Q1 can't have A as its answer (otherwise it would contradict itself).

If Q1's answer was C (meaning Q4's answer was A), then Q3's answer would be C. However, Q2's answer should now be A, but this isn't allowed by Q1 (as Q4 is the first answer with A). This is a contradiction.

Therefore Q1's answer is B (meaning Q3's answer is A).

For Q2's answer to be C, Q4's answer would have to be A, which would contradict Q4. Therefore Q2's answer is B, making Q4's answer B. QED.

Puzzle 3

Below are thirteen 5 lettered, everyday words, each of which has had two of its letters removed.

In total these 26 letters are A-Z. The remaining letters in each word are in the correct order.

There are no words which are spelled differently based upon location (favour/favor, etc) and there are no plurals.

Can you determine the original words?

APE
BAE
BOD
ANC
ROE
ELL
RAY
BUC
ORA
UMB
SUA
LOL
GES

APE + P + L >> APPLE
BAE + R + K >> BRAKE
BOD + X + E >> BOXED
ANC + M + I >> MANIC
ROE + F + Z >> FROZE
ELL + D + W >> DWELL
RAY + G + V >> GRAVY
BUC + N + H >> BUNCH
ORA + C + B >> COBRA
UMB + J + O >> JUMBO
SUA + Q + T >> SQUAT
LOL + Y + A >> LOYAL
GES + U + S >> GUESS

Puzzle 4

What 5 digit number am I thinking of.

It has two prime digits.

Digit 3 is the highest digit.

Digit 2 is lowest digit.

Digit 1 is higher than the sum of digits 4 and 5.

Digit 5 is half of digit 4.

Digit 1 is one smaller than digit 3.

Digit 5 is between digit 2 and digit 1

There are no duplicates and the digit 0 doesn't appear.

Since digit 5 is half of digit 4 we have four possibilities:

_ _ _ 2 1
_ _ _ 4 2
_ _ _ 6 3
_ _ _ 8 4

But since digit 1 is higher than the sum of digit 4 and digit 5 we can eliminate two possibilities leaving:

_ _ _ 2 1
_ _ _ 4 2

Since digit 5 is between digit 2 and digit 1, we can eliminate _ _ _ 2 1, leaving:

_ _ _ 4 2

Since digit 1 must be higher than the sum of digit 4 and digit 5 we have three possibilities:

7 _ _ 4 2
8 _ _ 4 2
9 _ _ 4 2

But digit 1 is 1 smaller than digit 3 we can eliminate 9 _ _ 4 2, leaving

7 _ 8 4 2
8 _ 9 4 2

Since digit 2 is the lowest number, it must be 1:

7 1 8 4 2
8 1 9 4 2

The answer has exactly 2 prime digits, which only leaves

7 1 8 4 2. QED.

Puzzle 5

Using the BrainTracker grid below, how many words can you find? Each word must contain the central C and no letter can be used twice, however, the letters do not have to be connected. Proper nouns are not allowed, however, plurals are. Can you find the nine letter word?