A short essay on what the BEST solutions for various N were. ============================================================ The contest winner is the contest winner ... my intent is to honor those runner-ups especially worthy of mention. 1. The choice of N=37, N=98, and N=153 were arbitrary - designed only to provide three different modulos base three since I was aware of the pigeonhole theory. The "qualifying" rounds were necessitated by the difficulty of checking "coverage" of the solution sets (which took much more time than actually computing the solutions!). I did NOT anticipate that so few entries would progress to the second round!!! RECOGNITION AWARD 1: To "The Happy Hacker" for DuckAndCover It is hard to dispute the efficacy of this entry. DuckAndCover found the SMALLEST number of tickets of all entries for all but four values up to N=96. Unfortunately, DuckAndCover does not work for N>96! Like several entries, DuckAndCover tries to find the "best" division of the N tickets into three subsets, and then finds a complete covering for each subset. DuckAndCover used published coverings at the La Jolla repository http://sdcc12.ucsd.edu/~xm3dg/cover.html to accomplish this feat - unfortunately these only cover up to M=32 and hence DuckAndCover only works up to N=96. While this stretches the definition of "precomputed solution" to its limits, I had previously ruled that the program was legal since the data it used did NOT solve the problem that was presented (even though it was EXTREMELY useful when N less than 97!). RECOGNITION AWARD 2: To Vincent Goffin for "ticketysplit" ticketysplit did not make use of the LaJolla tables in the program ... yet it managed to find about half of the DuckAndCover solutions for N less than 96 and even managed to find BETTER ones in four cases! Beyond N=96, nobody came close to ticketysplit's solutions (of those I looked at). Unfortunately, ticketysplit only found a 35 ticket solution for the N=37 case and did not advance to the second round. The following table has the best solutions that I found by running some of the programs. NOTE - the "coverage" for these cases was never checked for any of these runs although the approaches seemed to consistently cover at lower values of N. These are (I think) the best solutions available from the entries I looked at ... the starred solutions are from Vincent Goffin's "ticketysplit" and the others (most N<96) are from DuckAndCover from The Happy Hacker. 0 1 2 3 4 5 6 7 8 9 20 9 11 12 14 16 30 18 20 22 23 25 27 29 31 34 34 40 36 38 41 43 45 45 55 58 63 65 50 75 78 83 85 95 98 103 105 115 120 60 130 135 145 150 162 171 182 188 200 209 70 220 226 238 247 253* 258* 275* 292* 305 314 80 325 331 352 360 372 381 392 398 419 428 90 448 458 465 465 496 527 558 586* 614* 642* 100 669* 696* 723* 744* 765* 786* 821* 856* 891* 928* 110 965* 1002* 1039* 1076* 1113* 1150* 1187* 1224* 1261* 1298* 120 1335* 1375* 1415* 1455* 1495* 1535* 1575* 1577* 1579* 1581* 130 1599* 1617* 1635* 1647* 1659* 1671* 1696* 1721* 1746* 1761* 140 1776* 1791* 1813* 1835* 1857* 1865* 1873* 1881* 1892* 1903* 150 1914* The table stopped at 150 only because I got tired of it all! If you want more, the code for ticketysplit will be on the website. =Fred