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

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3
votes
1answer
77 views

A joke proof of a famous mathematician showing that a certain two-digit number is prime

There was a joke (highly sophisticated, non-elementary) proof of a famous mathematician showing that a certain two-digit number (like 43 or 83 but I forgot what) is prime. Could you remind me of a ...
1
vote
1answer
20 views

Number of algebraic solutions to a formula related to a square tiling problem

How can many different sets of prime-factors fit together so well in this formula? I am curious about the number of solutions to the following equation: $$ r_3 = \sqrt{2}\; \frac{ 1 + r_1 (r_2 ...
1
vote
2answers
26 views

What was the average speed for the whole journey?

Last weekend I went to London ... I calculated my average speed going to London was 30 mph On the return journey the traffic was terrible and I calculated an average speed of 20mph What was the ...
0
votes
0answers
31 views

Geometrical properties of tetrahedra under rotation

Consider two tetrahedra which share the same point of origin but differ in both scale and rotation over the X-axis. Can someone explain why the following points meet with these parameters? Both have ...
9
votes
4answers
327 views

Big List of examples of recreational finite unbounded games

What are some examples of mathematical games that can take an unbounded amount of time (a.k.a. there are starting positions such that for any number $n$, there is a line of play taking $>n$ times) ...
3
votes
0answers
31 views

Maximum number of points you can put on grid $ n\times m$ with no equidistant?

Assume we have a grid of $n\times m$ points. and the distance between two rows or two columns is 1 ( unit ). I have a couple of questions related to this grid:- What is the list of possible length ...
-1
votes
0answers
55 views

diophantine-equations

Why there are no solutions in positive coprime integers for the following diophantine equation $$2x^3 + y^2 = z^k$$ where, (x,y,z) are (pairwise) positive coprime integers, and k is positive integer ...
4
votes
3answers
83 views
+50

Recurrence relation for right-angled triangles stuck-together

Given the image: and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i , i = 1, 2, 3, · · ·$ can be arbitrarily picked. How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$? I ...
1
vote
2answers
131 views

The Probability Riddle

While working on a mathematical model we have come across a problem that seems easy yet has a bunch of intelligent, mathematically trained people start doubting themselves :). Riddle us this... ...
0
votes
1answer
17 views

Is a series that contains the index term a function of the same series without the index term?

Can it be shown that $U_{2} = \sum_{i=1}^{n} [i*g(Y_{i})]$ is a function of $U_{1}=\sum_{i=1}^{n} g(Y_{i})$ ? My intuition tells me that this is not true because of the changing (for lack of a ...
1
vote
1answer
29 views

Why is 438579088 a perfect digit-to-digit invariant?

A PDDI (perfect digit-to-digit invariant) is a number which is also the sum of each of it's digits raised to them self. My main problem with the number 438579088 being a PDDI is the 7th number - 0. ...
3
votes
1answer
43 views

Tiny arithmetic trigonometry anomaly

$1.96\sin(149^\circ) + 1.00842\sin(203^\circ) + 0.61446\sin(285^\circ) = 0.02193075901$ But if I calculated each of the terms separately, then add them together, I get a result that is a tiny bit ...
1
vote
1answer
26 views

Propability of M-faced dice rolling a greater result than an N-faced dice ( M<N) and expected value of the absolute difference?

Hello I have a game-mechanic which goes like this: Two-sides are rolling numbers. The defender rolls between [1,Armor_Value] , attacker rolls between [1,Damage_Value]. If the attacker_roll> ...
8
votes
0answers
150 views

Mathematical properties of two dimensional projection of three dimensional rotated object

Please be gentle as I do not have any degree in maths. By using a compass/straighedge method to construct Metatron's cube, a regular dodecahedron can be inferred from intersecting points. I'm looking ...
4
votes
1answer
84 views

Mystical looking graphs (three-dimensional rotating hearts)

Plop the following into Google: $$ 2-\sqrt{1-x^2-(y-|x|)^2}\cos(30(2-x^2-(y-|x|)^2)),\tag{1}\\ \text{$x$ is from $-1$ to $1$, $y$ is from $-1$ to $1.5$, $z$ is from $1$ to $2$} $$ Here is the result ...
3
votes
3answers
88 views

Choosing new teammates

My sister gave me a combinatorical riddle. It doesn't appear to be hard, but I ask you if my thoughts are right, just for certainty. Here it is. Assume you belong to a group of $100$ people, and ...
2
votes
6answers
82 views

Prove that the series $\sum_{1}^{\infty}\frac{k}{(k+1)(k+2)(k+3)}$ converges and find its limit

I try to split the summand into differences, but that seems to be a futile way in our case right here, because the numerator is $k$, instead of a given number. A closely-related series, say ...
32
votes
9answers
797 views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
2
votes
2answers
44 views

Are there integers a, b, c, d generating four right triangles with integer sides?

