Math Doubts from QBM037

Please tell me the answers and solutions of following questions Q3.A rod of length 1 is broken into four pieces. What is the probability that the four pieces are the sides of a trapezoid? Q4.Take a deck of 52 playing cards consecutively numbered from 1 to 52. A perfect shuffle occurs when you divide the deck in two, one half in each hand, and riffle them together in an alternating fashion; after the shuffle the cards in the deck will be ordered as follows: 1, 27, 2, 28, 3, 29, ... , 24, 50, 25, 51, 26, 52 Notice that the top card always stays on top and the bottom card always stay on the bottom. Imagine now that you repeatedly shuffle the cards, performing a perfect shuffle every time. How many times will you have to do this before all the cards in the deck are restored to their original position? Q5. Is the number 2438100000001 prime or composite? No calculators or computers allowed! Q14. An absentminded professor buys two boxes of matches and puts them in his pocket. Every time he needs a match, he selects at random (with equal probability) from one or other of the boxes. One day the professor opens a matchbox and finds that it is empty. (He must have absentmindedly put the empty box back in his pocket when he took the last match from it.) If each box originally contained n matches, what is the probability that the other box currently contains k matches? (Where 0 k n.) I am in TIME and performing quite well but these questions are forcing me to think I am far behind in the race :-((