Random Puzzles 

Workings

Hint
One of the words is AREA.

Answer

Workings

Hint
Start by labelling the number as ABCDE.

Answer
24,976.

Reasoning
We can start by labelling the digits as ABCDE.

We know that:

(i)  B = 2 x A

and:

     E = A + B

And using (i) we get:

     E = A + (2 x A)
(ii) E = 3 x A

If A = 1, this isn't allowed (as there are no 1's in the puzzle).

If A = 2, then B = 4, and E = 6.

If A = 3, then B = 6, and E = 9, but this isn't allowed (as C has to be the largest digit).

So, A = 2, B = 4, E = 6, and we now have to find C and D.

We also know that:

D = (A + B + C + D + E) ÷ 4

And using (i) and (ii) we get:

D = [A + (2 x A) + C + D + (3 x A)] ÷ 4

so:

3 x D = (6 x A) + C

so:

(iii) D = [(6 x A) + C] ÷ 3

C can only be 7, 8 or 9 (as it's the largest digit, and we've already found 6) and (iii) tells us that it must be a multiple of 3, which means that C = 9. Leaving D = 7.

So the final number is: 24976.

Double-Checking
The answer is 24976.

The fourth digit is a quarter of the total of all of the digits.
   A + B + C + D + E = 2 + 4 + 9 + 7 + 6 = 28, and 28 ÷ 4 = 7.

The second digit is twice the first digit.
   4 = 2 x 2.

The third digit is the largest.
   9 is the largest digit.

The last digit is the sum of the first two digits.
   6 = 2 + 4.

Workings

Hint
Try working backwards.

Answer
15.

Reasoning
One quarter of 360 is 90.

One third of 90 is 30.

One half of 30 is 15.

Workings

Hint
Pages 1 to 9 are one digit each.

Answer
There are 65 pages, and the number 9 appears six times.

Reasoning
The first 9 pages each require one digit per page, which leaves 112 digits remaining. This requires an additional 112 ÷ 2 = 56 more pages. For a total of 9 + 56 = 65 pages.

Each block of ten contains a single 9, for a total of six 9's (as we've only reached page 65, and we don't have to worry about the 9's in the 90's either).

Double-Checking
The 65 pages are (with the six 9's highlighted):

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65.

If we remove the spaces, we get:

1​2​3​4​5​6​7​8​9​1​0​1​1​1​2​1​3​1​4​1​5​1​6​1​7​1​8​1​9​2​0​2​1​2​2​2​3​2​4​2​5​2​6​2​7​2​8​2​9​3​0​3​1​3​2​3​3​3​4​3​5​3​6​3​7​3​8​3​9​4​0​4​1​4​2​4​3​4​4​4​5​4​6​4​7​4​8​4​9​5​0​5​1​5​2​5​3​5​4​5​5​5​6​5​7​5​8​5​9​6​0​6​1​6​2​6​3​6​4​6​5​. Which is 121 digits.