Puzzle Taken from Mind and Visual Puzzles! -
The parrot sequence
Gaston owns a clever parrot called Paco which can describe out loud numbers read from a piece of paper displayed in front of him. One day
Gaston tried this funny experiment: he wrote the digit 1 on a sheet and showed it to Paco, who answered “one 1”. Gaston wrote down the parrot’s answer “one 1” or ‘11’ on a new sheet and showed it to Paco, who readily answered “two 1’s”. Again, Gaston wrote down his reply “two 1’s” or ‘21’ on another sheet and showed it to Paco, who replied “one 2, one 1” or ‘1211’... Gaston continued this experiment until he obtained a long sequence of numbers 1, 11, 21, 1211, 111221, 312211, etc... in which each next term is obtained by describing the previous term.
Can you say how many sheets of paper Gaston will need until:
a) the string of digits 333 appears, or
b) the digit 4 occurs in the sequence?
Can you also confirm that the last digit of any term of the parrot sequence is always 1?
