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?
So I continued on in the sequence
13112221
1113213211
31131211131221
13211311123113112211
11131221133112132113212221
31131122212321121312211312113211
1321132132111213122112111311222113111221131221
111312211312111312311211131122211231132132211331222113112211
31131122211311123113111213211231132132211213211312111322212311322113212221
132113213221133112132113311211131221121321131211132221121113122113111231133211121321131222113121132
when will it stop? Am I close? Do you know the answer?
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