Random Puzzles 

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
One of the words is antelope.

Answers
rdvrk  = aardvark rndr   = reindeer prcpn  = porcupine rhncrs = rhinoceros sqrrl  = squirrel clt    = ocelot or colt frrt   = ferret sknk   = skunk or skink dnky   = donkey ntlp   = antelope

Workings

Hint
A little bit of algebra might help.

Answer
There are four boys and three girls.

If we call the number of boys B, and girls G.

From the first statement (we take one off as Billy is a boy):

              B − 1 = G

and similarly:

        2 x (G − 1) = B

Substitute G from the first equation into the second equation to give:

  2 x ((B − 1) − 1) = B

    2 x (B − 1) − 2 = B

      2 x B − 2 − 2 = B

          2 x B − 4 = B

                  B = 4

which means that B = 4 and hence G = 3.

Workings

Hint
There are no larger squares than size 4 x 4.

Answer
There are 110 squares without a rook.

Reasoning
There are 60 squares of size 1 x 1.
There are 35 squares of size 2 x 2.
There are 12 squares of size 3 x 3.
There are  3 squares of size 4 x 4.

Giving a total of 60 + 35 + 12 + 3 = 110 squares.