To make this more precise, we are looking for four (ETA: distinct) positive integers $a$, $b$, $c$, and $d$, such that $\sqrt{a^2+b^2}$, $\sqrt{b^2+c^2}$, $\sqrt{c^2+d^2}$, and $\sqrt{d^2+a^2}$ are ...
0
votes
1answer
21 views

How to get real value after discount.

Suppose my item real value is 100. and i have given discount 10 % to my customer. Now the changed value of item id 90. If i set value 110 and give 10% discount then i got result 99. but i need result ...
2
votes
1answer
32 views

Would the powerset of $\mathbb{Z}$ also not denumerable?

Would the powerset of $\mathbb{Z}$ also be not denumerable?, Since Cantor's theorem says that the $\mathbb{N}$ is denumberable but the powerset of $\mathbb{N}$ is not denumberable because there does ...
0
votes
2answers
82 views

How would you prove that $2^{n-1} > n!$?

How do i prove that $2^{n-1} < n!$ for all $n \ge 1$ This is my proof: Base case: Let n=1 then $2^{1-1} =1$ is the same on the right side so it holds Inductive step: let $k \le 1$ we assume that ...
0
votes
1answer
26 views

How to tell if a function and a composite function is onto or one to one

For each of the following, f : A → B, g : B → C. Which one are true and which ones are false? So far i have, f is onto but g ◦ f is not onto. (False) f is 1-1 but g ◦ f is not 1-1. (False) g is onto ...
1
vote
2answers
29 views

How to prove if $A \times C \subseteq B \times D \implies A \subseteq B$

My proof Given $(x,y) \in A \times C \implies x \in A$ and $y \in C$ since $A \times C \subseteq B \times D$ then $(x,y) \in B \times D$ then $x \in B$ and $y \in D$ since $x \in A$ and $x \in B$ ...
0
votes
2answers
22 views

How to prove intersections and subsets of sets

Simple proofs for these are pretty straight forward such as proving if two sets are equal then they are subsets of each other or if you want to show one set is a subset of the other just show that ...
0
votes
0answers
23 views

How can you prove that a collection of union set is equal to the set of natural numbers?

For example $\bigcup_{n\in \mathbb{N}}A_n=\mathbb{N}$ My proof, to prove that two sets are equal i must show that they are subsets of each other. I understand how to show $\bigcup_{n\in ...
2
votes
1answer
50 views

Find $a_{n}$ if $(a_{n})$ is a sequence such that $a_{1} := 1$ and $\frac{1}{a_{n+1}} = \frac{2}{a_{n}} + 3$ for $n \geq 2$?

This problem is weird. By the initial condition $a_{1} = 1$ we have $a_{2} = \frac{1}{5}$ and so on. But is there really a pattern for $a_{n}$? I guess this problem is that kind of problems that ...
1
vote
2answers
43 views

Find an integer $k$ such that $a_{k} = 2^{261}$?

Let $a_{1} := 2$ and $$a_{k} := \frac{2^{(k+1)(k+2)/2}}{\prod\limits_{j=1}^{k-1}a_{j}}$$ for all integers $k \geq 2.$ The problem is to find an integer $k$ such that $a_{k} = 2^{261}.$ The ...
15
votes
2answers
310 views

Do any of these sequences have infinitely-many distinct iterates under run-length substitution?

Let $$S = \{x \in \{1,2\}^\mathbb{N}: \ \text{every run in }x\text{ has finite length}\}$$ and define $$T: S\to \mathbb{N}^\mathbb{N} $$ such that for any $x\in S$, ${T}x$ is the sequence of ...
0
votes
1answer
49 views

How to prove that there are no integers a,b such that $b^2=4a+2$

How to prove that there are no integers a,b such that $b^2=4a+2$ This seems like a very simple prof but when i tried to work through it i keep on hitting walls. I tried to prove this by ...
0
votes
1answer
33 views

Anyone know of a good composite number counter?

I am looking for a chart that would show how many composite numbers there are under "n" broken out by how many factors they have. Has anyone seen a chart like this? Example information I am looking ...
0
votes
1answer
40 views

Implications of redefining base natural logarithm constant e

Disclaimer: I'm no math expert! I understand that the constant $$e$$ is expressed as follows: $$e = \sum_{n=0}^{\infty} \frac1{n!} = 1 + \frac1{1*1} + \frac1{1*2} + ...$$ What would be the ...
1
vote
1answer
37 views

Name of this type of plot? Does anyone know how to produce it

Does this type of polar plot have a name? Does anyone know how to produce it in octave 3.8.1 which is compatible with matlab? Link to site
0
votes
1answer
15 views

Price Difference Question. Unable to derive 3rd equation

A man buys Rs.7 apples. When after a few days the price of the apple goes down by Rs.2, he buys 6 more apples. Both the amounts were in whole numbers, were two digits and has the same digits. Find ...
-2
votes
1answer
37 views

When A was half B's age, B was one fourth A's age.

