Sunday, May 21

4*4*4 Elevator


A companion purchase to my Mean Cube, these are the six pieces of Jos Bergmans' 2010 4*4*4 Elevator. This one does have rotations (two, of the bottom-left piece). The two top-right pieces are shifted a couple of times in the assembly/disassembly of the cube to allow those rotations, making this a satisfyingly comprehensible construction toy. Visible in the top-left piece is one of three brass pins that help to reinforce that particular piece's joints.

Mean cube


These are the six pieces of Tom Jolly's 2004 Mean Cube that I recently acquired from Brian Menold. The small metallic circle embedded in the top of the bottom-left piece is a magnet whose complement resides on the bottom-right piece. It's a cheat meant to prevent the first piece out of the finished 4 by 4 by 4 cube from coming out too easily. There are no rotations in the assembly/disassembly but still a very challenging puzzle to own and appreciate.

Wednesday, May 17

Countdown primes

The concatenation of the integers from 1 to n have been called Smarandache numbers, whereby the concatenation of the integers from n to 1 would be reverse Smarandache numbers. No Smarandache numbers are yet known to be prime but we have two for the reverse. I prefer to call them countdown primes.

The first is 82818079787776757473727170696867666564636261605958575655545352515049484746454443424140393837363534333231302928272625242322212019181716151413121110987654321, first noted by Ralf Stephan in 1998. The second countdown prime was found by Eric Weisstein in 2010. We can call them countdown(82) and countdown(37765) for short.

Surprisingly, a tabulation of countdown primes in bases other than ten appears not to have been tackled by anyone so I shall remedy that herewith:

 2 — 2, 3, 4, 7, 11, 13, 25, 97, 110, 1939, ...
 3 — 2, 5, 13, 57, 109, 638, 3069, ...
 4 — 4, 106, 118, 130, 1690, ...
 5 — 2, 313, 505, ...
 6 — 2, 6, 17, 28, 33, 37, 81, ...
 7 — 373, 1825, ...
 8 — 2, 9, 47, 50, 99, ...
 9 — 2, 5, 346, ...
10 — 82, 37765, ...
11 — 2, ...
12 — 3, 4, 5, 7, 17, 58, 106, 303, ...
13 — ?