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

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0
votes
1answer
249 views

Sequence Question from past post

I recently saw a post about sequences. This made me remember some other post someone had posted here on Math.SE. He did not want answers but wanted general ways to tackle them. I did spend an hour or ...
4
votes
5answers
330 views

What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$,$F(1)=b$ and $a,b>0$?

What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$, $F(1)=b$ and $a,b>0$ ? It seems to be simple generalization of Fibonacci sequence but I can't find closed form for ...
5
votes
1answer
59 views

Proving upper limit on number of moves?

In a $(2n-1)\times(2n-1)$ square grid every small square is marked with Up or Right or Down or Left arrow. An example would be $$\begin{array}{|c|c|c|}\hline\\\leftarrow & \rightarrow & ...
2
votes
4answers
321 views

Can you simulate any probability with biased coin throws?

What you're given: $p \in (0,1)$, but you don't know the value of $p$. You have an algorithm $\mathcal{A}_p$ that returns $1$ with a probability of $p$ and $0$ with a probability of $(1-p)$. You ...
14
votes
3answers
320 views

Integer sequences which quickly become unimaginably large, then shrink down to “normal” size again?

There are a number of integer sequences which are known to have a few "ordinary" size values, and then to suddenly grow at unbelievably fast rates. The TREE sequence is one of these sequences, which ...
1
vote
1answer
123 views

A follow-up question on an arithmetic function satisfying a certain inequality

In the MSE question here, I asked whether the inequality: $$\frac{a}{b} + \frac{b}{a} < \frac{f(a)}{f(b)} + \frac{f(b)}{f(a)}$$ would imply $a < b$ (where $f(x) \in \mathbb{N}$ is a function ...
6
votes
2answers
259 views

How to solve this puzzle?

There are $N$ consecutive doors. Two players 'B' and 'J' plays a game. Both take turns alternately, and in each turn a player can open any one door. They define a block of 3 consecutive open doors as ...
7
votes
1answer
259 views

A good book for short problems

What is a good book for problems which can be done without much mathematical background? I don't mean IMO-level, since those questions generally require a fairly big amount of mathematical knowledge, ...
1
vote
1answer
140 views

Different number of apples are given to five children such that any $3$ receive more apples than the remaining $2$

A friend asked me this: A woman gave a different number of apples to each of her five children. Any three of her children together received more apples than the remaining two children. What is the ...
1
vote
1answer
350 views

What is the minimum number of moves of solve the puzzle?

There is board in which there are $m\times m$ boxes each assigned an a non zero integer except one box which is marked as $0$ and is treated as vacant. Only the vertical and horizontal neighbors of ...
1
vote
1answer
122 views

A quick question on general mathematics

I have the following question that I am currently unable to satisfactorily answer myself. My question is: Does the inequality $$\frac{a}{b} + \frac{b}{a} < \frac{f(a)}{f(b)} + ...
-2
votes
1answer
95 views

Solve for $x$: $\sqrt{12} - \sqrt[3\leftroot1]{720} = \sqrt{x}$

I want to solve for $x$ Here's the question $$\large \sqrt{12} - \sqrt[3\leftroot1]{720} = \sqrt{x}$$ I need to find the value of $x$ Help!
4
votes
1answer
1k views

How many different game situations has connect four?

In the game connect four with a $7 \times 6$ grid like in the image below, how many game situations can occur? Rules: Connect Four [...] is a two-player game in which the players first choose a ...
0
votes
1answer
76 views

In a set we have $a(b+c)=ab+c$. What is it?

suppose $A\subseteq \mathbb{N}$ and for any $a,b,c\in A$ with $a<b<c$ we have $$a(b+c)=ab+c$$ what are all $A$ with this property?! here $\mathbb{N}=\{1,2,3,...\}$.
3
votes
1answer
304 views

Triangle from a given rectangle

We are given a set of (marked) points in a 2D coordinate system and function $f(x,y)$ which counts number of points marked in the rectangle $(0 , 0), (x , y)$ - where $(0 , 0)$ if down-left corner, ...
5
votes
3answers
159 views

Does $p\mid f(m)+f(n)\leftrightarrow p\mid f(m+n)$ imply $f(m+n)=f(m)+f(n)$?

Let $f:\mathbb{N}\to\mathbb{N}$ be a function such that: $$(\forall p: \mathrm{~prime~})(\forall m,n\in\mathbb{N})(p\mid f(m)+f(n)\leftrightarrow p\mid f(m+n))$$ is $f$ linear? by linear I mean: ...
9
votes
0answers
181 views

What turmite runs the longest before becoming predictable?

