Table per patrol
Pen per patrol
Set up a table for each patrol/team at one end of the hall. Put a copy of the grid on each table
Line the patrols up at the opposite end of the hall in relay formation.
On the starting signal, the first team member runs to the table and signs their initials in the number 1 box on the grid.
They return to their place and the second team member runs up and signs box number 2.
The race continues until a fixed time (10 minutes?) and the team which has reached the highest sequential number wins.
Or you can continue until one team reaches 100.
You may wish to have a leader supervising each table so that only one box is signed on each turn.