(With help from Dr. Z's Experimental Mathematics Class )

Written: Nov. 30, 2004.

A (k,n)-*Perfect Rhythmic Tiling* (a concept due to
composer Tom JOHNSON) is a partition(!) of the set of integers 0 through nk-1
into n mutually-disjoint arithmetical progressions of length k, with
*all different spacings*. In Jean-Paul DELAHAYE's fascinating
article in *Pour La SCIENCE* (Novembre 2004, p. 93), it is
mentioned that none are known with k=4. Well, in yesterday's class,
the class (together!) wrote a Maple program,
TomJohnson
, that computes such Perfect Rhythmic Tilings.
That night, Lara Pudwell ran *Tom(15,4);*, and after
776 seconds of CPU time, got the following beautiful tiling.

[0, 16, 32, 48], [1, 3, 5, 7], [2, 13, 24, 35], [4, 22, 40, 58], [6, 21, 36, 51], [8, 14, 20, 26], [9, 10, 11, 12], [15, 29, 43, 57], [17, 25, 33, 41], [18, 30, 42, 54], [19, 23, 27, 31], [28, 37, 46, 55], [34, 39, 44, 49], [38, 45, 52, 59], [47, 50, 53, 56] .

Added Dec. 1, 2004: We were scooped!, see Tom Johnson's message to Lara Pudwell .

Personal Journal of Ekhad and Zeilberger .