No whirling knight's tour with winding number n/2 exists for n ≡ 4 or 6 mod 8, proven by exhibiting closed-form Farkas certificates showing infeasibility of the corresponding cycle-cover LP.
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Nonexistence of Whirling-Knight Tours at Half Coil Count for $n \equiv 4, 6 \pmod 8$
No whirling knight's tour with winding number n/2 exists for n ≡ 4 or 6 mod 8, proven by exhibiting closed-form Farkas certificates showing infeasibility of the corresponding cycle-cover LP.