When looking at 2D Turing machines, many of them eventually become predictable. For example, Langton's Ant, the champion 2-color 1-state turmite, develops a highway after 10,000 steps. Predictable ...
3
votes
2answers
367 views

Are numbers with repeating patterns in their decimal expansion (e.g. $0.123123123\ldots$) rational?

There's a question that I've been thinking about for quite some time now. We all know that numbers with infinite decimal expansion such as $0.\overline{3}$ or $0.\overline{1}$ are not necessarily ...
2
votes
1answer
40 views

least number of planes intersecting a finite number of points in space, but not intersecting origin.

Let $$\mathbb{R}^*=\mathbb{R}-\{0\}$$ and $$N=\{0,...,n\}$$ and $$\mathcal{M}=\{ A\subseteq \mathbb{R}^3\times\mathbb{R}^* \mid (\forall\mathbb{x}\in N^3:\mathbb{x}\ne 0)(\exists(\mathbb{a},d)\in ...
10
votes
4answers
357 views

A problem about symmetric relations on finite sets.

We have these assumptions: $X$ is a finite set. $\sim$ is an irreflexive symmetric relation on $X$. for any subset $Y\subseteq X$ we define $$\mathcal{Cl}(Y)=\{A\subseteq Y\mid(\forall a,b\in ...
3
votes
4answers
195 views

Are $4ab\pm 1 $ and $(4a^2\pm 1)^2$ coprime?

Let $a\ne b$ be two positive integers. Are $4ab+1$ and $(4a^2+1)^2$ coprime always? Can you find $a$ and $b$ such that they are not coprime? Edit: It has been proved that $4ab-1$ is not a divisor ...
2
votes
1answer
154 views

How can I solve an Euler-Lagrange equation that satisfies certain conditions

The idea comes from a recreational math problem-- Place two identical coins side by side and roll one along the circumference of another without slipping, how many revolutions will the rolling coin ...
10
votes
4answers
357 views

Number theory fun problem

Say $a,b > 2 $ are integers. Then we have that $2^a + 1$ is not divisible by $2^b - 1$. Any thoughts on how to tackle this problem???
3
votes
2answers
1k views

Why does a Penrose Stair seem to be correct?

Penrose Stairs seem to be a locally valid but globally inconsistent contraption. I have a couple of questions: Is it physically realizable? In other words, is it possible to build a 3-D structure of ...
0
votes
0answers
96 views

Upside down bell shaped graph moving with respect to $x$ axis

Basically, I wanted to create a "loser" graph. Along the $x$ axis we'd have time and along the $y$ axis is how much of a loser someone is. I want the graph to be an upside down bell shaped graph ...
0
votes
3answers
124 views

Given $n+1\mid2\sum_{k=1}^{n}{a_k}$, find $a_k$.

Let $m$ be a positive integer. There are only 2 finite sequences of positive integers like $a_1,a_2,...,a_m$ such that $$(\forall n \leq m)\left(n+1\mid2\sum_{k=1}^{n}{a_k}, \quad a_n\in [1,m],\quad ...
3
votes
2answers
280 views

Conjecture I came up with

For each number translated into binary $0$, $1$, $10$, $11$, $100$, $101$, $110$, $111$, $1000$, ... find a number where, when you take the length of the binary number, the binary number and the ...
3
votes
1answer
517 views

Correct Path To Castle Riddle [duplicate]

I'm working on the following riddle that I found to be kind of interesting, but I can't figure it out. The problem is as follows: A prince visits an island inhabited by two tribes. Members of one ...
1
vote
1answer
56 views

What is the max single day score on jeopardy?

So, I was trying to figure out what the max score on jeopardy is for a single day. what I did was account for the daily doubles at the very end with the lowest value category, (to save on the 1000's), ...
2
votes
1answer
92 views

What is the difference in chance, based on foreknowledge of the resolution?

I'm working to understand the differences between Odds, Probability and Chance. I've come up with a hypothetical situation to show where I'm having a bit of an issue. Chad shuffles a standard deck ...
5
votes
3answers
215 views

What is the most mathematically sound way to define the “damage per second” for a weapon?

