From MGRS grid reference which also can be found from https://www.maptools.com/tutorials/100km_square_id , the minimum column id for the first zone is 'A' and it's given when easting is smaller than 200km.

Is this mean that it is guaranteed that UTM easting value in a zone is always larger than 100km? Or, should I consider the width of first column is 200km?

Similarly, for the last column 'H' in the first zone, is it guaranteed that any UTM easting value in a zone is smaller than 900km? Or should I just handle any value larger than 800km as 'H' column?

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For UTM (and MGRS), the easting should always be bigger than 100 km and smaller than 900 km. I ran some test points on UTM 11 North.

easting northing  latitude  longitude
100000  0         0         -120.592
200000  0         0         -119.695
800000  0         0         -114.305
900000  0         0         -113.408

The central meridian of zone 11 is -117, so the range is -120 to -114. If I remember correctly, you can go outside the zone around 1/2 degree which we do if the easting is 100km or 900km. Away from the equator, those easting values will result in even farther away longitude values so the 'A' and 'H' boxes won't be used any more, except maybe the extended areas in Scandinavia.

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