85

I enjoyed the hexasphere by
A. V. Akopyan, J. Crowder, H. Edelsbrunner, R. Guseinov
from last year:
http://pub.ist.ac.at/~edels/hexasphere/
In the link, the sphere is animated, so you can look at it from all sides.

71

In 2009, it was discovered that the numerical value of $\pi$ has changed over time. This is a truly interdisciplinary work connecting the study of ancient cultures with string theory, cosmology and bicycle tires. Let me quote from the introduction:
Physicists have long speculated that the fundamental
constants might not, in fact, be constant, but ...

53

In 1975 Martin Gardner produced a map with 110 regions which he claimed required five colours:
http://mathworld.wolfram.com/Four-ColorTheorem.html

46

While the actual algorithm used in Tetris makes it impossible to achieve the combination of tetrominos necessary for the inescapable loss, in a perfect "abstract" Tetris, yes, Heidi Burgiel's first paper "How to Lose at Tetris" (which she wrote towards the end of our time in grad school in Seattle) answers Question 1 in the negative. It was published in ...

45

Here's a solution for the case of seven suspects that uses the Fano plane. Let the seven points of the Fano plane represent the seven suspects. Alice and Bob both reveal the name of the suspect completing a line with the two suspects on their list. There are now two cases to consider:
Alice and Bob did not name suspects on each other's lists. Then the ...

38

Someone (widely believed to be Henri Darmon) circulated the following email on April Fools' Day, 1994:
There has been a really amazing development today on Fermat's Last Theorem.
Noam Elkies has announced a counterexample, so that FLT is not true
after all! His spoke about this at the Institute today. The solution to
Fermat that he constructs involves an ...

38

Update. I made a blog post about Infinite Sudoku and the Sudoku game, following up on ideas in this post and the comments below.
I claim that the second player wins the even-sized empty Sudoku boards and the first player wins for odd-sized empty Sudoku boards, including the main $9\times 9$ case. (The odd-case solution uses a key idea of user orlp in the ...

answered Apr 15 '18 at 1:29

Joel David Hamkins

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37

Glad that MO is up and running again.
Following the suggestion by Michael Murray one can also produce more than just one sudden smiley:
I guess, that a higher number of images is also possible. But probably some structure may be visible from other directions as well in this case. By the way: The problem seems to be a bit related to tomography...

37

A few years ago several classic Nintendo games (including Mario, Donkey Kong, and Legend of Zelda) were examined from a computational complexity point of view. They proved that generalized versions of these games are NP-hard, and in some cases PSPACE-hard.
Greg Aloupis, Erik D. Demaine, Alan Guo, and Giovanni Viglietta, "Classic Nintendo games are (...

35

Surely you can draw the 2-D image in the XY plane so it consists of points of the form (x, y, 0) and then give each point in it a random non-zero Z co-ordinate. So it should look like a mess except viewed looking in along the Z-axis.

35

This one is my favorite (especially a mixture of anyons and morons with opposite spins):
> From: Enrico Bombieri <eb@IAS.EDU> Tue, 1 Apr 1997 12:35:12 -0500
> Date: Tue, 1 Apr 1997 12:35:12 -0500 To: eb@IAS.EDU,
> zeilberg@euclid.math.temple.edu
>
> Dear Doron,
>
> There are fantastic developments to Alain Connes's lecture at ...

31

The board game Monopoly is often used to explain Markov Chains to laypeople. Ian Stewart has a couple of Mathematical Recreations articles about it in Scientific American.
The first (in the April 1996 issue) has the title "How Fair is Monopoly?" (A copy it can be found here.)
The second (in the October 1996 issue) has the title "Monopoly Revisited" (A ...

27

Theorem. $\int_0^\infty \sin x \phantom. dx/x = \pi/2$.
Poof. For $x>0$ write $1/x = \int_0^\infty e^{-xt} \phantom. dt$,
and deduce that $\int_0^\infty \sin x \phantom. dx/x$ is
$$
\int_0^\infty \sin x \int_0^\infty e^{-xt} \phantom. dt \phantom. dx
= \int_0^\infty
\left( \int_0^\infty e^{-tx} \sin x \phantom. dx \right)
\phantom. dt
= \int_0^\infty ...

27

Hex (https://en.wikipedia.org/wiki/Hex_(board_game)) is a board game that was developed by John Nash (and independently and earlier, Piet Hein). It is interesting mathematically in a number of ways. For example, unlike something like chess, it is easy to see that under optimal play the first player will win a game of Hex. But though it is known that the ...

24

Lucas proved a congruence for binomial coefficients mod a prime $p$ that uses the base $p$ digits of the two numbers in the binomial coefficient. See http://en.wikipedia.org/wiki/Lucas%27_theorem. It was extended to multinomial coefficients by Dickson.
Stickelberger's congruence for Gauss sums (not to be confused with Stickelberger's congruence for the ...

