49
votes
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Is there winning strategy in Tetris ? What if Young diagrams are falling?
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 the ...
- 4,589
47
votes
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Is there mathematical significance to the LaGuardia floor tiles?
You can view this pattern as consisting of major and minor tiles. The major tiles are the union of four hexagons. These tiles are all identically subdivided into eleven minor tiles.
In the picture ...
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43
votes
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Which popular games have been studied mathematically?
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 ...
42
votes
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Who wins two player sudoku?
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 ...
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34
votes
Examples of interesting false proofs
I came across this one in a book of false proofs, the name of which I can't remember. It stuck out because it's not the usual hidden division by $0$ or unestablished base case in an induction.
...
34
votes
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Can an odd number of marbles jump to infinity?
With $5$ you can using the following moves:
...
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32
votes
Which popular games have been studied mathematically?
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. ...
30
votes
Which popular games have been studied mathematically?
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 ...
28
votes
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Does a function from $\mathbb R^2$ to $\mathbb R$ which sums to 0 on the corners of any unit square have to vanish everywhere?
The answer is yes, I learned this from the paper that appeared on arXiv literally yesterday https://arxiv.org/abs/2403.01279, they quote
R. Katz, M. Krebs and A. Shaheen, Zero sums on unit square ...
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27
votes
Is there mathematical significance to the LaGuardia floor tiles?
Concerning the secondary question who designed the pattern:
The tiled floor in LaGuardia terminal B was designed by HOK and installed by Consolidated Flooring. They received an award for this. The ...
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24
votes
Guessing each other's coins
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 ...
23
votes
Examples of interesting false proofs
Another subtle variant of the induction fallacy suggested by Fedor Petrov.
Theorem: every graph without isolated nodes is connected.
Proof Induction on the number of nodes. Clearly the result is ...
23
votes
How do you generate math figures for academic papers?
TikZ (a self-referential acronym for Tikz Ist Kein Zeichenprogramm) is an excellent, and extremely versatile drawing program. I highly recommend it.
There's an extensive manual. It might be ...
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22
votes
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Is there an open subset $A$ of $[0,1]^2$ with measure $>\frac{1}{100}$ that satisfies this property?
The answer is no: if $\varepsilon$ is small enough, then for every open $A \subset [0,1]^2$ of measure at least $1/100$, there exists a smooth curve $\gamma$ of length $\leq 1$ such that $\gamma+A$ ...
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22
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The $9$th tetration of $-\sqrt2$
This is not a huge coincidence: the idea is that the sequence $a_n={}^{n}(-\sqrt{2})$ has small norm until $n=6$, then it gets out of hand for $n=7$ ($a_7\sim-33+29i$), so that $a_8=e^{a_7\ln(-\sqrt{2}...
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20
votes
Which popular games have been studied mathematically?
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 ...
20
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Which popular games have been studied mathematically?
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,...
20
votes
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How do you generate math figures for academic papers?
To complement Kostya's answer, here is a way to turn hand-drawings into something looking vaguely professional using inkscape, very quickly (and for free).
Draw something on paper.
Take a photo in ...
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19
votes
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Covering a Cube with a Square
Four pieces, using the tessellation technique I learned from
Harry Lindgren's Geometric Dissections (1964):
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19
votes
Which popular games have been studied mathematically?
Winning Ways for your Mathematical Plays (Wikipedia link) by Berlekamp, Conway and Guy, 1982.
This is a book discussing two-player full-information games. It is very good. While most of the games ...
18
votes
Knight's tour problem
$k=3$ and $k=5$ are certainly possible, and I believe the procedure for $k=5$ can be extended to other odd $k$ as well. Here is a picture of $k=3$:
(click to enlarge; Java source code to generate ...
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17
votes
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Can we make 101 almost perfect banknotes from 100?
This (top half) is a way to cut 49 banknotes into pieces of at least 10% each and (bottom half) reassemble them to 50 "98%" banknotes. Let $w$ denote the width and $h$ the height. The bills below have ...
- 2,821
17
votes
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Runner's High (Speed)
The constant is $2$. Let $n=\lfloor d_2/d_1 \rfloor \geq 1$, and let $t_k$ be the time which
the long distance runner takes to arrive at the distance $kd_1$ from the origin,
$1\leq k\leq n$.
Proving ...
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17
votes
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Throwing a fair die until most recent roll is smaller than previous one
By doing casework on the second to last roll $m$, one has $$E_n = \sum_{k=2}^\infty k p_k^{(n)},$$ where $$p_k^{(n)} = \sum_{m=1}^n n^{-(k-1)}{m+k-3 \choose k-2}\frac{m-1}{n}.$$ Note, for any fixed $k$...
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17
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What is known in general about the liquid transfer problem?
These are also known as 'decanting problems' or water pouring puzzles. There is a list of literature references in that Wikipedia article.
They're quite popular among puzzling aficionados, and it ...
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17
votes
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Page-turning number of a graph
The page-turning number of a graph $G$ is also known as the bandwidth of $G$ (https://en.wikipedia.org/wiki/Graph_bandwidth).
The Wikipedia page also contains values of the bandwidth for some special ...
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16
votes
Covering a Cube with a Square
Just illustrating Noam Elkies' 4-piece solution:
Bottom face is mostly yellow (except for a little green); two hidden back faces ...
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16
votes
Guessing each other's coins
First, Alice chooses minimal $n_a$ divisible by 3 such that her bits at positions $n_a, n_a + 1, n_a + 2$ are not all the same, and Bob similarly chooses $n_b$.
Looking at triplet $A_{n_a}, A_{n_a + 1}...
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15
votes
Examples of interesting false proofs
A common mistake in using induction for statements concerning finite sets is the bad logic "prove it for 1-set, and if we have proved this for $n$-set, add an element and prove it for $(n+1)$-set&...
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