Random Puzzles 

Workings

Hint
Start by labelling the number as ABCDE.

Answer
24976.

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
One inch = 25.4mm.

Answer
Millimetres in a mile.

Reasoning
One inch = 25.4mm.

So:

1 mile = 1760 yards = 1760 x 3 feet = 1760 x 3 x 12 inches = 1760 x 3 x 12 x 25.4 millimetres = 1,609,344mm in 1 mile.

And:

2 kilometres = 2000 metres = 2000 x 100 centimetre = 2000 x 100 x 10 millimetres = 2000 x 100 x 10 ÷ 25.4 inches = 2000 x 100 x 10 ÷ 25.4 x 16 = sixteenths of an inch ≈ 1,259,843 sixteenths of an inch in 2 kilometres.

Workings

Hint
The country spans more than one word.

Answer
Peru.

A popular newspaper printed part of a page that pointed out that port officials use a pencil, not a pen, when they prepare paPER Updates to their port reports.

Workings

Hint
Some are much more common than others.

Answers
Common: vacuum, vacuumed, vacuuming, vacuums.

Less common: continuum, continuums, muumuu, muumuus.

Much less common: carduus, carduuses, continuua, duumvir, duumviral, duumvirate, duumvirates, duumviri, duumvirs, individuum, individuums, lituus, lituuses, menstruum, menstruums, mutuum, mutuums, paramenstruum, paramenstruums, residuum, residuums, squush, squushed, squushes, squushier, squushiest, squushing, squushy, triduum, triduums, ultravacuum, ultravacuums, weltanschauung, weltanschauungen, weltanschauungs.