May I leave this problem?
For all prime numbers above 3, the square of the prime minus 1 is always divisible by 24 precisely giving an integer answer.
The question is:- Is this result significant? If you think yes, say why; if you think it is not, say why not.
By the way, I know the answer and just thought it might tease some minds.
Contestant 1 Car Goat Goat
Contestant 2 Goat Car Goat
Contestant 3 Goat Goat Car
Let's apply the two differing strategies to the above situation.
Strategy # Choose a door and stick with your original choice.
All contestants choose door A
As expected only one contestant in three wins.
Strategy @ Choose a door and swap to the only remaining door after the goat is revealed.
All contestants choose door A.
The swap strategy proves unfortunate for contestant 1; yet notice that both the other two contestants now win.
In reality your chances of winning are doubled by using Strategy@.
A simple simulation running, say, 300,000 times will give an approximate result of 100,000 wins for Strategy #; and, of course, approximately 200,000 wins for Strategy @.
You may be interested in a book entitled: "Rapid Math ...tricks and tips" by Edward H. julius.
It contains numerous examples such as the ones you mentioned. Julius maintains that, if you do the exercises in the book over the course of 30 days, you will dramatically improve the speed of your mental arithmatic.