  Polyregular functions are the class of string-to-string functions definable
by pebble transducers, an extension of finite-state automata with outputs and
multiple two-way reading heads (pebbles) with a stack discipline. If a
polyregular function can be computed with $k$ pebbles, then its output length
is bounded by a polynomial of degree $k$ in the input length. But Boja\'nczyk
has shown that the converse fails.
  In this paper, we provide two alternative easier proofs. The first
establishes by elementary means that some quadratic polyregular function
requires 3 pebbles. The second proof - just as short, albeit less elementary -
shows a stronger statement: for every $k$, there exists some polyregular
function with quadratic growth whose output language differs from that of any
$k$-fold composition of macro tree transducers (and which therefore cannot be
computed by a $k$-pebble transducer). Along the way, we also refute a
conjectured logical characterization of polyblind functions.
