Mathematical Puzzles 

Workings

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 from the clues:

   A = B ÷ 2
   B = C − 6
   C = D x 2 ÷ 3
   E = D − A
   F = E − 14

Note that it could be E = A − D or E = D − A, but using A − D we'd end up with a negative value for E later, which isn't allowed (so we'd have to try again with D − A anyway).

As we have no information for D, 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), 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

Writing it as D = 3D ÷ 3 makes things slightly clearer in the next step.

We then use these in [1] to find that:

   (12D − 51) ÷ 3 = 103
         12D − 51 = 309
              12D = 360
                D = 30

Therefore, F = (2D − 33) ÷ 3 = 9.

Workings

Hint
Look at Digit 3 first.

Answer
6394.

Reasoning
Labelling the clues:

Digit 2 is smaller than Digit 4 Digit 4 is two thirds of Digit 1 Digit 1 is two thirds of Digit 3 Digit 3 is three times Digit 2
By (3), Digit 3 is divisible by 3:

   Digit 3 = 3, 6, 9

By (3), Digit 1 is two thirds of Digit 3:

   Digit 1 = 2, 4, 6.

But, by (2), Digit 1 is divisible by 3:

   Digit 1 = 6, and Digit 3 = 9.

By (4), Digit 3 is three times Digit 2:

   Digit 2 = 3.

By (2), Digit 4 is two thirds of Digit 1:

   Digit 4 = 4.

Workings

Hint
The answer involves a square root.

Answer
4.

The middle of each triangle contains the square root of the sum of the three surrounding numbers.

Workings

Hint
What is the total number of games played?

Answer
3.

Reasoning
There were 5 teams in the league, and every team played every other, so there were a total of 10 games played:
   Rugby     vs  Grimsby
   Rugby     vs  Hereford
   Rugby     vs  Lichfield
   Rugby     vs  Reading
   Grimsby   vs  Hereford
   Grimsby   vs  Lichfield
   Grimsby   vs  Reading
   Hereford  vs  Lichfield
   Hereford  vs  Reading
   Lichfield vs  Reading

We know that 2 points were allocated for each game played. Which means that there were a total of 20 points over the 10 games.

Since the given results add to 17, the remaining 3 points belong to Reading.