# Brain Teasers and Puzzles

## Puzzle Details

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 than the fifth player.

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

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

If the six player were A, B, C, D, E and F we know that:

A + B + C + D + E + F = 103     [1]

and

A = B ÷ 2
B = C - 6
C = D x 2 ÷ 3
E = D - A             (see note at end)
F = E - 14

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, but the result is:

A = (2D - 18) ÷ 6
B = (2D - 18) ÷ 3
C = 2D ÷ 3
E = (2D + 9) ÷ 3
F = (2D - 33) ÷ 3

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

Small note: if we chose E = A - D (instead of E = D - A) we'd end up with a negative value for E, which isn't allowed.