Let's analyze the problem step by step. ## Restating the Problem - **Two players** play Ping-Pong. - The first to **21 points** wins. - **Whenever the score (A-B) can be reduced to lower terms (i.e., gcd(A, B) > 1), it is immediately reduced** (divided by gcd). - For example: - 4-2 → 2-1 - 12-9 → 4-3 - 14-12 → 7-6 - Even the **final score** (e.g., 21-7) is reduced: 21-7 → 3-1. **Question:** **What are the possible final scores for this game?** --- ## Step 1: What does “final score” mean here? The final score is the **irreducible ratio** of the winner's and loser's point totals **at the moment someone reaches 21**. Let's say Player A wins, and the un-reduced score is \( 21-b \). The final score is the reduced pair \( (21', b') \) where \( 21' = \frac{21}{d} \), \( b' = \frac{b}{d} \), and \( d = \gcd(21, b) \). But the game **never displays the unreduced score**: as soon as the score becomes reducible, it's reduced. So **at every step**, the score is in reduced form. **Key:** **Every time a point is scored, the new score is reduced if possible.** --- ## Step 2: Which scores are possible as the final (reduced) score?** Suppose Player A wins. So (WLOG) the final (reduced) score is \( (a, b) \) with \( a > b \), and \( a, b \) are coprime. But **how does the score ever reach (21, b)?** But wait: **Because the score is reduced after every point, the only way to reach (21, b) is if, after some sequence of point-scoring and reductions, the winner gets to 21, and the score at that moment is irreducible.** But let's try to **track the score through the game's evolution**. --- ## Step 3: Try Small Cases ### Try to reach 21-0 - The only way to get there is if one player wins every point. - At each step, the score is (1-0), (2-0), (3-0), ..., (21-0). - All these have gcd = 1 or the score is (k, 0), so nothing to reduce. - **So (21-0) is a possible final score.** ### Try to reach 21-1 - Suppose the loser gets 1 point at some stage. - Let's see what happens: - At (1-0): winner - At (1-1): tie - At (2-1): gcd(2,1)=1, so no reduction - At (3-1): gcd(3,1)=1, no reduction - ... - At (21-1): gcd(21,1)=1, no reduction - **So (21-1) is a possible final score.** ### Try to reach 21-2 - At (21-2): gcd(21,2)=1. So, no reduction. - **So (21-2) is possible.** ### Try to reach 21-3 - At (21-3): gcd(21,3)=3, so it's reduced to (7-1). - So, we cannot have (21-3) as a final score: if you ever reach (21-3), it's immediately reduced to (7-1). **But can the game end at (7-1)?** - Let's consider: Is it ever possible for the score to be (7-1) at the end? - For that to happen, the winner must have scored 21, the loser 3, but reduced immediately to (7-1). - Let's see: (21-3) reduces to (7-1). - So, **(7-1) is a possible final score**. --- ## Step 4: Generalize ### For any final (unreduced) score (21, b): - If \(\gcd(21, b) = d\), then the final displayed score is \((21/d, b/d)\). - For all \(b\) from 0 to 20, the possible final scores are all **coprime pairs \((a, b)\) with \(a + b = 21\), or that arise as the reduction of (21, b)**. But the key is: **At each scoring event, the score is always reduced.** ### So, which pairs (a, b) can occur at the end? - The winner must have 21 points, but due to reduction, the displayed score is \((21/d, b/d)\), with \(d = \gcd(21, b)\). - Since the winner must reach 21, and the score is always reduced, the only possible final scores are **all coprime pairs \((m, n)\), with \(m > n\), such that \(m\) is a divisor of 21**. Why? Because **the only way to end the game is when the winner reaches 21 points.** The displayed score is \((21/d, b/d)\), for some \(b\), \(0 \leq b < 21\), and \(d = \gcd(21, b)\). So, \(21/d\) must be an integer, i.e., **must divide 21**. Let’s list all possible final scores. --- ## Step 5: List All Possible Final Scores Let’s let \(d\) be a divisor of 21. The divisors of 21 are: **1, 3, 7, 21**. - For each \(d\), the possible (reduced) final score is \((21/d, b/d)\) for some \(b < 21\) with \(\gcd(21, b) = d\). So, for each divisor \(d\): ### \(d = 1\): \((21, b)\), with \(\gcd(21, b) = 1\), \(0 \leq b < 21\) - \(b\) coprime to 21, in \(0 \leq b < 21\) - 21 = 3 × 7, so numbers less than 21 coprime to both 3 and 7. - \(b = 1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20\) (all numbers less than 21 and not divisible by 3 or 7) - So, possible final scores: \((21, b)\) with \(b\) as above. ### \(d = 3\): \((7, b/3)\), where \(b < 21\), \(\gcd(21, b) = 3\) - \(b\) is a multiple of 3 but not of 7, and \(b < 21\) - Multiples of 3 less than 21: 3, 6, 9, 12, 15, 18 - Remove those also divisible by 7 (21 is the only one, which is not <21) - So, \(b = 3, 6, 9, 12, 15, 18\) - So, final scores: (7, 1), (7, 2), (7, 3), (7, 4), (7, 5), (7, 6) ### \(d = 7\): \((3, b/7)\), where \(b < 21\), \(\gcd(21, b) = 7\) - \(b\) is a multiple of 7 but not of 3, and \(b < 21\) - Multiples of 7 less than 21: 7, 14 - Now, 7 and 14; check if divisible by 3. - 7: not divisible by 3 - 14: not divisible by 3 - So, \(b = 7, 14\) - So, final scores: (3, 1), (3, 2) ### \(d = 21\): \((1, b/21)\), where \(b < 21\), \(\gcd(21, b) = 21\) - Only \(b = 0\) (since 21 is not less than 21) - So, final score: (1, 0) --- ## Step 6: Collect All Possible Final Scores List all possible final scores for the winner: ### (A) Scores of the form (21, b), with \(b\) coprime to 21, \(0 \leq b < 21\): \[ (21, 1), (21, 2), (21,