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MGRS grid lines do not line up with UTM zone boundaries
UTM Coordinate System
The UTM coordinate system divides the world into 60 zones, each of them 6 degrees wide, that extend from 84 degrees North to 80 degrees South.
Because UTM zones are 6 degrees wide, they measure about 667,916 meters east to west at the equator. Because the east-west extent of a degree gets narrower as you move north or south of the equator, at 84 degrees North, the east-west extent of a degree is only about 11,675 meters. At 80 degrees South, the east-west distance is about 19,394 meters. The image below illustrates the decreasing distance across a UTM Zone in the state of Florida.
The length of a degree measured north-south changes very little and is about 111,670 meters, anywhere on earth.
Military Grid Reference System (MGRS)
The MGRS grid system is based on UTM, and can only be applied to an ArcMap Layout if the projection of the ArcMap data frame is set to the UTM Zone in which the data is located. The division in MGRS is based on a grid square that is 100,000 meters on a side. Six (6) degrees east-to-west is not evenly divisible by 100,000 meters, except in a few locations, as you move north or south from the Equator.
In the linked graphic at the end of this paragraph, note the tic marks which divide the 100,000 meter MGRS square "BE" into 10,000 meter squares. Also note that while square "BE" is complete, and measures 100,000 meters on a side, square AE is cut off at the boundary between UTM Zone 16 and UTM Zone 15.
The MGRS grid system is intended for use within the boundaries of a specific UTM Zone. Applying an MGRS grid to data which crosses UTM Zone boundaries results in inconsistent behavior, due to the design of the Military Grid Reference System.
For further information about the official military specifications for the MGRS grid system, click the link in the Related Information section below.