23

Perhaps the digital sundial of Falconer is what you need:
http://www.researchgate.net/publication/225574085_Digital_sundials_paradoxical_sets_and_vitushkins_conjecture
Here is a photograph of a working model:
http://apod.nasa.gov/apod/ap120626.html
(Image added by O'Rourke)

answered Oct 27 '12 at 3:06

Alexandre Eremenko

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23

As remarked by Joe Silverman, a natural way to look at this question is by phrasing it in terms of the map $f(x):=\frac{1}{2}x(x+1)$ on the $2$-adic integers $\mathbb{Z}_2$. We are then asking about the behaviour of the orbit $f^n(2)$ with respect to the partition of $\mathbb{Z}_2$ into two clopen sets $U_1:=2\mathbb{Z}_2$ and $U_2:=1+2\mathbb{Z}_2$. Most ...

nt.number-theory ds.dynamical-systems sequences-and-series recreational-mathematics arithmetic-dynamics

22

Theorem: Every bounded differentiable function $f\colon \mathbb{R}\to \mathbb{R}$ is constant.
Proof.
By assumption there exist real numbers $M,N$ such that
$$N\leq f(x) \leq M.$$
Taking derivatives we get
$$0\leq f'(x)\leq 0.$$
Hence $f'(x)=0$ so $f$ is constant. QED

22

I talked to Ronald Graham last night at the Joint Mathematics Meeting in San Diego and asked him the question here. He said he'd made up Graham's number when talking to Martin Gardner because 1) it was simpler to explain than his actual upper bound, the one that appears in his paper with Rothschild, and 2) it's bigger, so it's still an upper bound!
So, ...

22

I discussed this with Arvind Singh a while ago and I think we can show the non trivial inequality $p_{opt}\leqslant\frac{3}{8}$ with simple arguments.
The proof relies on the symmetry of the problem and the intuition is that one can not find a strategy wich is good simultaneously for a configuration and its inverse.
It will be simpler to work with the sets ...

20

Google "shadow sculptures".
Here are some links:
http://www.mymodernmet.com/profiles/blogs/incredible-shadow-scuptures-16
http://fractalenlightenment.com/722/artwork/shadow-optical-illusions-ix
http://www.subtielman.com/shadow-sculptures/
http://www.feeldesain.com/shadow-sculptures-tim-noble-sue-webster.html
http://www.technovelgy.com/ct/Science-...

19

Here's a nice false proof of the continuum hypothesis.
Consider the rational numbers $\mathbb{Q}$ as a totally ordered field. We can add an indeterminate $T_0$ and make it positive but infinitely small (i.e., smaller than positive any element of $\mathbb{Q}$), that is, order $\mathbb{Q}(T_0)$ by lexicographic order of the Laurent series expansion at $0$. ...

19

https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan.27s_constant
This concerns the number $e^{\pi\sqrt{163}}$, and says "In a 1975
April Fool article in Scientific American magazine,[7] "Mathematical
Games" columnist Martin Gardner made the (hoax) claim that the number
was in fact an integer, and that the Indian mathematical ...

18

Four pieces, using the tessellation technique I learned from
Harry Lindgren's Geometric Dissections (1964):

answered Apr 1 '17 at 7:01

Noam D. Elkies

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18

Well, there's the April 1, 1997 paper by Doron Zeilberger, The Transcendence of E plus Pi and E times Pi (the following quote is snipped a bit; full text available at the link).
The purpose of this note is to announce that Hermite's[H] celebrated
result that $e$ is transcendental, combined with an amazing
(but apparently overlooked) statement of ...

18

IMO one of the most important recent work related to the computational complexity of (puzzle) games is the Nondeterministic Constraint Logic model of computation developed by Robert A. Hearn and Erik D. Demaine:
Robert A. Hearn and Erik D. Demaine, "Games, Puzzles, and Computation", 2009
The framework can be used to easily prove the complexity of the ...

17

&
Yes, this is the title. Just "&". :-)
From Mulvey's homepage: "This paper, presented at the Topology Meeting in Taormina, Sicily in April, 1984, introduced the concept of quantale, outlining the programme of work in the spectral theory of C*-algebras and the constructive foundations of quantum mechanics to which it was expected to contribute. The ...

17

It is certainly true that some choice is required.
An isomorphic game (mapping (giraffes, scarves, lion) to (prisoners, hats, warden)) was considered by Hardin and Taylor in their stimulating (and elementary) paper
MR2501394 Hardin, Christopher S.; Taylor, Alan D.
An introduction to infinite hat problems.
Math. Intelligencer 30 (2008), no. 4, 20–25.
...

answered Oct 14 '14 at 22:08

Nate Eldredge

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17

Rubik's Cube puzzle https://www.youcandothecube.com/blog/puzzling-science-using-the-rubiks-cube-to-teach-problem-solving gives an excellent possibility for some musings in mathematics and physics. See, for example,
https://www.sciencedirect.com/science/article/pii/0378437182903624
https://arxiv.org/abs/1106.5736 https://arxiv.org/abs/1706.06708 https://...

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