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

learn more… | top users | synonyms (2)

6
votes
1answer
232 views

Average complexity of random-pick comparison sort

Motivation. Suppose we have a number of images that we want to arrange in a linear order from the prettiest to the ugliest. At our disposal we have a trained aesthete, whom we can show two pictures ...
7
votes
2answers
415 views

Real guessing puzzle

Allow me to propose a modification of a previously asked puzzle. I would like to replace 100 in that puzzle by $\omega$ and replace $99$ by "all but one". A version of the puzzle was also discussed on ...
1
vote
1answer
154 views

Intersecting Circles Theorem (about 1983 AIME #14's solution)

Please consider this problem: http://www.artofproblemsolving.com/Wiki/index.php/1983_AIME_Problems/Problem_14 Now look at solution 2 - it assumes that A, B, and R are co-linear, but does not prove ...
11
votes
2answers
1k views

Optimal Strategy for Bar Dice

This game is played in bars in Wisconsin, USA, but I'm sure variations are played many places around the world. The game has practical value, since once mathematicians figure out the best strategy, ...
1
vote
2answers
321 views

Is there any surprising elementary probability problem that result in surprising solution like the Monty Hall problem?

For recreational purpose, i haven't seen a interesting elemetary probability question quite a while. Is there any surprising elementary probability problem that result in surprising solution like the ...
0
votes
1answer
90 views

In Blackjack, why does the advantage shift to the player when the dealer shows the hole cards? [closed]

Blackjack is played with the dealer showing one card and having the other card turned over. While I understand that the player theoretically has a significantly larger advantage now, I don't know ...
0
votes
3answers
51 views

Different answers from different formulations of combinatorics problem

A certain men's club has sixty members; thirty are business men and thirty are professors. In how many ways can a committee of eight be selected if at least three must be business men and at least ...
4
votes
2answers
134 views

$2^5 \cdot a^b=2,5ab$

I came across this problem in an elementary number theory book, and I think I solved it. Well, the question is posed as $2^5 \cdot 9^2 = 2,592$. Are there any other pairs $a,b \in \mathbb{Z}$ such ...
1
vote
1answer
373 views

How many sepak takraw balls with distinct patterns can we make?

Sepak takraw is a sport native to the Malay-Thai Peninsula. A few days ago, a friend of mine taught me how to make a sepak takraw ball. The ball is related mathematically to a $32$-face semi-regular ...
0
votes
1answer
41 views

Probability, something easy!

Good afternoon people suppose the function $$f\left(x\right)=\frac{sx^{2}}{zx^{2}+1}$$ How many different functions can there be, if numbers s and z are taken randomly from the set $\left\{ ...
0
votes
0answers
40 views

What are the odds of reaching the end of a random choice path without repeat choices?

Given $m$ balls in a container, having $n\le m$ colors, if balls are chosen from the container randomly without replacement until all balls have been removed and the order of choices is considered, ...
2
votes
1answer
154 views

Very tricky probability

I just came by this "easy" question but i am abit worried, ill tell you why. ...
32
votes
5answers
1k views

Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
12
votes
1answer
176 views

Coloring 5 Largest Numbers in Each Row and Column Yields at Least 25 Double-Colored Numbers

This is a question from a very old American Mathematical Monthly, if I recall correctly. It has a very nice solution and illustrates an often useful technique. If it is unsolved after a while, I will ...
4
votes
1answer
2k views

Can you make an equilateral triangle from 3identical trapezoids?

Is it possible to make an equilateral triangle from 3 identical trapezoids? If so, what angles would be needed in the trapezoids?
0
votes
1answer
30 views

Range of Numbers

I have a list of 25 spots and I know Value #1 and Value #25. What is the best way to estimate the values in between: Example: 1: 250,000 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 ...
5
votes
5answers
1k views

Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms?

Why is 11 times the 7th term of a fibonacci series equal to the sum of 10 terms? I was watching scam-school on youtube the other day and this number trick just astonished me. Can someone please ...
1
vote
1answer
56 views

Any other prime numbers that satisfy this condition?

