Game Description

Players: Alice and Bob
They take turns choosing one number at a time from the following set, without replacement:

\[ { \frac{1}{16},\ \frac{1}{8},\ \frac{1}{4},\ 1,\ 2,\ 4,\ 8,\ 16} \]

The first player to obtain three numbers whose product is 1 wins.

Alice moves first.

Step 1: Change the Problem to Additive

Take the base-2 logarithm of all numbers:

Number \( \log_2 \) value
\( \frac{1}{16} \) \( -4 \)
\( \frac{1}{8} \) \( -3 \)
\( \frac{1}{4} \) \( -2 \)
\( 1 \) \( 0 \)
\( 2 \) \( 1 \)
\( 4 \) \( 2 \)
\( 8 \) \( 3 \)
\( 16 \) \( 4 \)

So instead of finding three numbers whose product is 1, we’re now finding three numbers whose sum of logs is:

\[ \log_2(a) + \log_2(b) + \log_2(c) = \log_2(abc) = \log_2(1) = 0 \]

Step 2: Reformulate

The problem becomes:

Alice and Bob take turns choosing one number from the set
\[ {-4, -3, -2, 0, 1, 2, 3, 4} \]
without replacement.
Whoever gets three numbers that sum to 0 wins.

Step 3: Analyze the Winning Lines

Let’s list all the possible triplets from this set that sum to 0:

  1. \( {-4, 0, 4} \)
  2. \( {-3, 0, 3} \)
  3. \( {-2, 0, 2} \)
  4. \( {-4, 1, 3} \)
  5. \( {-3, 1, 2} \)

There are exactly 5 winning lines.

(Note: This is often confused with the game of choosing numbers from 1 to 9 that sum to 15, which has 9 numbers, 8 winning lines, and is isomorphic to Tic-Tac-Toe. Our game here is much sparser!)

Step 4: Result

Because this game has only 8 numbers and 5 winning lines, it is significantly harder to form winning combinations than in Tic-Tac-Toe.

  • Alice cannot force a win. She has too few winning lines to easily set up an unblockable “double threat” if Bob plays optimally.
  • By the strategy-stealing argument (having an extra number is never a disadvantage), Alice can guarantee at least a draw.
  • If Bob plays optimally (e.g., picking central numbers like \( -3 \) or \( 0 \) to block multiple lines), the game will end in a draw.
  • But if Bob makes a mistake, Alice can win.

Conclusion

\[ \boxed{ \text{Alice cannot always win, but she can force at least a draw with optimal play.} } \]

Reference