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

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5
votes
1answer
233 views

Trisecting a paper using hand and without using a ruler or compass [duplicate]

This is a practical problem born while folding a paper. We can bisect a paper by using only hand. $\star$ Easy, fold it so that the two ends (of the length) coincide and press the paper to get ...
591
votes
25answers
106k views

A “simple” 3rd grade problem…or is it?

So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long ...
7
votes
1answer
166 views

Tiling an $n\times n$ Grid

Given an $n\times n$ grid, and $2\times 2$ checkered tiles (white in the upper left and bottom right corners, and black in the upper right and bottom left corners), what is the smallest number of ...
0
votes
0answers
62 views

What's the difference between a $2$-sided and $2$-sided strip polytan

There are $14$ $2$-sided tetratans and $13$ $2$-sided strip tetratans. The sets are identical, except the square is missing in the strip version. My best guess is that for strips, no vertex can have ...
3
votes
0answers
44 views

How to find the point in a closed geometrical figure which maximizes the “direct-line-of-sight function”

To expand upon the title, and put it in clear terms, I phrase the problem thusly: Consider the interior of any continuous, closed, non-self-intersecting curve in the plane. (I'm not sure if I'm ...
0
votes
2answers
148 views

Is $f(x)f(y)=f(x+y)$ enough to determin $f$? [duplicate]

I had a discussion with a friend and there it came up the question whether $f(x)f(y)=f(x+y)$, $f(0)=1$ and the existence of $f'(x)$ implies that $f(x)=\exp(a x)$. This seems very reasonable but I ...
1
vote
3answers
577 views

Weather station brain teaser

I am living in a world where tomorrow will either rain or not rain. There are two independent weather stations (A,B) that can predict the chance of raining tomorrow with equal probability 3/5. They ...
47
votes
14answers
2k views

How to entertain a crowd with mathematics? [closed]

I am a high school student who follows a university level curriculum, and recently my teacher asked me to hold a short lecture to a crowd of about 100 people (mostly parents of my classmates and such, ...
1
vote
2answers
156 views

How can be done by the method of mathematical induction?

We are given that $P(x+1)-P(x)=2x+1$ We also know that $P(0)=1$ We want to prove that $P(2004)=(2004)^2 +1$ Can someone explain how can be solved with mathematical induction? Thank you in advance!
54
votes
2answers
17k views

Predicting Real Numbers

Here is an astounding riddle that at first seems impossible to solve. I'm certain the axiom of choice is required in any solution, and I have an outline of one possible solution, but would like to ...
12
votes
1answer
169 views

Evaluation of a slow continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
3
votes
1answer
135 views

A game involving points in the integer plane - who wins?

I am running a workshop on puzzles and problem solving over the weekend and thought that it might be a good idea to get people engaged by phrasing some interesting mathematical results in terms of ...
5
votes
2answers
514 views

Why does the strategy-stealing argument for tic-tac-toe work?

On the Wikipedia page for strategy-stealing arguments, there is an example of such an argument applied to tic-tac-toe: A strategy-stealing argument for tic-tac-toe goes like this: suppose that the ...
5
votes
1answer
233 views

Does there exist a positive integer $n$ such that it will be twice of $n$ when its digits are reversed?

Does there exist a positive integer $n$ such that it will be twice of $n$ when its digits are reversed? We define $f(n)=m$ where the digits of $m$ and $n$ are reverse. Such as ...
1
vote
2answers
167 views

Gold Coins and a Balance

Suppose we know that exactly $1$ of $n$ gold coins is counterfeit, and weighs slightly less than the rest. The maximum number of weighings on a balance needed to identify the counterfeit coin can be ...
34
votes
1answer
687 views

Proving that $x$ is an integer, if the differences between any two of $x^{1919}$, $x^{1960}$, and $x^{2100}$ are integers

For a specific real number $x$, the difference between any two of $x^{1919}$, $x^{1960}$ , and $x^{2100}$ is always an integer. How would one prove that $x$ is an integer?
5
votes
3answers
701 views

palindromic squares of palindromes

This question is inspired by Google's recent programming competition (modified slightly for ease of exposition). For a given $n$, one of the problems was to find all positive "fair" integers $k$ less ...
1
vote
2answers
319 views

Lengths of increasing/decreasing subsequences of a finite sequence of real numbers

Let $x_1,\ldots,x_n$ be a finite sequence of real numbers. Let $f(\{x_i\}_{i=1}^n)=f(\{x_i\})$ be the length of the largest non-decreasing subsequence, and let $g(\{x_i\})$ be the length of the ...
5
votes
3answers
429 views

Distribution of palindromic numbers

We all know what a palindromic number is, it is a number which is the same, independent from which side we read it, for example 101, 202, 33733,.... It is also clear that there are infinity many ...
5
votes
3answers
404 views

Prove that $n+1$ elements of a set will contain a co-prime pair

Suppose $P$ is a set of $n + 1$ integers selected from $1,2,3,...,2n + 1$. Then how can we show $P$ contains two coprime integers? The result holds if $P$ contains only $n$ integers?? Added Let ...
2
votes
3answers
787 views

What is the value of this repeated square root: $\sqrt{1\sqrt{2\sqrt {3 \sqrt{4\cdots}}}}$

Find the value of $$\sqrt{1\sqrt{2\sqrt {3 \sqrt{4\sqrt{5\sqrt{6\cdots\sqrt{\infty}}}}}}}$$ What is the absolute value of the root in below question and what does it represent geometrically, I ...
3
votes
2answers
151 views

Recreational number theory problem

Suppose we have a positive integer $n$ that has exactly three distinct prime factors, say $p,q, r$. How can we find a formula for the number of positive integers $\leq n$ that are divisible by none of ...
2
votes
3answers
195 views

How to make a box which has the largest possible volume?

