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.

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August 17, 2015 at 4:56 am |

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