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The first 61 years of this rivalry were anything but that. From 1923-1984, UCLA went 23-2 against Arizona—and only seven of those games were played during John Wooden's 27 seasons with the Bruins. Imagine how lopsided this would be if they had played twice per season during those years.
Over the past 30 years, though, this has been the best West Coast rivalry and one of the most must-see pairings in the entire country.
Just how evenly matched have these teams been? They are dead even with 35 wins apiece over the last 70 games. Of those 70 games, 48 have been decided by 12 or fewer points; 15 were decided by a single possession.
Oddly enough, when there have been blowouts, Arizona always benefited. Nine of those 70 games were decided by 24 or more points—all nine were wins for the Wildcats—but it has been more than a decade since the last one of those ugly games.
Since 1988, Arizona and UCLA have combined to appear in 27 Sweet 16s and eight Final Fours while winning one national championship apiece. One or the other has won at least a share of the Pac-10/Pac-12 regular-season title in 22 of the past 29 years.
Despite all that recent history, there's no good nickname for the rivalry, and there certainly hasn't been a trophy awarded to its winner.
What if we just sort of mashed together their mascots? Instead of a bruin or a wildcat, how about a Bronze Bearcat? Sure, bearcats are indigenous to forests in Southeast Asia as opposed to deserts in Southwest U.S., but who doesn't love a good ironic trophy?