The perfect squares from $1$ through $2500,$ inclusive, are printed in a sequence of digits $1491625\ldots2500.$ How many digits are in the sequence?
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We consider it by four cases:
Case 1: There are
perfect squares that only have
digit,
and
Case 2: The smallest perfect square that has
digits is
and the largest is
so that's a total of
perfect squares with
digits.
Case 3: The smallest perfect square with
digits is
and the largest is
yielding a total of
Case 4: The smallest perfect square with
digits is
and the last one that is no greater than
is
giving a total of
So we have a total of digits.