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The parrot sequence

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Parul owns a clever parrot called Pasha which can call out loud numbers read from a piece of paper displayed in front of him. One day Parul wrote the digit 1 on a sheet and showed it to Pasha, who answered “one 1”. Parul wrote down the parrot’s answer “one 1” or ‘11’ on a new sheet and showed it to Pasha who readily answered “two 1’s”. Again, Parul wrote down his reply “two 1’s” or ‘21’ on another sheet and showed it to Pasha, who replied “one 2, one 1” or ‘1211’... Parul 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 Pasha 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?


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