From Newsgroup: rec.puzzles
As spoiler warning space, I show some of the Witches' lines
from the play MacBeth.
FIRST WITCH. When shall we three meet again?
In thunder, lightning, or in rain?
SECOND WITCH. When the hurlyburly's done,
When the battle's lost and won.
THIRD WITCH. That will be ere the set of sun.
FIRST WITCH. Where the place?
SECOND WITCH. Upon the heath.
THIRD WITCH. There to meet with Macbeth.
FIRST WITCH. I come, Graymalkin.
ALL. Paddock calls. Anon!
Fair is foul, and foul is fair.
Hover through the fog and filthy air. Exeunt.
Enter the three Witches.
FIRST WITCH. Where hast thou been, sister?
SECOND WITCH. Killing swine.
THIRD WITCH. Sister, where thou?
FIRST WITCH. A sailor's wife had chestnuts in her lap,
And mounch'd, and mounch'd, and mounch'd. "Give me," quoth I.
"Aroint thee, witch!" the rump-fed ronyon cries.
Her husband's to Aleppo gone, master the Tiger;
But in a sieve I'll thither sail,
And, like a rat without a tail,
I'll do, I'll do, and I'll do.
SECOND WITCH. I'll give thee a wind.
FIRST WITCH. Thou'rt kind.
THIRD WITCH. And I another.
FIRST WITCH. I myself have all the other,
And the very ports they blow,
All the quarters that they know
I' the shipman's card.
I will drain him dry as hay:
Sleep shall neither night nor day
Hang upon his penthouse lid;
He shall live a man forbid.
Weary se'nnights nine times nine
Shall he dwindle, peak, and pine;
Though his bark cannot be lost,
Yet it shall be tempest-toss'd.
Look what I have.
SECOND WITCH. Show me, show me.
FIRST WITCH. Here I have a pilot's thumb,
Wreck'd as homeward he did come. Drum within.
THIRD WITCH. A drum, a drum!
Macbeth doth come.
ALL. The weird sisters, hand in hand,
Posters of the sea and land,
Thus do go about, about,
Thrice to thine, and thrice to mine,
And thrice again, to make up nine.
Peace! The charm's wound up.
---------------------------------------------------
David Entwistle <
qnivq.ragjvfgyr@ogvagrearg.pbz> posted:
1,2,2,4,8,3,2,2,4,6,4,8,2,4,2,4,3,2,1,6,8...
Gur irel arkg gjb qvtvgf va guvf frdhrapr ner rnpu rvtug,
guhf cebqhpvat guerr pbafrphgvir rvtug'f -- frr orybj.
Show that the numbers 0, 5, 7, and 9 never appear, but arbitrarily long sequences of 8's appear.
Jr svefg cebir gung gur "onq" qvtvgf -- 0, 5, 7, 9 -- pnaabg bpphe. Vg vf fhssvpvrag gb cebir gung gur svefg onq qvtvg qbrfa'g bpphe. Sbe rknzcyr
(9,8) --> (7,2) ohg gung onq 7 vf bs ab jbeel fvapr vg jnf cebqhprq ol
na rneyvre onq 9. Vafcrpgvba fubjf gung (3,3) --> (9) vf gur bayl jbeel.
Ab tbbq qvtvgf cebqhpr (k3) fb (3)(32) naq (3)(3) ner gur bayl jnlf gb
trg gur haqrfverq (3,3).
(3) vf cebqhprq bayl ivn 1&3 naq arvgure (1,k) abe (3,k) cebqhpr (32).
(3,3) pna or cebqhprq ol (1,3,1) be (3,1,3), rvgure bs juvpu erdhverf na rneyvre (3,1).
Fb ab onq qvtvgf pna nevfr.
X pbafrphgvir vafgnaprf bs (8) cebqhpr X-1 vafgnaprf bs (64) juvpu gura
lvryq 2X-3 pbafrphgvir vafgnaprf bs (24) juvpu cebqhpr 4X-7 pbafrphgvir
(8)f. Vs X=3, 4X-7 vf 5, fb 3 8'f lvryq 5 8'f juvpu lvryq 13 8'f
lvryqvat 45 8'f naq fb ba. Naq jr QB unir 888 nf jnf zragvbarq nobir.
I think I'm okay with those questions, but I'm not sure about this - can
you prove that neither sequence is periodic? I think I can, as I've seen
it demonstrated for another related sequence.
Tvira gung OBGU gur frdhraprf lbh qrsvar raq hc jvgu rire-tebjvat
frdhraprf bs 8'f, V guvax guvf vf gur rnfvrfg dhrfgvba! Nal crevbq
zhfg or n svavgr crevbq ohg gur frdhraprf bs pbafrphgvir 8'f tebj
jvgubhg yvzvg.
Znlor gurer'f n cebbs gung qbrfa'g eryl ba fcrpvsvp tebjvat frdhraprf,
ohg V'z pbagrag ...
Purref,
wnzrfqbjnyyra @ Tznvy
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