If $a=2$ and $b=3$, then $a^2-1$ is an integer multiple of $b$. Is there any other pair of primes $a$ and $b$ that satisfy this relationship? I don't think so, but can't figure out why not.
9
votes
4answers
319 views

Math games for car journeys

On long car journeys with kids we are all familiar with "I spy" or "Twenty questions". What math related games can one play on a car journey instead that are fun and offer similar variety?
7
votes
1answer
64 views

Halving One in Odd Size Rings

Consider the rings $\mathbb{Z} /n \mathbb{Z}$ where $n$ is odd. Every number is even in such rings. Assume we start with $1$ and keep "halving" until we get back to $1$. What can be said about the ...
4
votes
0answers
154 views

Card game probability

Suppose the following solitaire with a standard deck. I turn four cards visible on the board and on each turn, I remove those suits that appears more than once in the board. Then I fill the board such ...
5
votes
4answers
145 views

How to minimize $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + |z_3 - z_4|^2$?

If $z_1,z_2,z_3,z_4 \in \mathbb{C}$ satisfy $z_1 + z_2 + z_3 + z_4 = 0$ and $|z_1|^2 + |z_2|^2 + |z_3|^2 + |z_4|^2 = 1$, then the least value of $|z_1 - z_2|^2 + |z_1 - z_4|^2 + |z_2 - z_3|^2 + ...
1
vote
0answers
54 views

Finding code in 5-guesses only

Suppose you want to crack a code composed of 4 digits (each between 0 to 5 when repetitions allowed) and you get feedback like in mastermind, how can you find it in less than 5 guesses in an ...
1
vote
2answers
45 views

Can you find nice parameters for a point that has two closest points on a quadratic function?

When you have a quadratic function $$f(x) = a x^2 + bx + c$$ and a point $$P = (x_p,y_p)$$ you can find one or two points on the graph of $f$ that have minimum (euclidean) distance $$d_{P,f}(x) ...
2
votes
2answers
423 views

Aid solving logical puzzle

I have a logical puzzle here that I need to solve. I'm not after just the answer of this puzzle, but also the reasoning and thought-process behind solving it. Any assistance is greatly appreciated. ...
10
votes
1answer
224 views

A congruence in the number of certain ternary strings

Let $a_n$ be the number of ternary strings of length $n$ which do not contain three consecutive symbols that are all different. That is, $$a_n = \Bigl|\bigl\{\,(b_k)_{1\leq k\leq n}\in ...
0
votes
1answer
695 views

Equation For Multiples

So this is probably a super easy question for Math Stack Exchangers. Anyways I can determine the multiples of 3 up to 10 doing this. multiples of 3 up to 10 ...
4
votes
2answers
251 views

Number of cells inside a circle

Suppose we have a circle with diameter `r˙ whose center is in the center of a cell. I would like to calculate how many cells are inside this circle (even if only a fraction of the cell is inside). How ...
16
votes
3answers
590 views

Counting the number of polygons in a figure

I often come across figures like this on the net, or as contest problems, asking to find the number of a specific type of polygon in the figure (triangles, in this case). But I've never really found ...
3
votes
1answer
754 views

Solving circle's radius only knowing angle & lengths of external triangle OR solving for sides of a triangle partial side lengths

Is this possible? Given that I know the length of Y and Z and the angle of X can I figure out the radius A? If I can't without more information, I can produce another set of data X Y Z at a ...
3
votes
1answer
61 views

If a function is smooth is 1 over the function also smooth

If $f(x):\mathbb{R}\rightarrow\mathbb{C}$ is $C^\infty$-smooth. Is $1/f(x)$ also $C^\infty$-smooth? $f(x)\neq0$
3
votes
0answers
88 views

Why is this proof false? (Why is $e^i \neq 1$?) [duplicate]

I found this on MathOverflow: $$e^i = (e^i)^{(2\pi/2\pi)} = (e^{2\pi i})^{1/2\pi} = 1^{1/2\pi} = > 1.$$ I first saw this one many years ago, written on the wall of a bathroom stall in ...
2
votes
1answer
62 views

Area of a right angled triangle is an even integer

Actual Question is : Let $ABC$ be a triangle in the plane such that $BC$ is perpendicular to $AC$. Let $a,b,c$ be the lengths of $BC$, $AC$, and $AB$ respectively. Suppose that $a,b,c$ are integers ...
32
votes
1answer
371 views

Zero-avoiding integers

Let's say an integer $n>2$ is zero-avoiding if, for every $2\leq b < n$, the representation of $n$ in base $b$ has no $0$ digits. (Obviously every $n$ has a $0$ when written in base $n$ and no ...
6
votes
0answers
193 views

Math Behind the Dragon Illusion!

