# [r-t] All runs in a quarter, proper version!

Alexander Holroyd holroyd at math.ubc.ca
Fri Jan 30 09:53:32 UTC 2015

```On Fri, 16 Jan 2015, Philip Earis wrote:

> As the overall aim here is to get all 480 run rows, I’m tempted to
> generalise and rephrase accordingly...what’s the shortest length
> 8-part anyone can produce which contains all 480 runs? Elegance and
> maximal symmetry are desirable.
>
> Posing the question like this lends itself to interesting architectured
> approaches...previous 480-run solutions (whilst superb) have applied
> traditional constraints such as consisting of a number of different,
> treble-dominated methods.

Thanks for this challenge, Philip.  After considerable effort I believe I
have a good solution.  All 480 runs (including 1234s this time) in a
longish quarter length (3.2 runs per row on average!).  16-part structure
plus exact rotation symmetry.  At most a 2-row gap between "potential
runs" (including things like 7812).  No long places.  No single or double
changes.  Lots of variety in run types, and lots of movement across 45.

Also the simplicity and obvious structure make it the perfect thing for
any fledgling band to toss off on the way to the pub.

1536 Major
Alexander E Holroyd

12345678
------------
x   34562871 )
18  35426781 )A
z   45678123 )
------------
2A  23456781
------------
x   45673182 )
38  54637812 )
y   65437281 )B
16  64573218 )
z   54321876 )
------------
x   32184675 )
38  23148765 )
y   12348576 )
36  21384567 )C
y   32184756 )
16  31248765 )
z   21876543 )
------------
6C  87654321
------------
B   56781234
------------
4A  12345678
------------

x = 56-4.56.1.56.1.34.5-36.5.34-4.3.6-56-5.34.1.34.5-5-3.4.3.56-5-
y = 4-4-6.3.56.34.1.34-5.4-56.1.56.34.6.3-5-5
z = -4-34.6.5.6-4-4.56.1.56.4-34-3.6.5-56.4.36-4.56.1.34.1.34.5-34

Blocks A, B and C each have rotational symmetry

24 each 1234s, 2345s, 3456s, 4567s, 5678s, forwards and backwards, front
and back (3.2 rows per run!)

Block B grid:

http://www.boojum.org.uk/cgi-bin/line.pl?bells=8&pn=56-4.56.1.56.1.34.5-36.5.34-4.3.6-56-5.34.1.34.5-5-3.4.3.56-5-3.4-4-6.3.56.34.1.34-5.%204-56.1.56.34.6.3-5-5.6-4-34.6.5.6-4-4.56.1.56.4-34-3.6.5-56.4.36-4.56.1.34.1.34.5-34&title=Block%20B&qs=Grid&lines=1&line=2&line1=a&line2=n&style=1&place-bellsx=on&miss-numbers=never&rule=36%3B61&action.x=1

Block A line:

http://www.boojum.org.uk/cgi-bin/line.pl?bells=8&pn=56-4.56.1.56.1.34.5-36.5.34-4.3.6-56-5.34.1.34.5-5-3.4.3.56-5.1.-4-34.6.5.6-4-4.56.1.56.4-34-3.6.5-56.4.36-4.56.1.34.1.34.5-34&title=Block%20A&qs=Line&action.x=1

```