1. Kb2? Kb8!
Same file, odd number of squares in between. 2. Kc3 Kc7 3. Kd4 Kd6
White is stuck.
1. Kb1? also loses to 1…Kb7!, and if White tries to be sneaky by sticking to the first rank with 2. Kc1 We know 2. Kc2 gets nowhere: 2… Kc6 3. Kd3 Kd5 kaput. 2…Kc7! Keeping the odd-numbered distance. 3. Kd1 Kd7 4. Ke1 Ke7 5. Kf2 Kf6 6. Kg3 Kg5 Black wins.
Why does any of this king vs. king exercise matter? Because this position is most critical: