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Twenty six words put G first and M third, and seven of them repeat a letter.
A takes the second box in thirteen of the twenty six, fifty percent. E and I take four each, U takes three and O takes two.
GAMES, GAMER, GAMED and GAMEY come off one stem and differ only at the end. GAMUT, GAMIN and GAMIC step outside it, and GAMMA and GAMMY double the M.
GAMES and GAMUT are the two most people would produce. GAMUT is the better opening because it avoids the crowded GAME stem entirely.
GAMMA, GAMMY, GEMMA, GEMMY, GUMMA and GUMMY all put a second M in the fourth box. M takes that box in seven of the twenty six, tied with E on seven.
So a visible double points at the M branch, and the vowel then decides. Those words differ from each other in only the second and last boxes.
Y closes five of the page, nineteen percent, with A and S on four each. A double plus a Y leaves you a very short list.
GAMUT. Both are common, but GAMES sits in a stem with three near duplicates while GAMUT tests U and T at once.
A pair of matched things, often used of windows or rings. It sits in the E branch and is the least reachable entry there.
Both appear on this list as variant spellings of the same stew. They differ only in the second box.