Consider a weapon firing shots every $f^{-1}$ seconds (i.e. $f$ is the weapon's fire rate). Each shot deals $n$ damage to is target. Consider another weapon firing every $3f^{-1}$ second, but dealing ...
0
votes
1answer
70 views

Problem with probability

If we choose randomly an infinite, countable set of disks in $\mathbb R^2$, what is the probability that intersection of every pair of disks from the set is an empty set? EDIT: Because the problem in ...
78
votes
1answer
3k views

$4494410$ and friends

The number $4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
6
votes
2answers
410 views

How to Construct orthogonal circles?

Let $C_{1}$ be a circle of unit radius. Let A and B be two points inside $C_{1}$. Now I want to construct another circle $C_{2}$ such that A and B lie on $C_{2}$ and $C_{2}$ is orthogonal to $C_{1}$ ...
1
vote
0answers
39 views

maximising the frequency of mode.

I have 4 numbers 5,5,3,1. Now I have the number 5, which I can distribute in any manner to ...
0
votes
1answer
86 views

Compute Birthdate From Age

I have an interesting problem. I want to compute a birthdate (Month Date, Year) from a given age. I understand that subtracting the given age from the current date will supply the year of birth, but ...
2
votes
0answers
140 views

Dates and times with no repeated digits?

I have a digital clock that shows the date and time like this: $$ \mathsf{YYYY-(M)M-(D)D\qquad (H)H:MM \; [:SS]} $$ That is, the seconds display is optional, and if the month or day or hour is ...
5
votes
2answers
830 views

Sailors, monkey and coconuts

Five sailors and a monkey were shipwrecked on a deserted island, and they spent the first day gathering coconuts for food, piled them all up together and went to bed. But when they were all asleep one ...
1
vote
2answers
113 views

Help on French Math Education Paper

I am looking for very basic (probably I should say very elementary) papers in french designed for elementary school teachers and elementary school educators. I would appreciate if someone can provide ...
2
votes
0answers
121 views

A system of linear equations in integer squares - solvable?

I consider this more a "recreational math" problem, possibly lacking a solution (because it stems from the question of "magic square of squares") or simply intractable with reasonable effort. ...
11
votes
3answers
530 views

Arc sums for a circle of $k$ positive integers whose total sum is $n$

This problem got me thinking about the following more general scenario: Suppose you have $k$ positive integers with total sum $n$, and you arrange them in a circle. Given such an arrangement, you ...
6
votes
2answers
263 views

Concise naming-scheme for polyominoes

There is a neat naming scheme for pentominoes based on letters they resemble. Is there a generalized naming scheme for polyominoes? If there isn't a canonical one, can you think of a good one? ...
6
votes
1answer
3k views

Expected number of calls for bingo win

Before I begin, I did a search through math.stackexchange and came across two previous attempts to get people to solve probability problems involving bingo. Neither produced a response. So what ...
-1
votes
1answer
181 views

Buddhabrot Sewing machine [closed]

The Buddhabrot fractal traces the orbits of the points outside the Mandelbrot set. What design considerations need to be taken into account to create a computerised sewing machine that traces out ...
9
votes
1answer
231 views

Is the number of alternating primes infinite?

I'm not sure if the recreational-mathematics tag is appropriate, but this problem came up during a practice Putnam seminar so maybe? The problem: Say that a positive integer is alternating if ...
5
votes
3answers
1k views

Cyclic sums — How do you use them?

Can someone give me an example of how cyclic sums are used? I don't really understand how they're used in problem-solving. For example, $$\sum_{a,b,c}a^2$$ Any help would be appreciated, and I'm not ...
11
votes
2answers
744 views

Is there a collection of alternative mathematical notation? (Semi-soft Question)

I'm interested in alternative systems of notation for mathematics. I've often heard how mathematical notation is illogical, inconsistent, filled with grandfather clauses that serve no purpose, and ...
2
votes
1answer
75 views

Question on pathological sine function

Some years ago I came across what was defined as "pathological" function defined as: $$ f(x)=\sum_{k=1}^\infty \frac{1}{k^2}\cdot \sin\left(k!\cdot x\right) $$ It was mentioned (in an article I ...
15
votes
5answers
1k views

Is high school contest math useful after high school?

I've been prepping for a lot of high school math competitions this year, and I was just wondering if all the math I learn would actually mean something in college. There is a chance that all of it ...
5
votes
4answers
153 views

Example of a question that would seem not to have enough information for an answer

Looking for an example of a question that would seem not to have sufficient information for an answer, or a question that the solution would not require (or maybe even maybe hindered ) by the extra ...