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

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 (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

Writing it as 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.

Therefore, F = 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
Using (4), Digit 3 can only be 3, 6 or 9 (as it's 3 x Digit 2).

Therefore, using (4), Digit 2 can only be 1, 2 or 3.

Using (3), Digit 1 can only be 2, 4 or 6 (as it's two thirds of Digit 3).

However, using (2), the only possible value for Digit 1 that we can find two thirds of is 6.

So, using (2), Digit 4 is 4.

So far, we have 6??4.

Using (3), Digit 1 is two thirds of Digit 3, so Digit 3 must be 9.

And, by (4), Digit 2 must be 3.

Giving 6394.

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
How many games took place?

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.