Reasoning
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.
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Puzzle 6
How many people must be at a party before you are likely to have two having the same birthday (but not necessarily the same year)?
Reasoning
By likely, we mean greater than 50% chance.
With one person there is a 0 percent chance that you'll have two people with the same birthday.
With two people the probability that they won't share a birthday is 364 / 365. The probability that they will share a birthday is therefore 1 - (364 / 365).
With three people the probability that they won't share a birthday is the same as for two people, times 363 / 365. So the probability that three people will share a birthday is 1 - (364 / 365) x (363 / 365).
Notice that with each additional person added, the probability that they shares a birthday with one of the previous persons goes up, because there are fewer "free" days remaining.
We keep adding people until the %age is greater than 50%.
When we have 23 people the %age is 50.729%.
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Puzzle 7
Can you complete this grid of letters?
Each line is a valid 5-letter word with its first and last letters missing.
The missing letters form a word when read downwards, and the same word appears both at the start and at the end.
What is the missing word?
I
N
E
R
O
D
Y
L
O
O
A
S
E
A
T
Note: this puzzle is not interactive, and the squares cannot be clicked.
