Some relatively easy math problems for your consideration and enjoyment:


  1) The classical proof that there are infinitely many primes is a proof by
contradiction: Let's assume that there's a largest prime. If we multiply
all the primes up to that largest prime and add 1, we get a number which
is not divisible by any of the primes, and thus the assumption we made is
false: There was a prime which is larger than the one we assumed was the
largest.

  However, consider this: 2*3*5*7*11*13 + 1 = 30031, which is a composite
number.

  Isn't this a contradiction to the proof? It clearly doesn't hold that the
product of the first n primes plus 1 is a prime.

  How to explain this apparent contradiction?


  2) Assume you have an array of 24 integers. Each element of that array
can get a value between 0 and 7. Thus the total amount of different contents
for such as an array is 8^24, which is approximately 4*10^21.

  Now assume that you fill the array with some values and then calculate
all the possible permutations of that array. The amount of permutations
for 24 elements is 24!, which is approximately 6*10^23.

  Now here's the apparent paradox: The total amount of different contents is
about 4*10^21, and naturally all those permutations should be among them as
well. How come the total amount of permutations, 6*10^23, is way larger
than the total amount of possible different array contents?


  3) Assume two people, person A and person B, who want to decide who gets
a price by tossing a coin.

  Person A is a bad loser and a bully, so if he loses he says "I said it's
two out of three". So they play it like that. If A loses again, he says
"I said it's three out of five", and so on, until he wins.

  How many tosses is this game expected to last, in average?


  4) By their nature, factorials tend to accumulate trailing zeroes. For
example, 5! (120) has one trailing zero, 10! (3628800) has two trailing
zeroes, 25! (15511210043330985984000000) has six trailing zeroes, etc.

  Give a (non-recursive) mathematical function which, for an integer n,
gives the number of trailing zeroes in n!.

-- 
                                                          - Warp

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