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

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26
votes
6answers
2k views

Where is the flaw in this argument of a proof that 1=2? (Derivative of repeated addition)

Consider the following: $1 = 1^2$ $2 + 2 = 2^2$ $3 + 3 + 3 = 3^2$ Therefore, $\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$ Take the derivative of lhs and rhs and we get: ...
386
votes
131answers
23k 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 ...
40
votes
2answers
7k views

Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints

I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented: The King has $1000$ wines, $1$ of which is poisoned. He needs to ...
38
votes
6answers
4k views

How come $32.5 = 31.5$?

Below is a visual proof (!) that $32.5 = 31.5$. How could that be?
25
votes
7answers
10k views

How many triangles

I saw this riddle today, it asks how many triangles are in this picture . I don't know how to solve this (without counting directly), though I guess it has something to do with some recurrence. ...
113
votes
6answers
22k views

Deleting any digit yields a prime… is there a name for this?

My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100: But ...
24
votes
2answers
1k views

Proof of recursive formula for “fusible numbers”

The set of fusible numbers is a fantastic set of rational numbers defined by a simple rule. The story is well told here but I'll repeat the definitions. It's the formula on slide 17 that I'm trying to ...
23
votes
5answers
10k views

What is the math behind the game Spot It?

I just purchased the game Spot It. As per this site, the structure of the game is as follows: Game has 55 round playing cards. Each card has eight randomly placed symbols. There are a total of 50 ...
15
votes
4answers
787 views

Fun math for young, bored kids?

For 6 months, I'll be organizing, as part as my volunteer work in an NGO, math classes with small groups (~10 students, aged 16 or 17). These classes are not compulsory, but students willing to stay ...
18
votes
2answers
4k views

How to tell if a Rubik's cube is solvable

How can I determine if a certain Rubik's cube, that is in a certain state, is solveable? By "certain state" it could mean that the cube has been dismantled and put together again. And in my experience ...
20
votes
1answer
878 views

Rubik's Cube Not a Group?

I read online that although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all. How can that be true? There is obviously an identity and it is closed, so ...
3
votes
1answer
145 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 ...
15
votes
1answer
392 views

Choosing points in fractions of the unit interval

How long a series of points in (0,1) can be chosen such that the first two are in different halves, the first three are in different thirds, ... the first $n$ are in different $n^{\text{th}}$s? My ...
5
votes
3answers
376 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 ...
218
votes
6answers
6k views

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
13
votes
3answers
648 views

How many trees in a forest?

Some time ago I met a forester. He told that there are only larches and spruces in his forest. He also said that there are exactly $10$ spruces at the distance of exactly 1 km from each larch. Next, ...
11
votes
1answer
251 views

Is there a real-valued function $f$ such that $f(f(x)) = -x$?

Is there a function $f\colon \mathbb{R} \to\mathbb{R} $ such that $ f(f(x)) = -x$ ?
8
votes
4answers
331 views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
2
votes
2answers
276 views

So close yet so far Finding $\int \frac {\sec x \tan x}{3x+5} dx$

Cruising the old questions I came across juantheron asking for $\int \frac {\sec x\tan x}{3x+5}\,dx$ He tried using $(3x+5)^{-1}$ for $U$ and $\sec x \tan x$ for $dv$while integrating by parts. below ...
12
votes
8answers
810 views

Approximation of $e$ using $\pi$ and $\phi$?

$$e \approx \frac{4 \phi +3 \pi-5}{4}$$ where $~\phi~$ is a Golden ratio . Is it possible to construct better approximation of $e$ using $\pi$ , $\phi$ and integers ?
8
votes
3answers
621 views

What is the Probability that a Knight stays on chessboard after N hops?

Say a $8 \times 8$ chessboard as per picture. A position is represented here by co-ordinates $(x,y)$. A move is aslo considered as valid, where the Knight lands outside the chessboard [ For eg. ...
3
votes
2answers
232 views

A less challenging trivia problem

There are 25 people sitting around a table and each person has two cards. One of the numbers 1,2,..., 25 is written on each card, and each number occurs on exactly two cards. At a signal, each person ...
1
vote
3answers
456 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 ...
172
votes
4answers
10k views

The Mathematics of Tetris

I am a big fan of the oldschool games and I once noticed that there is a sort parity associated to one and only one Tetris piece, the $\color{purple}{\text{T}}$ piece. This parity is found with no ...
41
votes
26answers
3k views

Big List of Fun Math Books

To be on this list the book must satisfy the following conditions: It doesn't require an enormous amount of background material to understand. It must be a fun book, either in recreational math (or ...
138
votes
2answers
13k views

Proving you *can't* make $2011$ out of $1,2,3,4$: nice twist on the usual

An undergraduate was telling me about a puzzle he'd found: the idea was to make $2011$ out of the numbers $1, 2, 3, 4, \ldots, n$ with the following rules/constraints: the numbers must stay in order, ...
47
votes
2answers
14k 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 ...
52
votes
12answers
7k views

Can the golden ratio accurately be expressed in terms of e and $\pi$

I was playing around with numbers when I noticed that $\sqrt e$ was very somewhat close to $\phi$ And so, I took it upon myself to try to find a way to express the golden ratio in terms of the ...
87
votes
10answers
10k views

How can a piece of A4 paper be folded in exactly three equal parts?

