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Fifty nine words start with Q, and fifty five of them put a U straight after it.
U follows the Q in fifty five of the fifty nine, ninety three percent. QAIDS, QANAT, QIBLA and QOPHS are the only exceptions and none is a word you would produce unprompted, so the second box is settled before you begin.
The third box is the live one. I and A take nineteen words each, E takes ten and O takes seven. That is a genuine open split and it decides more than anything else on the page.
QUEEN, QUICK, QUIET and QUOTE are the entries most people would reach for, with QUEST, QUILT and QUIRK close behind. That is a strong haul for a letter this rare.
S closes thirteen of the fifty nine, twenty two percent, with T on ten and E on nine. QUEST tests two of those at once and sits in the smaller E branch, which makes it a better opening than QUEEN despite being the less common word.
Four of them. QAIDS, QANAT, QIBLA and QOPHS are the entries here without one, all borrowed from other languages.
I or A. They take nineteen words each, which is thirty eight of the fifty nine between them.
Seven words carry one, mostly QUEEN, QUELL, QUILL and QUASS. That is a modest share, so a repeat narrows usefully without settling the page.