(;C[White to play.]AW[ip]AW[lo]AW[jp]AB[qr]AB[qq]AB[qp]AW[po]AW[pp]AW[pr]AW[pq]AW[nq]AW[np]AW[mp]AB[mq]AB[nr]AB[ms]AB[lr]AW[ko]AW[ks]AW[kr]AB[jq]AB[kq]AB[kp]AB[jr]AB[js]AW[iq]AW[ir]AW[is]AW[jo]AB[rn]AP[goproblems]
(;W[or];B[lq]C[Black lives.])
(;W[ps];B[lq]C[Black lives.])
(;W[qs];B[lq]C[Black lives. The profit of damaging black's corner is not enough. This line is inferior.])
(;W[os];B[ns]
(;W[qs];B[lq]C[Black lives. Profit from damage to black's corner is not enough. This is not the best result for white.])
(;W[or];B[lq]C[Black lives.])
(;W[lp];B[lq]C[Black lives.])
(;W[lq];B[ls]
(;W[kr];B[lp]C[])
(;W[lp]C[Black is dead.RIGHT])
(;W[mr]C[This is correct. Black dies. However, M4 instead of N2 is slightly better because N2 gives black a ko threat at M4.RIGHT]MA[mq])))
(;W[lp];B[lq]C[Black lives.])
(;W[ns];B[os]C[This move does not help. Black lives.])
(;W[lq];B[ls]
(;W[kr];B[lp])
(;W[os];B[lp])
(;W[lp];B[os]
(;W[or];B[ps]C[Black escapes.])
(;W[mr];B[ps]C[Black escapes.])
(;W[ps];B[or];W[mr]MA[mq]C[Ko is not best.]))
(;W[mr]MA[mq]C[];B[mm]C[Ko is not best. If this is not clear to you that this is a ko, please play at A to simulate responding to a black ko threat.]LB[nm:A];W[nm];B[mq]MA[mr]LB[lp:A]C[Assume that black has many more ko threats than white. Please connect at A to see how black can stop white from avoiding the ko.];W[lp];B[os]
(;W[or];B[ps]C[Black connects.])
(;W[ps];B[or];W[mr]MA[mq]C[White cannot avoid the ko. This is not the best line for white.]))))