Gill has shown that if there exists a finite-memory n-state sequential machine with finite memory Â¿, then Â¿ cannot exceed Â¿n(n-1)Â¿Nn. He has further shown that there exists an n-state Nn input-binary output machine with memory Â¿= Nn for every n. The question of whether a tighter upper bound might be placed on Â¿ by the order of the input alphabet was raised by Gill. Massey recently has shown that there exists a ternary input-binary output finite-memory machine with memory Â¿=Nn for every n. The primary purpose of this note is to show that for every n there exists an n-state binary input-binary output finite-memory machine with memory Â¿= Nn, and thus Â¿ is shown not to be limited by the order of the input alphabet. It is shown that for every n there are actually at least two different machines with memory Â¿ = Nn. It will also be shown that for every n there exists a binary input-binary output n-state finite-memory machine with Â¿ = Nn-1.