Medium 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
Try: soot + tire, most + mare, much + pink.

Answers
soot + tire = tortoise
most + mare = marmoset
much + pink = chipmunk
area + tent = anteater
leap + then = elephant

Workings

Hint
The 1^st word begins with the letter F.

Answers
-e-er      = fever      (FV) --eue      = queue      (QU) --o        = zoo        (ZO) ma-        = mat        (T) -p-a-e-    = speaker    (SEKR) -erso-     = person     (PN) ---k-am-on = backgammon (BACGM) -ouse      = house      (H) -a-        = wax        (WX) -ur-       = jury       (JY) ---igent   = diligent   (DIL)
This puzzle isn't guaranteed to have a unique answer, can you find another?

Workings

Hint
Look at Q2 first.

Answer
Q1. d)
Q2. c)
Q3. a)
Q4. b)

Reasoning
Let's look at Q1 first, and check each possible option:

If Q1's answer was a):
...we'll leave checking this for now, because...

If Q1's answer was b):
...we'll leave checking this for now, because...

If Q1's answer was c):
it would reference itself and we'd learn nothing new, and we'd then have to check Q2.

So, let's start with Q2 first instead of Q1.

If Q2's answer was a):
   an immediate contradiction, as Q4 can't be the first answer with a) as we've assumed Q2's answer is a).

If Q2's answer was b):
   an immediate contradiction, as Q2 can't be the first answer with a) as we've assumed Q2's answer is b).

If Q2's answer was c):
   Q3's answer would be a)
   Q1's answer would be d)
   confirming Q2's answer as c)
   leaving Q4's only possible answer as b)
   no contradiction.

If Q2's answer was d):
   Q1's answer would be a)   (*)
   Q3's answer would be c)
   Q4's answer would be d)
   Q1's answer would be b)
   but this contradicts the section marked (*).

Therefore, the only possible answer with no contradiction is when Q2's answer is c).