problem: If you continue this pattern in which each line-segment extremity is replaced by a gradually smaller Y in the next figure, in the manner shown, how many endpoints will Figure 5 have?
  
  [asy]
  draw((0,0)--(0,-3),linewidth(.75));
  draw((0,0)--(-2,2),linewidth(.75));
  draw((0,0)--(2,2),linewidth(.75));
  label("Figure 1",(0,-3),S);
  
  draw((5,0)--(5,-2),linewidth(.75));
  draw((4,-3)--(5,-2)--(6,-3),linewidth(.75));
  draw((4,1)--(5,0)--(6,1),linewidth(.75));
  draw((3,1)--(4,1)--(4,2),linewidth(.75));
  draw((6,2)--(6,1)--(7,1),linewidth(.75));
  label("Figure 2",(5,-3),S);
  
  draw((10,0)--(10,-2),linewidth(.75));
  draw((9.5,-2.5)--(10,-2)--(10.5,-2.5),linewidth(.75));
  draw((9,-2.5)--(9.5,-2.5)--(9.5,-3),linewidth(.75));
  draw((11,-2.5)--(10.5,-2.5)--(10.5,-3),linewidth(.75));
  
  draw((9,1)--(10,0)--(11,1),linewidth(.75));
  draw((8.5,1)--(9,1)--(9,1.5),linewidth(.75));
  draw((11.5,1)--(11,1)--(11,1.5),linewidth(.75));
  draw((8.25,.75)--(8.5,1)--(8.25,1.25),linewidth(.75));
  draw((8.75,1.75)--(9,1.5)--(9.25,1.75),linewidth(.75));
  draw((10.75,1.75)--(11,1.5)--(11.25,1.75),linewidth(.75));
  draw((11.75,1.25)--(11.5,1)--(11.75,.75),linewidth(.75));
  label("Figure 3",(10,-3),S);
  
  [/asy]
solution: In the transition between figures, each endpoint splits into two new segments, creating two new endpoints, so the number of endpoints doubles.  Figure 1 has $3$ endpoints, so Figure $n$ has $3*2^{n-1}$ endpoints.  Thus Figure 5 has $\boxed{48}$ endpoints.