During a recent BrainBashers thinking contest(!), the total number of points scored by the first six players was 103 and every score was above zero.

The first player scored half the points of the second player, who in turn scored 6 points fewer than the third player.

The third player in turn scored two thirds the points of the fourth player.

The fifth player managed to score the same number of points as the difference between the first and fourth player's points.

Finally, the sixth player scored 14 fewer points than the fifth player.

Can you determine how many points the sixth player managed to score?

Puzzle Copyright © Kevin Stone

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Hint

The fourth player is the key to this tricky question.

Answer

9 points.

Respectively the scores were 7, 14, 20, 30, 23, 9.

Reasoning

If we label the six players A, B, C, D, E, and F we know that:

[1] A + B + C + D + E + F = 103

and:

A = B ÷ 2

B = C - 6

C = D x 2 ÷ 3

E = D - A

F = E - 14

Note: if instead we choose E = A - D, we'd later see that we end up with a negative value for E, which isn't allowed.

Since D is the letter we're missing information for, it's best to find all of the other letters in terms of D. These steps are left as an exercise (e.g. use C in the equation for B, etc), but the result is:

A = ( D - 9) ÷ 3

B = (2D - 18) ÷ 3

C = (2D ) ÷ 3

D = (3D ) ÷ 3

E = (2D + 9) ÷ 3

F = (2D - 33) ÷ 3

Setting D = 3D ÷ 3 makes it slightly clearer to see when adding in the next step.

We can then use these in [1] to find that 12D = 360, so D = 30. Which allows us to find F = 9.