Random Puzzles 

Workings

Hint
You might need an old calculator.

Answer
Coil.

Reasoning
This is the only word that cannot be spelled using a calculator.

For example, if you enter 376616 into an old-fashioned calculator, and look at the number upside down, you can read GIGGLE.
376616  = giggle
7105    = soil
57738   = bells
378806  = gobble
4506    = gosh
5508    = boss
3704    = hole
5355378 = blesses
coil    = ????

Workings

Hint
Look to the skies for the answer.

Answer
neerg.

These are the colours of the rainbow reversed.

Workings

Hint
The first change makes SOFT into LOFT.

Answer
SOFT, LOFT, LEFT, LENT, LENS.

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!).