Weekender numbers

A Weekender number is, for any integers k ( > 1) and n ( > 0)

W = ( n / 2 ) ( ( ( k – 2 ) n ) + 4 – k ) where W is divisible by k

k is described as the order of the Weekender number. W/k is the count.

There is a trivial solution, W = k for any k, when n = 1.

Weekender numbers of count <= 8 are considered feasible.

The Weekender Theorem is that there are no non-trivial, feasible Weekender numbers of order > 8. I believe there is a proof of this, but it currently escapes my recollection.

Weekender numbers were first proposed by Patrick Heesom and Rupert Morrish ~1992 as an exercise in arranging beer bottles in k-gons where k drinkers are drinking equal numbers of beers.

Advertisements

One Response to “Weekender numbers”

  1. deinotes Says:

    The feasible and non-trivial Weekender numbers (with k and n less than 2^32):
    k = 3; n = 3; W = 6; count = 2
    k = 3; n = 5; W = 15; count = 5
    k = 3; n = 6; W = 21; count = 7
    k = 4; n = 4; W = 16; count = 4
    k = 5; n = 5; W = 35; count = 7
    k = 8; n = 4; W = 40; count = 5
    Raising the feasibility threshold to 9 gives us 2 more:
    k = 4; n = 6; W = 36; count = 9
    k = 15; n = 5; W = 135; count = 9
    There are none at 10, but another 9 under 17:
    k = 6; n = 6; W = 66; count = 11
    k = 3; n = 8; W = 36; count = 12
    k = 8; n = 6; W = 96; count = 12
    k = 12; n = 6; W = 156; count = 13
    k = 5; n = 7; W = 70; count = 14
    k = 24; n = 6; W = 336; count = 14
    k = 3; n = 9; W = 45; count = 15
    k = 4; n = 8; W = 64; count = 16
    k = 7; n = 7; W = 112; count = 16
    And a couple more at 20:
    k = 6; n = 8; W = 120; count = 20
    k = 35; n = 7; W = 700; count = 20

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s


%d bloggers like this: