Random Puzzles 

Workings

Hint
A little bit of algebra might help.

Answer
64 pounds.

The head weighs 24 pounds, the body weighs 32 pounds, and the tail weighs 8 pounds.

Reasoning
The trout is made up of the head, body and tail, H, B and T.

We are told that:

    T = 8

(1) H = T + B ÷ 2

(2) B = H + T

Placing T into (1) and (2) we get:

(3) H = 8 + B ÷ 2

(4) B = H + 8

Using (4) in (3) we get:

    H = 8 + B ÷ 2

(5) H = 8 + (H + 8) ÷ 2

Multiplying (5) by 2 we get:

 2H = 16 + H + 8

 2H = 24 + H

  H = 24

Therefore, by (2):

 B = H + T

 B = 24 + 8

 B = 32

Giving:

 H = 24
 B = 32
 T = 8

And a total of 24 + 32 + 8 = 64 pounds.

Workings

Hint
One of the countries begins with the letter M.

Answers
Chad, China, Togo, Italy, Malta.

Domini[c had] a certain way with animals, he could almost talk to them. By tickling it under the [chin, a] rabbit would roll over and let its belly be tickled. This technique was sometimes used [to go]od effect in gaining trust, and allowed a v[ital y]early treatment to be performed safely. The removal of nits. Using a vet approved shampoo to clean the fur (and not a blend of [malt a]nd water!).

Workings

Hints
The first letters of the words are: A, P, T, I, P, U, A, R, R, W, D, C, R, B, E.

Answers
----nst = against ----nip = parsnip ----fth = twelfth ----rno = inferno ----dox = paradox ----sil = utensil ----nac = almanac ----iem = requiem ----arb = rhubarb ----ior = warrior ----bey = disobey ----van = caravan ----ipt = receipt ----dth = breadth ----nox = equinox

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!).