbers. Give an upper bound for f(n)? I get somewhere T(n) = Θ(n^2), is that correct?
13
votes
1answer
662 views

Any other Caltrops?

This question has been edited. The regular tetrahedron is a caltrop. When it lands on a face, one vertex points straight up, ready to jab the foot of anyone stepping on it. Define a caltrop as a ...
1
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1answer
34 views

Reflected rays bouncing in a regular polygon?

Suppose we have the following scenario: You are standing in a room that is in the shape of a regular n sided polygon with mirrors for walls. You shine a light, a single ray of light, in a random ...
2
votes
0answers
42 views

The hardest game of mahjongg

I was playing Mahjongg solitaire the other day. It got me thinking... The board has $2n$ pieces at the beginning and assuming that the game is winnable. The game would be trivial if there would be ...
2
votes
1answer
47 views

Variant of “prisoners and hats” puzzle with more than two colors

There are $n$ prisoners and $n$ hats. Each hat is colored with one of $k$ given colors. Each prisoner is assigned a random hat, but the number of each color hat is not known to the prisoners. The ...
1
vote
0answers
19 views

Oscillations in a Discrete Dynamical System.

If you are familiar with SingingBanana on youtube, he posted the following question: There is a 10 digit number where the first digit tells me how many 0 there are in the number, the second digit ...
21
votes
1answer
567 views

Is this a way to prove there are infinitely many primes?

Someone gave me the following fun proof of the fact there are infinitely many primes. I wonder if this is valid, if it should be formalized more or if there is a falsehood in this proof that has to do ...
14
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3answers
225 views

Finding the common integer solutions to $a + b = c \cdot d$ and $a \cdot b = c + d$

I find nice that $$ 1+5=2 \cdot 3 \qquad 1 \cdot 5=2 + 3 .$$ Do you know if there are other integer solutions to $$ a+b=c \cdot d \quad \text{ and } \quad a \cdot b=c+d$$ besides the trivial ...
4
votes
4answers
210 views

Reflected rays /lines bouncing in a circle?

Consider the following situation. You are standing in a room that is perfectly circular with mirrors for walls. You shine a light, a single ray of light, in a random direction. Will the light ever ...
9
votes
1answer
149 views

Interesting shapes using probability and discrete view of a problem

Suppose we have a circle of radius $r$, we show the distance between a point and the center of the circle by $d$. We then choose each point inside the circle with probability $\frac{d}{r}$ , and turn ...
13
votes
3answers
180 views

Can exist an even number greater than $36$ with more even divisors than $36$, all of them being a prime$-1$?

I did a little test today looking for all the numbers such as their even divisors are exactly all of them a prime number minus 1, to verify possible properties of them. These are the first terms, it ...
3
votes
1answer
82 views

What does it mean to suppress a number in math?

What does it mean to suppress a number in math? I was doing a math problem and it said to "suppress a term of a sequence." Does this mean to decrease or get rid of the term? Problem: Let ...
2
votes
0answers
135 views

Number of ways to color a grid?

I have a $N \times M $ grid and I am trying to calculate the number of ways I can color this grid in maximum $k$ colors (I can use only $2$ colors or all $k$ colors) with the exception that two ...
13
votes
10answers
1k views

Small Representations of $2016$

It's the new year at least in my timezone, and to welcome it in, I ask for small representations of the number $2016$. Rules: Choose a single decimal digit ($1,2,\dots,9$), and use this chosen digit, ...
25
votes
10answers
5k views

Is it possible to draw this picture without lifting the pen?

Some days ago, our math teacher said that he would give a good grade to the first one that will manage to draw this: To draw this without lifting the pen and without tracing the same line more than ...
15
votes
1answer
343 views

New Year Summation 2016: $\displaystyle\sum_{r=3}^{\; 3^2}r^3$

Decode the following summation to welcome the new year! Find integer $n$ such that $$\large\color{darkblue}{\sum_{\qquad \qquad r={\sum_{m=0}^\infty\left(\frac{n-1}n\right)^m }}^{\qquad \qquad ...
5
votes
2answers
68 views

Issues solving equations involving $x^{x^x…}$?

I stumbled across this problem: $x^{x^{x^{...}}}=2$ Obviously, I used the substitution trick and I got $x^2=2$ and thus, $x=\pm\sqrt{2}$. I have tested that this works. However, I tried to ...
10
votes
1answer
105 views

Relationship between primes and practical numbers

This is my first post here. I am a musician, and not a mathematician, but I enjoy doing things to prime numbers and seeing what comes out. I have defined a sequence which takes the following values ...
-1
votes
1answer
239 views

New year incoming, 2016 [closed]

Here we are, another year is going to finish. Then, what are the "good" or "funny" properties you can find about the number $2016$? Or, is there a natural problem having $2016$ as answer? I tried to ...
-10
votes
3answers
181 views

Interesting patterns to the algebraic solutions of polynomials [closed]

In yet another attempt to find the solution to the quintic polynomial, I started looking backwards at the solutions to the quartic, cubic, quadratic, and linear polynomials to see if I could pick up ...
14
votes
4answers
1k views

What's the smallest number that we can multiply with a given one to get the result only zeros and ones?

I have the following set of numbers, $$4, 198, 4356, 10296, 14454, 25542, 31779, 51252, 53946, 99999$$ Let's take $3,4$ as an examples: The smallest number to multiply with $4$ to get the result ...
2
votes
0answers
223 views

How can all players in the Starcraft 2 Grandmaster league win more than they lose?

Starcraft 2 is a competitive online strategy game where players compete in leagues with other players of similar skill. The most difficult and highest league is the Grandmaster (GM) league, which ...
0
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1answer
40 views

What is differential time ratio? [closed]

Alright, so I discovered this website detailing some far-out time travel theories. Now, before you say anything let me be clear, I'm not interested in time travel so much as the creative writing ...