There was a football tournament of three teams, A, B & C and each team played two matches. There are some patterns of each match; (i) Each team get two points for scoring a goal to opponent team. (ii) Each team get one point extra scoring goal outside form D area. (iii) Each team get penalty of one point if it conceded any goal. Goal conceded means goal is scored by opponent team. (iv) Only three players from each team scored goal. Match of team A to B: B is winner in this match and B gets four points. Team A scored two goals in match against B. No player scored goal outside form D area. Match of team A to C: C has zero points in this match. Only one player from team A scored goal outside form D area. A has secured four points from this match. Match of team B to C : B gets six points from the match and team C scored one goal more than B. . Only one player from team B scored goal outside form D area.
Question: What is the sum of the maximum goals scored by a single player in each of the three matches?

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**Solution:**

Let's analyze each match step-by-step using the given rules.

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**Match A vs B:**
- B is the winner and gets 4 points.
- Team A scored 2 goals.
- No goals outside D area.

Let *G_A* = goals scored by A, *G_B* = goals scored by B.

Given *G_A = 2*.

Points for B:
- Goals scored: $2 	imes G_B$
- Goals outside D area: 0 (no extra points)
- Penalty for goals conceded: $-1 	imes G_A = -2$

Total points for B = $2 	imes G_B - 2 = 4$

So,
$$2G_B - 2 = 4 \ 2G_B = 6 \ G_B = 3$$

Thus, B scored 3 goals.

Maximum goals by a single player in this match:
- Total goals by A: 2 (all by 3 players, but max by one player unknown yet)
- Total goals by B: 3

Since only 3 players scored goals, maximum goals by a single player can be at most 3 (all goals by one player).

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**Match A vs C:**
- C scored 0 points.
- Only one player from A scored a goal outside D area.
- A earned 4 points.

Let *G_A* = goals by A, *G_C* = goals by C.

Points for C:
- Goals scored: $2 	imes G_C$
- Goals outside D area: 0
- Penalty for goals conceded: $-1 	imes G_A$

Total points for C = $2G_C - G_A = 0$

Points for A:
- Goals scored: $2 	imes G_A$
- Goals outside D area: 1 extra point (since only one player scored outside D area)
- Penalty for goals conceded: $-1 	imes G_C$

Total points for A = $2G_A + 1 - G_C = 4$

From C's points:
$$2G_C - G_A = 0 \ 2G_C = G_A \ G_A = 2G_C$$

Substitute in A's points:
$$2G_A + 1 - G_C = 4 \ 2(2G_C) + 1 - G_C = 4 \ 4G_C + 1 - G_C = 4 \ 3G_C = 3 \ G_C = 1$$

Then,
$$G_A = 2 	imes 1 = 2$$

Maximum goals by a single player in this match:
- A scored 2 goals, with one goal outside D area by a single player.
- C scored 1 goal.

Maximum goals by a single player can be 2 (all goals by one player from A).

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**Match B vs C:**
- B earned 6 points.
- C scored one goal more than B.
- Only one player from B scored a goal outside D area.

Let *G_B* = goals by B, *G_C* = goals by C.

Given:
$$G_C = G_B + 1$$

Points for B:
- Goals scored: $2 	imes G_B$
- Goals outside D area: 1 extra point
- Penalty for goals conceded: $-1 	imes G_C = -(G_B + 1)$

Total points for B:
$$2G_B + 1 - (G_B + 1) = 6 \ 2G_B + 1 - G_B - 1 = 6 \ G_B = 6$$

Then,
$$G_C = G_B + 1 = 7$$

Maximum goals by a single player in this match:
- B scored 6 goals, with one goal outside D area by a single player.
- C scored 7 goals.

Since only 3 players scored goals, maximum goals by a single player can be 6 (all goals by one player from B) or 7 (all goals by one player from C).

We want the maximum goals scored by a single player in this match. Since C scored 7 goals, and only 3 players scored, the maximum by one player can be 7.

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**Sum of maximum goals scored by a single player in each match:**
- Match A vs B: 3
- Match A vs C: 2
- Match B vs C: 7

Sum = $3 + 2 + 7 = 12$

**Answer:** 12

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