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!).