Cordial Minuet Resurrected

You and a stranger see the same cursed grid — the same 36 numbers, every row, column, and diagonal summing to 111 — but rotated 90 degrees between you. Each of you secretly chooses one column for yourself, and one column to hand to your opponent. When both picks are sealed, they’re revealed together: your column and the one they chose for you intersect at a single cell. That number is your score. Higher score wins. Neither of you sees the other’s choice until both are locked in.