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

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2
votes
2answers
92 views

What is sum of occurrences of zeros, at the end of integers, up to number $n$?

What is sum of occurrences of zeros, at the end of integers, up to number $n$ ? Let's call this function $O(n)$ Examples : $1,2,3,4,5,6,7,8,9,10,$so $O(10)=1$ $1,...,20,$ so $O(20)=2$ $O(100)=11$ ...
6
votes
3answers
621 views

Using the digits $7,7,7,7,1$ and the operators $+,-,*,/$ to make a formula which equals $100$

I know the answer is $(7+7)*(7+(1/7))$ or a more ghetto answer is $177-77$. I'm not interested in the answer, more in the problem itself. What is the name of this class of problem? Is there a ...
1
vote
2answers
153 views

Interesting and irritating problem.

How to deal this problem. I found this problem in math competation in 2012. But, I could not solve. Could you help me... Uncle John has taken blood pressure drops for a long time according to the ...
2
votes
4answers
684 views

Number of positive integers $\le n$ with different digits.

An integer $m$ is acceptable iff in it's decimal representation all digits are different. For example $9876543210$ is the largest acceptable integer. For each $n\in \Bbb N$, $\theta(n)$ is the number ...
2
votes
1answer
107 views

Does this Graph converge in a finite number of steps? How fast is it?

Suppose you have a finite, planar Graph $G = (V,E)$ and a function $f_0:V \rightarrow \mathbb{Q}$. Now you define the function $f_{i \in \mathbb{N}}$ like this: $$f_i(v) := \frac{f_{i-1}(v) + ...
1
vote
2answers
173 views

Easy Probability Problem

I was told the following probability problem: While doing a math problem today at the contest the probability of Annie, Tom and Karen getting the problem correct first is 1/7, 1/2, and 5/14 ...
7
votes
2answers
150 views

Determining the strength of an abstract army

Lets imagine we have two armies, represented by lists of pairs of positive numbers, like this: [($attack1$,$defence1$),($a2$,$d2$)...($an$,$dn$)] face each other in combat. The rules of combat are ...
14
votes
4answers
501 views

Gambling puzzle

A math friend of mine showed me this strange gambling puzzle. There is a button in a casino and every time you press it you can win either $1$ or $0$ dollars. The probability of winning $1$ dollar ...
4
votes
0answers
80 views

Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)

Background: This question arose purely recreationally and doesn't really fit into any context that I know of. Let $A \sqcup B = \mathbb{R}$ be a partition of the ...
16
votes
1answer
630 views

What are the symmetries of a colored rubiks cube?

Technically the symmetry group of the rubiks cube is the symmetry group of the cube with all its label peeled off. The normal rubiks cube with all its faces painted different colors has trivial ...
2
votes
1answer
183 views

Checking Sudoku - sufficient sums

Are the following condition sufficient for checking if solution of Sudoku with (extended output) is valide : sum of values in each row, column and subsquare is equal to 45 and sum of squares of ...
2
votes
1answer
1k views

How to calculate $ 1^k+2^k+3^k+\cdots+N^k $ with given values of $N$ and $k$? [duplicate]

Here $ 1<N<10^9$ and $0<k<50$ So we have to calculate it in order of $O(\log N)$.
4
votes
3answers
278 views

Dissecting an equilateral triangle into equilateral triangles of pairwise different sizes

It is know that a square can be dissected into other square such that no two of the squares have the same size. This is the simplest dissection of that kind: Is it also possible to dissect an ...
3
votes
2answers
166 views

Method of expressing the product of first n integers

I am trying to show a pattern whereby the first term is 140 the next term is $140\times139$ and the next $140\times139\times138\dots$ I can do this as follows: $\frac{140!}{(140-n)!}$ but that doesnt ...
12
votes
1answer
298 views

Algebraic structures associated to flexagons?

Flexagons strike me as objects that would admit investigation in a first course in modern algebra. I'm surprised to be unable to find a reference discussing flexagons using modern algebra language. ...
2
votes
1answer
46 views

Total length of pieces after splitting

If I consider the number line from $0$ to $n$ and cut it into $x$ pieces, it is well known that there is at least one stretch of length at least $n/x$. My question is what is the minimum total length ...
538
votes
152answers
34k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's ...
2
votes
2answers
281 views

Help me, was given a Mathematical problem to figure out, but the answers told me it was just demorgans law in c code.

Was given this to figure out. He said its mathematical, no its not homework. More of one guy trying to prove he is smarter than me. The code: ...
6
votes
5answers
206 views

How can I find the value of $a^n+b^n$, given the value of $a+b$, $ab$, and $n$?

I have been given the value of $a+b$ , $ab$ and $n$. I've to calculate the value of $a^n+b^n$. How can I do it? I would like to find out a general solution. Because the value of $n$ , $a+b$ and $ab$ ...
28
votes
2answers
445 views

Can a collection of points be recovered from its multiset of distances?

Consider $n$ distinct points $x_1,\dots,x_n$ on $\mathbb{R}$. Associated to these points is the multiset of all distances $d(x_i,x_j)$ between two points. Suppose one is only handed this multiset (you ...
1
vote
2answers
279 views

Calculating the same exchange rate to make an investor indifferent

If we consider an American investor in 2009 with 160 million Dollars to place in a bank deposit in either America or the UK. The (1 yr) interest rate on bank deposits is 6 percent in the UK and 1 ...
5
votes
2answers
156 views

Show that there is a star for which there are less stars in its column than in its row.

In a table there are n columns and m rows, n > m. Some cells are marked by stars, and in each column there's at least one star. Show that there is a star for which there are less stars in its column ...
1
vote
1answer
152 views

Primes in a certain arithmetic progression

Primes $=_m 1 $. For any positive integer m, prove that arithmetic progression $$1 +m, 1 + 2m, 1 + 3m, ... $$ contains infinitely many primes. How can we prove that it suffices to show that for ...
3
votes
0answers
780 views

Turning radius of a vehicle

What's the minimum turning radius of a vehicle, rectangular in shape, with length l units and width w units? One key point to consider, would be that, the inclination of the front wheels can be ...
18
votes
3answers
2k views

Why should Rubik's cube get attention from mathematicians?

I've seen a lot of math debate, calculations and other stuff related to Rubik's cube lately, but I don't really understand why is it important, why should anyone spend time asking and answering ...
0
votes
1answer
255 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
423 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
61 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 & ...
3
votes
4answers
421 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
352 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
128 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
273 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
292 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
141 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
522 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
126 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)} + ...
-1
votes
1answer
98 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
2k 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
81 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
355 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
161 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: ...
10
votes
0answers
202 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
416 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
41 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
366 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
203 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
168 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
406 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
2k 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
105 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 ...