We should mention Bob Floyd's times 4 solution, http://merganser.math.gvsu.edu/calculus/summation/cubes.html from Knuth, Vol 1. Also, I suspect the late Robert Ammann could have cooked this problem with a magic frac-tile. --rwg Retraction: Vijayaraghavan's name is not on the nested radical constant, except on some crank pages that were apparently confused by Google-glimpsing [...] No closed-form expression is known for this constant. "It was discovered by T. Vijayaraghavan [...] in http://oeis.org/search?q=A072449 . SPOILER WARNING I got zero responses to the √(x + √(4 x + √(16 x + √(64 x + √(256 x + √...))))) evaluation puzzle, leaving me perplexed as to whether it was too easy or too hard, (and whether or not it's old). The proof should be easy, once you see the answer: 1 + √x. On 2013-11-03 07:06, Michael Kleber wrote:

Since Joerg mentioned Ventrella's book _Brain-Filling Curves: A Fractal Bestiary_, let me take this opportunity to re-mention an open problem I'm fond of. (At least, I think it's still open.)

Recall that 1^3 + 2^3 + 3^3 + ... + n^3 = (1+2+...+n)^2. That means that one 1x1 square, two 2x2 squares, ..., n nxn squares together all have the same total area as one square of side length 1+2+...+n. Robert Wainright's "Partridge problem" asks whether you can realize this numerical identity as a dissection of a big square into the requisite set of smaller squares. It turns out to be possible for all n >= 8; you can see some pictures of solutions on pages by Funsters at http://www2.stetson.edu/~efriedma/mathmagic/0802.html or http://www.mathpuzzle.com/partridge.html. (The "all n >=" part is very not obvious even once you have an n=8 solution.)

The name "partridge problem" comes from "four calling birds, three french hens, two turtledoves, and a partridge in a square tree", of course.

This identity can manifest itself as a tiling problem not just for squares, but for an arbitrary planar figure P. If kP means a copy of P scaled linearly by a factor of k, then the question is whether you can use one P, two 2P's, ..., n nP's to cover a (1+2+...+n)P. If so then we say P has "partridge number" n (or maybe only for the minimal such n, depending on who you ask). For example, the square has partridge number 8, the equilateral triangle has partridge number 7, the 30-60-90 right triangle has a partridge number of 4, and so on. The 30-60-90 find is by Patrick Hamlyn, and is a truly beautiful puzzle; I've had a physical copy sitting on my desk for years.

Anyway, the open problem part of all this!

Patrick Hamlyn's 30-60-90 is the polygon with the smallest known partridge number. As far as I know, it is open whether there are polygons OR fractals with partridge numbers 2 or 3.

The fact that 2 is open, even for polygons, really bugs me. I really want to know whether there is a polygon P such that a P and two 2P's fit together to make a 3P.

(For a while I thought I had a proof of impossibility, and I was all set to write a paper called "On the Necessity of French Hens", and that my proof fell apart is a great disappointment, though I have to admit more for the title of the non-paper than for its theorem.)

Anyway, the fractal version of the partridge-2 or partridge-3 problem is very much along the lines of Ventrella's book. In particular, I think that if a solution exists then then must be a space-filling curve that realizes it.

--Michael

On Sun, Nov 3, 2013 at 9:19 AM, Joerg Arndt <arndt@jjj.de> wrote:

...

A pointer to a neat picture book featuring very many "space filling" curves:

Jeffrey J. Ventrella: Brain-Filling Curves: A Fractal Bestiary, LuLu.com, (2012).

The book can be downloaded for free at

http://archive.org/download/BrainfillingCurves-AFractalBestiary/BrainFilling...

...

Warning: it is a hefty 235 Megabytes!

Two pertinent URLs are http://www.fractalcurves.com/HorrorVacui.html (book as html) http://www.brainfillingcurves.com/ (needs scripting activated)

On page 107 (top) there is a curve corresponding to the L-system (and turns of 120 degrees) Start: F Rules: F --> F-F+F+FF-FF + --> + - --> -

Here is a rendering http://jjj.de/3frac/R7-x1-curve.pdf

What Ventrella does not show is how neatly 3 of these combine into either http://jjj.de/3frac/R7-x1-closed-trihook.pdf or http://jjj.de/3frac/R7-x1-closed-gosper-island.pdf

The better known splittings of Gosper's island are http://jjj.de/3frac/gosper-split7.pdf and possibly (3 renderings) http://jjj.de/3frac/R7-2-gosper.pdf http://jjj.de/3frac/R7-2-sty1-gosper.pdf http://jjj.de/3frac/R7-alt-hex-gosper.pdf Here is a bubbly one: http://jjj.de/3frac/R7-bubble-gosper.pdf

(cf. http://www.fractalcurves.com/Root7.html )

For the truly bored there are more images under http://jjj.de/3frac/

Best, jj