For my 100th toot, I have a new puzzle page to introduce: "Port-and-Sweep Solitaire!" I wrote about it previously for Math Horizons, but it's so much better experienced in an interactive format:

homepages.gac.edu/~jsiehler/ga

Check the bottom of the page for further links.

One of my favorite KöMaL problems (B. 4463): "In the Four-square Round Forest, trees form a regular triangular lattice. Is it possible to build a fence around a rectangular part of the forest such that the vertices of the rectangle are lattice points and the number of lattice points on the boundary of the rectangle is the same as in the interior?"

Does anyone know if Recreational Mathematics Magazine is still alive and kicking? Nothing has appeared so far in 2019, the text about their publishing schedule is missing, and the site just feels dead.
rmm.ludus-opuscula.org/

Wikipedia's entry for harmonic numbers gives the formula \[-\sum_1^n (-1)^k{n\choose k}\frac{1}{k}\] for \(H_n\), with a derivation via a clever trick involving a geometric sum and a substitution integral. Is this clever trick an isolated thing, or is it an instance of some general method that everybody except me knows about?

en.wikipedia.org/wiki/Harmonic

A design for coloring, if you're in to that sort of thing. Based on the orbit of a single point under the action of two rotation groups with different centers.

(*) By a theorem of Richard Savage (1994), if the numbers from 1-24 are partitioned into three sets of eight numbers like this, then at least one of the winning probabilities (R>G, G>B, or B>R) must be less than (√5-1)/2. Thus, if all three are to be equal, then 39/64 is the highest possible common value, and this partition achieves it.

A partition the numbers 1-24 into an optimal(*) symmetric, nontransitive set of three octahedral dice:
red = {4,5,6,7,9,22,23,24}
green = {8,10,11,12,13,14,15,17}
blue = {1,18,3,20,2,16,19,21}.
Red beats green, green beats blue, and blue beats red with probability 39/64.

If you're interested in the game of Hex, Matthew Seymour's online strategy book is the best one-stop guide to practical play that I know of (and it's not too easy to stumble across):
mseymour.ca/hex_book/hexstrat.

HopeSort algorithm.

while (list is not sorted) {
do nothing;
hope things sort themselves
}

jsiehler boosted

@joshmillard wouldn't you say there's a set

a set that has... everything?

How do you handle the punctuation collision at the end of sentences like "How many zeroes are at the end of 20!?"gracefully?

TeX ignores everything following \bye, and so, since 2008, my TeX files have ended with \bye bye beautiful.

Snap Cube Puzzle: Snap cubes have a post on one side, and holes (into which a post can fit) on the other five sides. How many ways to build a 2x2x2 cube out of eight snap cubes such that (1) the entire structure is connected, and (2) all the posts are in the interior – no posts sticking out of the surface? (All the different structures look the same from the outside; the difference is in the orientation of the eight posts internally.)

jsiehler boosted

What's the "most scalene" right triangle – that is, how should the edges be proportioned so that the minimum difference over all pairs of sides is as large as possible? Let's assume we've normalized so that the hypotenuse is 1 unit.

(If you enjoy the answer for right triangles, also try the problem for triangles with a 120-degree angle.)

Silly arithmetic amusement:
\(8^2-4^2\)
\(848^2-484^2\)
\(84848^2-48484^2\), etc.

(I constructed this problem myself, but I still don't know a really *nice* way to solve it. I hope somebody finds a pretty solution.)

O is an interior point in the triangle 10, 26, and 36 units away from the three vertices respectively. Determine the side lengths a, b, and c, given that they are integers.

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