The Roman army problem
 

The Roman army has 2018 units guarding their provinces. The Emperor was worried that when there are at least 64 units in a province, they might get together and overthrow the Emperor. So on each day, he visited one such potentially troublesome province and sent all its units to other provinces, no two going to the same provinces. Prove that after 64 days, there were no more provinces with at least 64 units.