Random Puzzles 

Workings

Hint
How many squares of size 1 x 1 are there?

Answer
There are 116 squares without a rook.

Reasoning
There are 61 squares of size 1 x 1.
There are 37 squares of size 2 x 2.
There are 15 squares of size 3 x 3.
There are  3 squares of size 4 x 4.

Giving a total of 61 + 37 + 15 + 3 = 116 squares.

Workings

Hint
How many miles were left at the start of yesterday?

Answer
336 miles.

Reasoning
To work these numbers out, it is perhaps easier to start at the end and work backwards.

Yesterday I travelled ¹/₃ of the distance to leave 21 miles, therefore there were 31.5 miles (=21 ÷ ²/₃) at the start of yesterday.

On day 3 I travelled ³/₄ of the distance to leave 31.5 miles, therefore there were 126 miles (=31.5 ÷ ¹/₄) at the start of day 3.

On day 2 I travelled ¹/₂ of the distance to leave 126 miles, therefore there were 252 miles (=126 ÷ ¹/₂) at the start of day 2.

On the first day, I travelled ¹/₄ of the distance to leave 252 miles, therefore there were 336 miles (=252 ÷ ³/₄) at the start of the first day.

Double-Checking
On the first day, I travelled one quarter of the distance (84, leaving 252).

On day two, I travelled one half of the remaining distance (126, leaving 126).

On day three, I travelled three quarters of the remaining distance (94.5, leaving 31.5).

Yesterday I travelled one third of the remaining distance (10.5).

I now have 21 miles left to travel.

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
The words are all vegetables.

Answers
cbbg   = cabbage clflwr = cauliflower ccmbr  = cucumber crgtt  = courgette crrt   = carrot clrc   = celeriac