Random Puzzles 

Workings

Hint
Start with the possible answers where the first digit is a prime number, and then look at the third digit.

Answer
23,475.

Reasoning
By (1), the first digit is prime:
   2----
   3----
   5----
   7----

By (3), the third digit is twice the first digit, so we can eliminate 5---- and 7----:
   2-4--
   3-6--

By (4), the fourth digit is the third digit plus three:
   2-47-
   3-69-

By (5) the fifth digit is the difference between the first digit and the fourth digit:
   2-475
   3-696

We know from the introduction that no digit is repeated, so we can eliminate 3-696. And, by (2) the second digit is the fifth digit minus the first digit:
   23475

Workings

Hint

Answer
6: the numbers denote the number of letters each month has starting with January, therefore August has 6 letters.

Workings

Hint
A tricky base 8 question.

Answer
61.

Each term is a square expressed in base eight, 7 2=49 = 61 in base 8.

Workings

Hint
The answer is not 31.

Answer
The largest sum that can be made is 34.