Thanks to Fokatukc, DarkSycthe, Traxtin, and Blaster097 for showing the correct solution and the focus of this problem.
(;AB[gc]AB[ge]AB[cd]AB[ce]AB[df]AB[ef]AB[ff]AB[eg]AB[eh]AW[de]AW[ee]AW[fe]AW[gf]AW[gg]AW[fg]AW[dg]AW[cg]AW[cf]C[Black and White each have 3 liberties. Black can save 5 stones by capturing 3 White stones.]SZ[9]AP[goproblems]
(;B[fd]
(;W[ei]C[Fokatukc's nose tesuji to connect the 2 White groups.]
(;B[ed];W[fh]
(;B[dd];W[dh]C[RIGHT])
(;B[dh];W[ch];B[dd];W[di]C[RIGHT]))
(;B[dd];W[fh]
(;B[ed];W[dh]C[RIGHT])
(;B[dh];W[ch];B[ed];W[di]C[RIGHT]))
(;B[dh];W[ch]
(;B[fh];W[gh]
(;B[ed];W[fi];B[dd];W[di]C[RIGHT])
(;B[dd];W[fi];B[ed];W[di]C[RIGHT]))
(;B[di];W[fh];B[fi];W[ci];B[ei];W[gi])
(;B[ed];W[fh];B[dd];W[di]C[RIGHT]))
(;B[fh];W[gh];B[dh];W[ch]
(;B[ed];W[fi];B[dd];W[di]C[RIGHT])
(;B[dd];W[fi];B[ed];W[di]C[RIGHT])))
(;W[fh]
(;B[ei])
(;B[ed]
(;W[ei];B[dh];W[ch];B[dd];W[di]C[RIGHT])
(;W[dh]
(;B[ei];W[di];B[dd]C[Whites 2 groups are divided.])
(;B[dd];W[ei]C[Black should have descended to the first line.])))
(;B[dd];W[dh]
(;B[ei];W[fi];B[ed]C[Black's 2 groups are divided.])
(;B[ed];W[ei]C[Black should have descended to the first line.]))))
(;B[gd]C[White has 3 liberties and Black has 3. But when White moves to capture Black, Black will lose the capture race.]
(;W[fh]C[Black has fewer liberties than the 3 White stones.]
(;B[dh])
(;B[ei]))
(;W[ec]C[The Crane's Nest tesuji is the wrong move for White.];B[ed];W[fd];B[fc];W[dd];B[dc];W[ed];B[eb]C[The Crane's Nest tesuji works for Black, but it was the wrong move for White.]))
(;B[dd];W[gd];B[he];W[fd];B[fc];W[hd];B[hf];W[hg];B[hc];W[ie];B[ed];W[if])
(;B[ed];W[gd]
(;B[he];W[fd];B[fc];W[hd];B[hf];W[hg];B[hc];W[ie];B[dd];W[if])
(;B[fd];W[he];B[dd];W[ge]C[Black's 5 stones can't survive.]TR[df]TR[ef]TR[ff]TR[eg]TR[eh]))
(;B[ei];W[gd];B[fd])
(;B[dh])
(;B[fh])
(;B[gh]))
