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

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40
votes
9answers
6k views

Where is the flaw in this “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: $\...
57
votes
6answers
6k views

How come $32.5 = 31.5$?

Below is a visual proof (!) that $32.5 = 31.5$. How could that be?
63
votes
7answers
15k 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 ...
7
votes
3answers
1k views

Horse Race question: how to find the 3 fastest horses?

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 ...
53
votes
10answers
2k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
15
votes
4answers
1k 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 ...
39
votes
7answers
41k 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. ...
28
votes
2answers
2k 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 ...
151
votes
12answers
24k 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 ...
42
votes
7answers
30k 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 ...
139
votes
6answers
25k 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 ...
73
votes
24answers
15k 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 ...
13
votes
1answer
348 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$ ?
29
votes
2answers
13k 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 ...
13
votes
8answers
1k 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 ?
2
votes
3answers
839 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 had ...
58
votes
2answers
18k 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 ...
51
votes
10answers
2k views

Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
15
votes
4answers
771 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, ...
16
votes
4answers
2k views

9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. Arriving on a deserted island, they now have to split up the ...
16
votes
4answers
1k 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 ...
29
votes
2answers
2k 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 ...
5
votes
2answers
275 views

Prisoners Problem

We have an infinite number of prisoners enumerated $\{1, 2, \dots\}$, and on each prisoner there is a hat of either blue or red color. The $n$th prisoner sees the hats of prisoners $\{n+1, n+2, \dots\}...
3
votes
2answers
495 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 ...
5
votes
1answer
997 views

Social Golfer Problem - Quintets

I wrote an article on the Social Golfer Problem, which has questions like: Each day, 16 people play Munchkin in foursomes simultaneously. How many days can they play with no two people playing with ...
3
votes
0answers
350 views

An interesting way to visualize the Mandelbrot Set. Proofs? Simplifications? Extensions?

This is a multi-part Question. Please chime in with any interesting insights in addition to Answers. I have noticed some interesting properties of Mandelbrot series that lead to a different way to ...
2
votes
2answers
160 views

A non-trivial, non-negative, function bounded below by its derivative with $f(0)=0$?

I did not know what to search to see if this existed elsewhere. But, I could not find it. Here's the question: Does there exist a continuously differentiable function, $f: [0,1] \rightarrow [0, \...
2
votes
3answers
9k views

Smallest multiple whose digits are only ones and zeros [duplicate]

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 ...
1
vote
2answers
1k views

Convert Equilateral triangle to Isosceles triangle

Let an equilateral triangle have the length of each side an integer $N$. I need to find if it is possible to transform the triangle keeping two sides fixed and alter the third side such that it still ...
233
votes
8answers
14k views

The length of toilet roll

Fun with Math time. My mom gave me a roll of toilet paper to put it in the bathroom, and looking at it I immediately wondered about this: is it possible, through very simple math, to calculate (with ...
178
votes
2answers
14k 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, ...
77
votes
14answers
10k 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 ...
85
votes
5answers
5k views

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
49
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, ...
25
votes
11answers
3k 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. ...
44
votes
4answers
2k views

How does the divisibility graphs work?

I came across this graphic method for checking divisibility by $7$. $\hskip1.5in$ Write down a number $n$. Start at the small white node at the bottom of the graph. For each digit $d$ in $...
14
votes
3answers
2k 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. ...
18
votes
3answers
235 views

Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
15
votes
2answers
4k views

What is the most efficient numerical base system?

I remember reading somewhere that base $e$ is the most "efficient" base system because of its ratio of possible characters to number length. For example, binary is "inefficient" because each ...
15
votes
6answers
778 views

Fascinating induction problem with numerous interpretations

Problem: Suppose you begin with a pile of $n$ stones and split this pile into $n$ piles of one stone each by successively splitting a pile of stones into two smaller piles. Each time you split a pile, ...
12
votes
1answer
18k views

Expected Ratio of Coin Flips

If you flip a coin until you decide to stop and you want to maximize the ratio of heads to total flips, what is that expected ratio? Assuming that you want to maximize the ratio, meaning whether ...
2
votes
2answers
430 views

Puzzle with pirates

That one I'm pretty low on ideas of how to approach it. Five pirates of different ages have a treasure of 50 gold coins. On their ship, they decide to split the coins using this scheme: The oldest ...
6
votes
1answer
271 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
5
votes
4answers
2k views

A 3rd grade math problem: fill in blanks with numbers to obtain a valid equation

Even though this is a 3rd-grade math problem, people found it extremely hard. Any people have a solution, or algorithm is welcome. I'll try make a program base on the algorithm and see if it's correct....
4
votes
2answers
859 views

What is the parity of permutation in the 15 puzzle?

You might know the 15 puzzle: $\hskip1.4in$ Concerning the solvability, Wiki says: The invariant is the parity of the permutation of all 16 squares plus the parity of the taxicab distance (...
3
votes
2answers
429 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 ...
3
votes
2answers
789 views

How to choose between an odd number of options with a fair coin

It is possible to choose between three equally desirable outcomes by tossing a fair coin as follows: Choose option 1 if the first head appears on an even toss Choose option 2 if the first tail ...
0
votes
1answer
2k views

Can this crate have even numbers in all rows and columns?

A milk crate holds 24 bottles in four rows and six columns. Can you put 18 bottles of milk in the crate so that each row and each column of the crate have an even number of bottles in it?
0
votes
0answers
60 views

Horse Race Math: Determine the top fastest 3 of 25 horses in the fewest number of races possible. Help! :) [duplicate]

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 ...
356
votes
7answers
13k 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 $f_n(x)$ ...