Dragon illusion has been one of the items presented in the 3rd "Gathering for Gardner". This video shows the illusion. What does it have to do with mathematics?
1
vote
2answers
125 views

Is there an algebraic method to concat two numbers?

I'm searching an algebraic way to concat numbers in base $10$. Concatening two numbers is to put side by side their notations. Let $c$ a concatenating function. $c(2,2) = 22$ $c(8,9) = 89$ ...
2
votes
1answer
78 views

Can we make rectangle from this parts?

I have next problem: Can we using all parts from picture (every part exactly one time) to make rectangle? I was thinking like: we have $20$ small square, so we have three possibility: $1 \times ...
7
votes
2answers
621 views

Come up with some fun “equation Limericks”

We were discussing "Limericks" in my Calculus class. Specifically, "equation Limericks". A Limerick is a poem with five lines. The first, second, and fifth lines should have nine syllables each and ...
3
votes
0answers
225 views

Help explain why (or why not) the solution for a in $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)=0$ is 1-2$C$MRB

$C$MRB is approximately 0.1878596424620671202485179340542732. See this and this. $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)$ is formally convergent only when $a =1$. However, if you extend the ...
3
votes
1answer
247 views

Number theory: 2 numbers within a set with same difference

You have the numbers 1,2,3...,99,100. From that set you have to choose 55 different numbers. Show that: There are 2 numbers with a difference 9,10,12,13 Show that there aren't ...
2
votes
0answers
64 views

How to prove $\sum_{n=1}^\infty (-1)^n(x n^{1/n}+y n)=(c-1/2) x-1/4 y$?

Could someone help me prove this theorem where regularization is used so that the sum that formally diverges returns a result that can be interpreted as evaluation of the analytic extension of the ...
4
votes
3answers
120 views

Is there a $3\times 3$ magic square adding up to $7$.

I suspect that there is no magic square with natural number entries (matrix where each row, column and long diagonal add up to the same number) which would add up to $7$. There is no restriction on ...
8
votes
1answer
343 views

Evaluation of the integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$

As it says in the title, I'd like to know how to solve the definite integral $\int_0^1 \log{\Gamma(x+1)}\mathrm dx$. Mathematica gives the answer $\frac{1}{2}\log (2\pi)-1$ but I have no idea how one ...
2
votes
3answers
437 views

hours minutes and seconds hand will make equilateral triangle

When I am going through some aptitude questions I have got this problem How many times the hours minutes and seconds hand will make equilateral triangle in 12 hours of clock I can't understand how ...
3
votes
1answer
120 views

Fair Division: Making the Differences in Players' Valuations Believable

When teaching basic fair division algorithms, the students always propose some simple and (at the first glance) correct solutions for $n$ players, which unfortunately are not correct! The only way I ...
6
votes
2answers
212 views

If $x^3+\frac1{x^2}=1$, what is $x^3+\frac1{x^3}$?

$x^3 + \frac1{x^2} = 1$. Then, $x^3 + \frac1{x^3} = ~?$ $p + \frac1{p^2} = 47$. Then, $p + \frac1p = ~?$
11
votes
1answer
404 views

What am I getting for Christmas? Secret Santa and Graph theory

I live with four people, who thankfully don't spend much time on maths.se. We decided this year that we'd do a Secret Santa. We can represent the arrangement of who's buying for whom using a directed ...
3
votes
2answers
122 views

Argument over an induction proof

My friend gave me a problem. Define a sequence $\langle a_n \rangle$ by the recurrence relation :$$ a_{n+2} - 6a_{n+1} + 8a_n = 0 $$ and $a_1 = 4, a_2 = 8 $. Find the general term $a_n$ in ...
3
votes
5answers
3k views

Find the four digit number?

Find a four digit number which is an exact square such that the first two digits are the same and also its last two digits are also the same.
0
votes
1answer
35 views

Calculating variable in math equation

I am not good with math, I have this equation (very simple to most) but I need help on how to get the value of x 10 = x - (1.29 + 4.99% of x) my question is how ...