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2007 March 03

Restricted Patterns
Life Digits

Life Digits Dean Hickerson and Eric Angelini have investigated a "recreational Life" question, that of Life patterns from from the digits of numbers. The following is excerpted from some messages from Hickerson on this subject, and quoted with permission.

Although this is an unnatural restriction to put on patterns, and looking at such patterns is unlikely to lead to anything of great significance, I've had some fun playing around with them. Some of our discussion is on Angelini's website. [Note: much of the commentary is in French, while correspondence is in English.] More information can also be found at my web page.

1125344743766111111111947742 ...22222222222222... Eric first asked if infinitely many such patterns die. It's easy to see that the answer is "yes"; e.g. numbers of the form 1444...444 all die in 9 gens. Arbitrarily slow death is also possible; e.g. numbers of the form 1125344743766111...111947742, with an odd number of 1's, die by crashing a LWSS and a MWSS (Shown here at Gen 50). Numbers of the form 1125344743766189077900222...2220066748424 are even more amusing; here the number of 2's in the middle must be divisible by 3 and >= 9. The 2's in the middle form two lines of blinkers, which decay from opposite ends (Shown here at Gen 100).

Spaceship producers Eric also asked about numbers N which die in exactly N generations. A trivial case is N=10, and I'm not sure we'll ever see another example. However, I've pretty much convinced myself that there are larger ones: There are digit strings which can exist within a longer number and which produce spaceships traveling east or west. Here are the ones that I've found.

Using things like this, we can do "slow *WSS constructions", similar to the slow glider constructions which it is believed can build just about anything. (Or, if we find an appropriate collision, we can crash the spaceships into each other to form gliders, and then use those to do slow glider constructions.)

So presumably there's a number M that produces a computer that runs the particular program that I'll describe. Now form a larger number N consisting of the digits of M followed by a string of digits that produce a binary representation of M. (For example, a long string of 1's with an occasional 3 in the middle forms a pair of blocks at each 3; the presence or absence of such a pair represents one bit.) The computer is programmed to read this representation of M, use it to compute both M and the digit string that produced the pairs of blocks, concatenate them to form the value of N, and then self-destruct after N generations.

Objects from digits Spaceships from digits Turning to patterns that don't die, I've found numbers that produce many of the small named objects and patterns. (I'm especially fond of the Pentadecathlon.)

Switch engines The smallest number which shows infinite population growth is 154299. In Gen 539 it produces a block-laying Corderman Switch Engine. Also shown here is 4114073236, which produces the glider shooting Switch Engine. (Both shown here at Gen 1500.)

Hickerson's webpage on this subject gives more information on additional topics such as

Logic Elements
New Herschel Conduit Discoveries

Herschel-controlled glider demultiplexer
Brice Due, 23 August 2006
Last August, Brice Due ran some interesting searches with Paul Callahan's catalyst search program, ptbsearch. His first discovery was a compact 'demultiplexer' -- a Herschel-to-boat converter where the boat can be used to reflect a glider. Unlike previously known Herschel-to-boat converters, the glider has a clear path through the circuit if the boat is not present:

F171 Herschel conduit discovered by Brice Due on 31 Aug 2006
The next discovery was a previously unknown F171 Herschel conduit -- the first new addition to the elemental Herschel conduit list in almost eight years:

glider #22: Brice Due, 2 September 2006
Another unrelated ptbsearch discovery was a Herschel-to-glider converter, #22, with a new output lane:

oscillator-supported versions of Brice Due's F171 conduit --
sparks in several possible locations suppress an extra blinker
The base F171 conduit consists of just three eaters, and in this form is unusually slow to recover (it takes 227 ticks before a second Herschel can follow the first.) It is possible to improve the recovery time to 120 ticks with any of a number of oscillators to suppress the transient blinker -- plus an extra eater to suppress the glider that ordinarily removes the blinker. Here are p2, p3, p5, p7, and p15 variants of the F171:

The F171 also produces an extra glider, making it H-to-G converter #23.

glider #24: boojum cleanup of converter by Paul Callahan, 1996
A few more miscellaneous H-to-Gs have also been added to the revised H-to-G collection (not related to recent ptbsearch searches). H-to-G #24, like #21, is a foray into composite technology; a "dirty" H-to-G converter discovered by Paul Callahan leaves an extra block which can be cleaned up by a boojum reflector:

glider #25: boojum cleanup of alternate dirty converter
Like #24, H-to-G #25 is a "dirty" conduit that requires an extra cleanup step, and it uses the identical boojum reflector to do it -- but its output glider and block appear in different places:

A glider following on the same path as #24's output glider would not hit the extra block; in fact, if the first block-suppressing eater is removed, the glider misses both blocks. In #25, by contrast, the following glider cleanly annihilates both blocks. This converter, without the boojum reflector, provides the timing adjustment in the p103079214841 oscillator from Golly 1.1's pattern collection.