The question goes like this : A told B, "When I was half your present age, you were one -fourth my present age". If A is currently 60 yrs, how old is B ? I am unable to form equations for the ...
1
vote
0answers
16 views

Discrete Model Finding Stability

For the discrete model $$x_{t+1} = (\lambda +1)x_t +x_t^3$$ Draw a bifurcation diagram (expressing the equilibrium vs $\lambda$ for values of $\lambda$ near zero. I have the bifurcation diagram. It ...
2
votes
1answer
45 views

Convergence in probability iff convergence for every bounded continuous function

I'm trying to show the following: $X_n \overset{p}{\to}X \iff f \circ X_n \overset{p}{\to} f \circ X$ for every continuous, bounded function $f$. I can show ($\Rightarrow$) already using the usual ...
8
votes
5answers
215 views

What multiples of $d$ are still multiples of $d$ when they have their digits reversed?

I teach at a school for 11 to 18 year olds. Every term I put up a Challenge on the wall outside my classroom. This question is one that I have devised for that audience. I think that it is quite ...
3
votes
0answers
45 views

Conway's Game OF Life maximum periods on a set x by x game board.

I have taken interest in Conway's Game of Life and want to know if you guys can help me with a mathematical problem :) That is what this website is for right? You need to be familiar with the rules ...
4
votes
5answers
98 views

Find all possible values of $ a^3 + b^3$ if $a^2+b^2=ab=4$.

Find all possible values of $a^3 + b^3$ if $a^2+b^2=ab=4$. From $a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(4-4)=(a+b)0$. Then we know $a^3+b^3=0$. If $a=b=0$, it is conflict with $a^2+b^2=ab=4$. If $a\neq0$ ...
0
votes
1answer
32 views

How to prove that A $\subseteq$B $\implies$ |A|$\le$|B|?

How to prove that A $\subseteq$B $\implies$ |A|$\le$|B|? I know that for |A|$\le$|B| there has to be a function f:A $\mapsto$B which is an injective function. But i get stuck because the sets A and B ...
10
votes
4answers
213 views

Tricky 3d geometry problem

We have a cube with edge length $L$, now rotate it around its major diagonal (a complete turn, that is to say, the angle is 360 degrees), which object are we gonna get? Astoundingly the answer is D. ...
2
votes
0answers
49 views

What is the best way to master my algebra skills without taking an algebra class?

I was in advanced math my entire life. I got through all the math I needed for my original degree. 8 years later here I am changing degrees and I need more math. I just took calculus I and I passed ...
-1
votes
1answer
21 views

How to prove by induction with a set of equivalence sets?

For example Prove by induction that the operation of raising to the power m$\in$ $\mathbb{N}$ is well defined in $\mathbb{Z}_n$ $\forall$m$\in$ $\mathbb{N}$ $\forall$[x]$\in$ ...
0
votes
2answers
26 views

how to prove that $x^2 + y^2 =1$ is injective and surjective depending on the restrictions?

Suppose we have $S=\{(x,y) \in [-1,1]\times[0,1]: x^2 + y^2 = 1\}$ I know this is a function since the domain(s)= $[-1,1]$ and I know this should be surjective and injective since the restriction ...
0
votes
2answers
34 views

There's addition, multiplication and exponentiation. Is there another “level” after exponentiation? [duplicate]

I guess they all can be broken back down into addition but I just have always had this burning question if there was some other mystery operator after exponentiation.
-2
votes
0answers
45 views

A Flea And A Wallaby [duplicate]

. The tail of a giant wallaby is attached by a giant rubber band to a stake in the ground. A flea is sitting on top of the stake eyeing the wallaby (hungrily). The wallaby sees the flea leaps into the ...
-2
votes
1answer
66 views

Puzzle: Players A,B,C,D are in a line

Players A,B,C,D stands in a line. Players A, D do not move. round 1: player B moves one distance closer to the midpoint of A,C round 2: player c moves one distance closer to the midpoint of B,D ...
0
votes
1answer
26 views

How to find a relation when given the distinct equivalence classes?

For example I am not sure how to approach this type of problem. I know that the equivalence classes partition $A$. Suppose $[a]= \{1,4,5\}$, $[b]=\{2,6\}$ and $[c]= \{3\}$. $[a]\bigcap[b]= ...
0
votes
1answer
17 views

How to show that a relation is an equivalence relation given a defined relation

I understand for the most part the conceptual aspects of an equivalence relation. A relation is considered a equivalence relation if it satisfies reflexive, symmetric and transitive properties but Im ...