Random Puzzles 

Workings

Hint
The second word is CORD.

Answer
COLD, CORD, WORD, WARD, WARM

or

COLD, CORD, WORD, WORM, WARM

or

COLD, CORD, CARD, WARD, WARM.

Workings

Hint
The word APPLES appears 34 times.

Answer
The word APPLES appears 34 times.

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!).

Workings

Hint
The question is written in English, but not in its usual order.

Answer
32.

Reading one letter from the top row and then one from the bottom row, the puzzle reads: 'How many letters does this puzzle have'.