WASHINGTON — Colin Allred of Texas brought his pink quartz from the campaign trail. Sharice Davids of Kansas dropped to the floor for a lightning round of push-ups.

And Chrissy Houlahan of Pennsylvania, in her lucky jeans, paused to rub a colleague’s bald head for an extra boost of support. (“He’s got a lucky head,” she said later of Max Rose of New York.)

Of everything the 85 newly elected representatives had endured in the last two weeks of congressional orientation — welcoming parties, the maze of Capitol Hill tunnels, leadership elections — the lottery on Friday had perhaps the highest stakes.

Who gets the best office?

Forget coveted committee assignments and leadership positions. The concern of the hour was over who could land a picturesque view, or a spacious lobby to woo colleagues and constituents alike. Not to mention the quickest commute to votes or, more important, the Dunkin’ Donuts in the basement of the Longworth House Building.