puzzle #2: magic number
Your task is to identify the smallest positive whole number $N$ with the two following properties.
$N$ has exactly $144$ factors (including itself and $1$).
Among the factors of $N$ there are at least $10$ consecutive numbers.
puzzle #1: coin tossing
This problem is about tossing coins, H denotes a Head and T denotes a Tail.
Anne tosses a fair coin twice. What is the probability that she obtains HH?
What is the probability that she obtains HT?
Next, she plays a different game. She repeatedly tosses the same fair coin until she obtains H. What is the 'average' (i.e. expected) number of tosses this will take?
In games of this kind, she repeatedly tosses the same fair coin until a certain pattern comes up. For example, if that pattern is HH then she keeps tossing the coin until the two most recent tosses have both been Heads.
What is the expected number of tosses when she is seeking the pattern HH?
Finally, what is the expected number of tosses when she is seeking the pattern HT?