Medium Puzzles 

Workings

Hint
Alan is married to Julie.

Answer
Alan is married to Julie and they had a new kitchen.
Brian is married to Maud and they had a new bathroom.
Charles is married to Laura and they had a new conservatory.
David is married to Kara and they had new windows.

Workings

Hint
The first letters of the words are: E, S, P, C, I, A, P, A.

Answers
-----yee = employee -----egy = strategy -----igm = paradigm -----lse = convulse -----com = intercom -----bet = alphabet -----lel = parallel -----dee = attendee

Workings

Hint
Try to find out where Mrs Umbrella lives first.

Answer
From, left to right:
#1 Mrs Scott    - glass
#2 Mrs Umbrella - straw
#3 Mr Tidsley   - brick
#4 Mr Wilshaw   - wood

Reasoning
If we separate and label the clues, and label the bungalows #1, #2, #3, #4 from left to right we can see that:
Mrs Scott's bungalow is somewhere to the left of the wooden one. The third one along is brick. Mrs Umbrella owns a straw bungalow Mr Tidsley does not live at either end. Mr Tidsley lives somewhere to the right of the glass bungalow. Mr Wilshaw lives in the fourth bungalow. The first bungalow is not made from straw.
By (6) Mr Wilshaw lives in the fourth bungalow.
   #1
   #2
   #3
   #4 Mr Wilshaw

By (3), Mrs Umbrella owns the straw bungalow. By (7) it isn't #1, by (2) it isn't #3, so it must be #2.
   #1
   #2 Mrs Umbrella - straw
   #3
   #4 Mr Wilshaw

By (4), Mr Tidsley must therefore live in #3, which, by (2) is the brick bungalow.
   #1
   #2 Mrs Umbrella - straw
   #3 Mr Tidsley   - brick
   #4 Mr Wilshaw

By (1), Mrs Scott's bungalow must be #1, and #4 must be the wooden one.
   #1 Mrs Scott
   #2 Mrs Umbrella - straw
   #3 Mr Tidsley   - brick
   #4 Mr Wilshaw   - wood

This leaves #1 as the glass bungalow.
   #1 Mrs Scott    - glass
   #2 Mrs Umbrella - straw
   #3 Mr Tidsley   - brick
   #4 Mr Wilshaw   - wood

Workings

Hint
How many leaves will I have collected on day 5?

Answer
On the 23^rd.

Reasoning
We could simply keep adding until we get the required number:

   1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23

   = 276 leaves.

But a more mathematical method might help to answer the Bonus Question, as this might take a while if we keep adding!

So, let's create a method by imagining that we are adding the numbers from 1 to 30.

   1 + 2 + 3 + … + 28 + 29 + 30

If we now take the numbers in pairs, taking one from each end, we have:

   (1 + 30) + (2 + 29) + (3 + 28) + … + (15 + 16)

Each pair adds to 31, and we have 15 pairs. So the total sum is 31 x 15 = 465.

The total sum from 1 to any number (N) can be found using this technique, and we will have:

   Each pair adds to (1 + N), and there are N ÷ 2 pairs. So the total is:

   (1 + N) x N
             —
             2

In this puzzle, we know that this equals 276.

So:

   (1 + N) x N = 276
             —
             2

We can expand the brackets, and multiply both sides by 2, to give:

   N + N² = 552

Rearranging we get:

   N² + N – 552 = 0

And 552 = 2 x 2 x 2 x 3 x 23, so this can be factorised as:

   (N + 24) x (N – 23) = 0

Because we need to find a positive number of days, the only possible answer is:

   (N – 23) = 0

So N = 23 days.
Bonus Question

To answer the bonus question, we have:

   (1 + N) x N = 56616
             —
             2

Rearranging we get:

   N² + N – 113232 = 0

And 113232 = 2⁴ x 3 x 7 x 337, so this can be factorised as:

   (N – 336) x (N + 337) = 0

Because we need to find a positive number of days, the only possible answer is:

   (N – 336) = 0

So N = 336 days (I did say that I liked collecting leaves!).