This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on ...
41
votes
14answers
1k 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, ...
42
votes
24answers
4k views

Blue eyes: a logic puzzle

Today I read the Blue Eyes puzzle here. I also read the solution which I find quite interesting. But there are three follow up questions which I don't know the answer to: What is the quantified ...
22
votes
10answers
1k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
17
votes
5answers
701 views

Fun math outreach/social activities

What are some great math social activities for students? I'm looking for things that bring people together with a "light" mathematical touch. The goal is to create a stronger mathematical community in ...
28
votes
7answers
2k views

Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”

In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It ...
51
votes
1answer
2k views

Are there infinitely many “super-palindromes”?

Let me first explain what I call a "super-palindrome": Consider the number $99999999$. That number is obviously a palindrome. ${}{}{}{}$ The largest prime factor of $99999999$ is $137$. If you divide ...
14
votes
2answers
850 views

any pattern here ? (revised 2)

for any positive number $k$, I have a $(k+1)*(k+1)$ matrix. I wonder if these matrices follow any "obvious" pattern. My goal is to guess the elements for matrix with $k=5$ and above (most probably in ...
6
votes
3answers
2k views

Optimal Strategy for Deal or No Deal

When I have watched Deal or No Deal (I try not to make a habit of it) I always do little sums in my head to work out if the banker is offering a good deal. Where odds drop below "evens" it's easy to ...
9
votes
3answers
967 views

Secret santa problem

We decided to do secret Santa in our office. And this brought up a whole heap of problems that nobody could think of solutions for - bear with me here.. this is an important problem. We have 4 people ...
5
votes
1answer
849 views

Handshake problem

I was given the following math puzzle which (I thought) has an interesting solution. A mathematician and her husband attended a party with $n-1$ other couples. As is normal at parties, handshaking ...
1
vote
2answers
132 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 ...
16
votes
3answers
728 views

Is the Collatz conjecture in $\Sigma_1 / \Pi_1$?

Prompted by some of the comments on this question, I'm wondering if anything is known about the place of the Collatz Conjecture in the arithmetic hierarchy. More specifically, is Collatz known to be ...
2
votes
2answers
2k views

Winning strategy for a matchstick game

There are $N$ matchsticks at the table. Two players play the game. Rules: (i) A player in his or her turn can pick $a$ or $b$ match sticks. (ii) The player who picks the last matchstick loses the ...
0
votes
3answers
787 views

Smallest multiple whose digits are only ones and zeros

I have a collection of typewritten pages that formed the basis of a third year problem solving course offered about 25 years ago at U. Waterloo. I've been slowly working through the problems and have ...
9
votes
1answer
2k views

What are the 2125922464947725402112000 symmetries of a Rubik's Cube?

In a recent talk, Marcus du Sautoy says there are 2125922464947725402112000 (2.1*10^24) symmetries of a Rubik's cube, but doesn't explicitly identify what qualifies as a symmetry. What counts as a ...
8
votes
3answers
712 views

What is the probability that GCD of $(a,b)$ is $b$?

My question is quite simple. I have been googling a lot lately trying to find a solution to this: Given a sequence of n integers $[1,2,...,n]$. If we pick two numbers randomly from the set say, a and ...
7
votes
2answers
289 views

What is the most unfair set of three nontransitive dice?

In a set nontransitive dice, each die is superior to another die, but is inferior to a third. It is similar to the game of rock-paper-scissors. Here is one example: ...
6
votes
5answers
671 views

Cauchy-Schwarz inequality and three-letter identities (exercise 1.4 from “The Cauchy-Schwarz Master Class”)

Exercise 1.4 from a great book The Cauchy-Schwarz Master Class asks to prove the following: For all positive $x$, $y$ and $z$, one has $$x+y+z \leq 2 \left(\frac{x^2}{y+z} + \frac{y^2}{x+z} + ...
5
votes
2answers
574 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 ...
3
votes
2answers
116 views

Horse Race Math question

There are 25 horses. You can take 5 of the horses at a time and race them. Each horse always finishes the race in the same amount of time, and there are no ties. The only information you get from each ...
3
votes
1answer
140 views

Truchet tiles on a flattened cube

We randomly place copies of the tiles into faces of the flattened cube. 1.Find the probability that the circular arcs on the Truchet tiles will form one loop, two loops, three loops and four loops? ...