I have sheet metal in form of an equilateral triangle and I want to fold it to make a container for the screws. How should I cut and fold to make the a box with largest volume? Basically I cut the ...
91
votes
8answers
3k views

Probability that a stick randomly broken in five places can form a tetrahedron

Edit (June. 2015) This question has been moved to MathOverflow, where a recent write-up finds a similar approximation as leonbloy's post below; see here. Randomly break a stick in five places. ...
19
votes
2answers
464 views

Evaluation of a continued fraction

Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
3
votes
2answers
93 views

Number theory Exercise

for positive integer $n$, how can we show $$ \sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)} $$ where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)} $ if $n$ is square ...
1
vote
2answers
290 views

How many cubes are left after removing the surface of a cube?

$n^{3}$ cubes are glued together to form one solid cube which is then hung in the air. As time proceeds, the most outer layer of this solid cube begins to dissolve and eventually those smaller cubes ...
6
votes
1answer
233 views

Question on Q&A's

My friend gave a fun problem to me that went as follows: A. For how many of these questions is zero the answer? B. For how many of these questions is one the answer? C. For how many of ...
0
votes
1answer
56 views

Dividing an arbitrary $2-D$ shape with integer area into arbitrary shapes of unit area

The name explains it all. I searched for it in MSE and came across a similar [one] but more simpler1. I was interested to know if we can prove that, i.e., given an arbitrary shape (closed and ...
2
votes
1answer
147 views

License plate consisting of 4 letters and 4 numbers

While doing homework today, the following question popped into my head: Can you easily calculate the amount of unique license plates consisting of 4 letters and 4 numbers in any order? It doesn't ...
4
votes
2answers
254 views

Wolves and chicks puzzle

This problem is from the handheld video game, Professor Layton and the Curious Village. I think the solution is very cool, but more than that, I want to know how to show that the minimum number of ...
1
vote
3answers
780 views

Intersection of chord with circle knowing the length and a point

Let's take a circle with radius R, and center in O (0, 0). We take on this circle a point A with coordinates xA and yA. We know that point A is one of the endings of a chord with length l. Which is ...
3
votes
2answers
362 views

How to form pairs in a group so that each element is coupled with each other only once

My question is derived from real life but I think it's a classic mathematical problem. My uncle wants to organize a group activity with 12 people and wants to start by pairing each person with each ...
3
votes
1answer
569 views

Another puzzle with locks

There is a safe with three locks, like the ones in the hotel rooms that are opened with a "key" which is similar to a credit card. There are three keys, a correct one for one for each of the locks, ...
3
votes
2answers
495 views

Combinatorics riddle: keys and a safe

There are 8 crew members, The leading member wants that only a crew of 5 people or more could open a safe he bought, To each member he gave equal amount of keys, and locked the safe with several ...
0
votes
4answers
516 views

How to verify method used to solve integral was actually the fastest?

Is there any way to verify if the method I chose to integrate (by hand) was indeed fastest, or if there exists some better technique? Can a computer tell me or show me what the fastest method was, ...
0
votes
1answer
313 views

LOVES+LIVE=THERE. How many “loves” are “there”?

This is a problem from Mathematical Circles ( Chapter 0, Problem#17 ). It goes like this:- The answer is that there are 95343 "loves" in "there". Now, this is something that I am unable to ...
6
votes
2answers
935 views

Fun Math Topics, Activities, and Riddles

I'll be teaching college algebra this summer. Last summer when I taught the same course, I finished lectures early (thanks mainly to LaTeX's Beamer package). I want to fill these gaps this time ...
34
votes
2answers
1k views

Do all natural numbers have a nonzero multiple that is a palindrome in base 10?

Some natural numbers have a nonzero multiple that is a palindrome in base 10. For example, $106 \times 2 = 212$, which is a palindrome, and $29 \times 8 = 232$, which is also a palindrome. Aside ...
3
votes
1answer
404 views

line of mathematicians guess their own hat color out of c colors

There is a common problem in which a long line of N mathematicians are each given a hat that is either red or blue. They cannot see their own hat but can see all in front of time and can hear any ...
0
votes
3answers
114 views

Problem - Sum of digits

For every numbers with $3$ digits you calculate the product of the numbers. After that you take the sum of the products, what number do you get? I didn't know how to do this exactly. What would be ...
1
vote
2answers
113 views

What is the angle the car turned?

A car drives a route through town as indicated alongside, crossing square M five times while doing so. How large is the total angle his car turned through when it has completed the route? I just ...
1
vote
1answer
80 views

Is there efficient way of finding last number in following sequence

"Imagine the sequence lying on a circle. Take every second number in the sequence. Continue the process until you finish" Is there efficient way of finding last number in following sequence : we ...
2
votes
2answers
159 views

Are there constraint problem calculators?

So I just remembered Lincoln Logs exist, so I found ten giant sets of them on ebay for Buy It Now, and I'm trying to decide what combination of purchases gives me the most logs for the least money if ...
2
votes
0answers
37 views

For which minimal $k$ true is that ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$, ${a}_{i}\in {1,2,3,4,5,6}$?

I've got the following inequality, which bounds Minkowski distance. ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$ and values of ${a}_{i}\in {1,2,3,4,5,6}$ We know all ...
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
645 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
156 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
704 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
108 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) + ...