Question 1184829: Three players play the game “rock-paper-scissors”. In every round, each
player simultaneously shows one of these shapes. Rock beats scissors, scissors beat
paper, while paper beats rock. If in a round exactly two distinct shapes are shown
(and thus one of them is shown twice) then 1 point is added to the score of the
player(s) who showed the winning shape, otherwise no point is added. After several
rounds it occurred that each shape had been shown the same number of times. Prove
that the total sum of points at this moment was a multiple of 3.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's a proof that the total sum of points in this rock-paper-scissors game is a multiple of 3, given that each shape has been shown the same number of times:
**1. Consider the possible outcomes of a single round:**
* **All three shapes are different:** No points are awarded.
* **Two shapes are the same, one is different:** One or two players score a point.
* **All three shapes are the same:** No points are awarded.
**2. Focus on the rounds where points are awarded:**
Points are only awarded when exactly two different shapes are shown. Let's analyze the three possible scenarios:
* **Two rocks, one scissors:** Two players get a point (2 total points awarded).
* **Two papers, one rock:** Two players get a point (2 total points awarded).
* **Two scissors, one paper:** Two players get a point (2 total points awarded).
Notice that in each case, *two* points are awarded in each scoring round.
**3. Relate this to the overall game:**
Let 'n' be the number of times each shape is shown. The total number of rounds played is 3n (since each of the three shapes must be shown n times).
**4. Consider the total points awarded:**
Since each scoring round awards 2 points, the *total* number of points awarded across all scoring rounds will *always* be an even number. However, this doesn't directly prove it's a multiple of 3.
**5. The key insight:**
Consider all possible combinations of three moves. There are 27 possible combinations (3 choices for each of the 3 players: 3*3*3 = 27).
* 3 of these combinations result in all three players playing the same move (rock, rock, rock; paper, paper, paper; scissors, scissors, scissors). No points are awarded.
* 6 of these combinations result in all three moves being different (rock, paper, scissors; rock, scissors, paper; paper, rock, scissors; paper, scissors, rock; scissors, rock, paper; scissors, paper, rock). No points are awarded.
* This leaves 18 combinations where exactly two moves are the same. In each of these 18 combinations, two points are awarded.
Since there are 18 scoring combinations, and each awards 2 points, there are a total of 36 potential points across all possible combinations. Crucially, 36 is divisible by 3.
**6. Applying this to the game:**
Since each shape is shown the same number of times, the distribution of rounds played *must* mirror the overall distribution of possible combinations. For every instance of "rock, rock, paper," there must eventually be a corresponding "rock, paper, rock" and "paper, rock, rock" (and so on for all the other scoring combinations).
Therefore, the total number of points awarded in the actual game, which is some subset of the 36 potential points, must also be a multiple of 3.
**Conclusion:**
Because the total possible points across all combinations are a multiple of 3, and the actual game's point distribution mirrors this overall structure (due to each shape being shown the same number of times), the total sum of points in the game must also be a multiple of 3.
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