Answer: No

Can Alice Always Win?

With perfect play, neither player can force a win. The game ends in a draw.

The key trick: use powers of 2

Write every number as 2k. Then three chosen numbers have product 1 exactly when their exponents add to 0.

1/16−4
1/8−3
1/4−2
10
2+1
4+2
8+3
16+4

All winning triples

1/16 · 1 · 16 = 1   (−4 + 0 + 4 = 0)
1/16 · 2 · 8 = 1   (−4 + 1 + 3 = 0)
1/8 · 1 · 8 = 1   (−3 + 0 + 3 = 0)
1/8 · 2 · 4 = 1   (−3 + 1 + 2 = 0)
1/4 · 1 · 4 = 1   (−2 + 0 + 2 = 0)

Play Alice vs. perfect Bob

You are Alice. Choose any number.
Alice
—
Bob
—

Why Alice cannot guarantee a win

This is a finite strategy game. Checking every legal continuation with minimax shows that Alice has no first move that forces a win. For every opening, Bob can respond so that Alice never completes one of the five product-1 triples. Alice can also prevent Bob from forcing a win, so perfect play gives a draw.

Interview shortcut: transform products into exponent sums, enumerate the five winning triples, and recognize the game as a tic-tac-toe-style hypergraph game. Then ask whether the first player has a forced winning line. Exhaustive minimax says no.
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