Random Puzzles 

Workings

Hint
Another example, PEN is a complete word because:
   -en = ten
   p-n = pin
   pe- = pea

Answer
Two of the words are not complete: bugs, lean.

Reasoning
The remaining 8 words can be changed as follows:
hard:  -ard = card
 h-rd = herd
 ha-d = hand
 har- = harp

port:  -ort = fort
 p-rt = part
 po-t = poet
 por- = pore

duck:  -uck = muck
 d-ck = deck
 du-k = dusk
 duc- = duct

bugs:  -ugs = hugs
 b-gs = bags
 bu-s = buys
 bug- = ---- X

bail:  -ail = hail
 b-il = boil
 ba-l = bawl
 bai- = bait

mast:  -ast = cast
 m-st = most
 ma-t = malt
 mas- = mask

thin:  -hin = shin
 t-in = twin
 th-n = than
 thi- = this

lean:  -ean = mean
 l-an = loan
 le-n = ---- X
 lea- = leaf

fire:  -ire = hire
 f-re = fare
 fi-e = five
 fir- = firm

read:  -ead = bead
 r-ad = road
 re-d = reed
 rea- = real

Workings

Hint
The first change makes SOFT into LOFT.

Answer
SOFT, LOFT, LEFT, LENT, LENS.

Workings

Hint
A little algebra might help.

Answer
Pineapple = 83p, coconut = 57p, melon = 62p.

Reasoning
If we assign the letters P, C and M to the fruits, and work in pence we have:

(1)   M + C = 119
(2)   M + P = 145
(3)   C + P = 140

Using (2) − (1) we have:

(4)   P − C = 26

And (3) + (4) gives:

         2P = 166
          P = 83

This can then be substituted back into (2) and (3).

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!).