Brain Teasers and Puzzles 

Workings

Hint
The first letters of the names are: G, R, E, P, M, E, A, A.

Answer
gl    = Gail
rs    = Rose or Iris
dth   = Edith
ptsy  = Patsy
mxn   = Maxine
dwn   = Edwina or Dawn
gth   = Agatha
dry   = Audrey

Workings

Hint
The country spans more than one word.

Answer
England.

Alex's aches and pains felt worse because of swollEN GLANDs that had flared up during the night, perhaps a tonic would help?

Workings

Hint
There are 3 P's, so there are 3 different ways to start the word.

Answer
960.

Reasoning
Given: P P P U U Z Z Z Z Z L L L L E S S.

The order of the Z's is important. For example, we could chose a red one and a green one, so the order they appear in the sign has to be taken into consideration.

There are:
   3 different P's that could be chosen.
   2 different U's that could be chosen.
   5 different Z's that could be chosen.
   4 different Z's that could be chosen, as one has already been chosen.
   4 different L's that could be chosen.
   1 E that could be chosen.
   2 different S's that could be chosen.

So there are 3 x 2 x 5 x 4 x 4 x 1 x 2 = 960 different ways to make the sign.

Workings

Hint
It doesn't matter what the distance is, so try a random distance.

Answer
This cannot be done.

It does not matter how far the trip is, let's assume that it is random distance, say 60 miles, and remembering that Time = Distance ÷ Speed.

The morning trip took 60 miles ÷ 30 mph = 2 hours.

To average the required 60 mph, we would have to travel the total distance (of 120 miles) in 2 hours. However, we've already used all of the two hours for the morning trip!

Choosing any random distance would also show that it can't be done.