Puzzles, curiosities, brain teasers and other mathematics done "just for fun".
295
votes
17answers
81k 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:
I don't understand what's wrong with this question. I think the student answered the question wrong, yet my ...
266
votes
115answers
13k 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 ...
156
votes
6answers
4k 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 $f_n(x)$ ...
153
votes
4answers
8k 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 ...
121
votes
2answers
12k 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, ...
120
votes
4answers
4k views
Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?
My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question:
I am thinking of a number...
It ...
91
votes
6answers
17k 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 ...
70
votes
3answers
3k 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
...
68
votes
17answers
4k views
How do you find the center of a circle with a pencil and a book?
Given a circle on a paper, a pencil and a book. Can you find the center of the circle with the pencil and book?
66
votes
5answers
3k views
Help find hard integrals that evaluate to $59$?
My father and I, on birthday cards, give mathematical equations for each others new age. This year, my father will be turning $59$.
I want to try and make a definite integral that equals $59$. So ...
58
votes
4answers
1k views
Probability that a stick randomly broken in five places can form a tetrahedron
Randomly break a stick in five places.
Question: What is the probability that the resulting six pieces can form a tetrahedron?
Clearly satisfying the triangle inequality on each face is a necessary ...
46
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 ...
40
votes
26answers
2k 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 ...
36
votes
1answer
1k views
What's the largest possible volume of a taco, and how do I make one that big?
Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
36
votes
3answers
5k 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 ...
34
votes
6answers
3k views
33
votes
6answers
1k views
A variant of the Monty Hall problem
Everybody knows the famous Monty Hall problem; way too much ink has been spilled over it already. Let's take it as a given and consider the following variant of the problem that I thought up this ...
32
votes
2answers
677 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 ...
30
votes
14answers
605 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, ...
30
votes
1answer
549 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?
28
votes
2answers
558 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 ...
27
votes
3answers
1k views
For which number does multiplying it by 99 add a 1 to each end of its decimal representation?
This was asked by my maths lecturer a couple of years ago and ive been wracking my brains ever since:
Find a number that, when multiplied by
99 will give the original number but
with a 1 at ...
25
votes
5answers
2k views
Is Mega Millions Positive Expected Value?
Given the rapid rise of the Mega Millions jackpot in the US (now advertised at \$640 million and equivalent to a "cash" prize of about \$448 million), I was wondering if there was ever a point at ...
25
votes
1answer
834 views
Dividing a square into equal-area rectangles
How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$?
The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
24
votes
2answers
346 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 ...
23
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
0answers
833 views
$4494410$ and friends
$4494410$ has the property that when converted to base $16$ it is $44944A_{16}$, then if the $A$ is expanded to $10$ in the string we get back the original number. ...
22
votes
4answers
1k views
Which is bigger?
In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
22
votes
4answers
1k views
Something that I found, and would like to see if it's known.
Well I am quite sure it's known (I mean number theory exists thousands of years), warning beforehand, it may look like numerology, but I try not to go to mysticism.
So I was in a bus, and from ...
22
votes
1answer
1k views
Going to the Movies!
I was looking at movie times today and was struck by the oddly-spaced showing times. For example, at the local Loew's Theater "Tron: Legacy 3D" (127 min.) is playing on two screens at the following ...
21
votes
3answers
904 views
Guaranteed Checkmate with Rooks in High-Dimensional Chess
Given an infinite (in all directions), $n$-dimensional chess board $\mathbb{Z^n}$, and a black king. What is the minimum number of white rooks necessary that can guarantee a checkmate in a finite ...
20
votes
1answer
555 views
Expert Minesweeper Probability Question
This is just a question I thought of while playing minesweeper. I think that finding the solution might be kind of fun, so I'm sharing it with you guys. If you have no concept of what minesweeper is, ...
19
votes
5answers
719 views
a big number that is obviously prime?
I once heard it asserted that $91$ is the smallest composite number that is not obviously composite. The reasoning was that any composite number divisible by $2$, $3$, or $5$ is obviously composite, ...
19
votes
6answers
6k 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.
...
18
votes
3answers
681 views
Brute force method of solving the cube: How many moves would it take?
Given that Rubik's cube has finitely many positions, one possible "brute force" method to solve it would be to determine once a sequence of moves which eventually reaches every possible position of ...
18
votes
8answers
465 views
Contemporary Mathematical Columns in Magazines
In good old days, Scientific American was host to some legendary mathematical (and computer science) oriented columns that inspired generations of scientists and engineers. Douglas Hofstadter, Martin ...
17
votes
1answer
641 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 ...
17
votes
5answers
5k 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 ...
17
votes
2answers
495 views
Question about a program generating palindromic prime numbers
I'm a programmer and software designer. I'm definitely not a mathematician and my maths is quite basic.
One of my colleagues challenged me to generate a palindromic prime number, at least 1000 digits ...
17
votes
5answers
555 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 ...
17
votes
1answer
1k views
The $n$ Immortals problem.
I saw this riddle posted on reddit a long time ago, called the "Seven Immortals."
In the beginning, the world is inhabited by seven immortals, ageless and sexless, who begin to multiply and ...
16
votes
5answers
1k 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:
...
16
votes
3answers
814 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 ...
16
votes
2answers
666 views
What is the millionth decimal digit of the (10^10^10^10)th prime?
What is the millionth decimal digit of the $10^{10^{10^{10}}}$th prime?
(This prime is, of course, far larger than the largest currently "known" prime, the latter having nearly 13 million ...
16
votes
1answer
386 views
How to create mazes on the hyperbolic plane?
I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the ...
16
votes
3answers
631 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 ...
16
votes
1answer
287 views
Extracting individual race results from Mario Kart final scores
In Mario Kart, one "cup" involves 4 races, and after every race each racer gets points awarded based on what place they came in (better rank means more points). After playing it enough I grew curious ...
14
votes
6answers
706 views
Properties of the number 50
I will shortly be engaging with my 50th (!) birthday.
50 = 1+49 = 25+25 can perhaps be described as a "sub-Ramanujan" number.
I'm trying to put together a quiz including some mathematical content. ...
14
votes
5answers
646 views
Is high school contest math useful after high school?
I've been prepping for a lot of high school math competitions this year, and I was just wondering if all the math I learn would actually mean something in college. There is a chance that all of it ...
14
votes
3answers
3k views
Average Scrabble graph structure: diameter?
Tonight a game of Scrabble ended in what I consider a very unusual graph structure,
unlike this generic web image, which seems more typical:
...
