If you have five unit fractions (a unit fraction is a fraction with 1 on top and a positive whole number on the bottom, such as 1/2, 1/3 or 1/1000), and you add them together, how close can you get to a sum of 1 without actually equalling 1 or going beyond it?

For instance, if the first four fractions were each 1/5, the most the last one could be is 1/6, which would make the sum 29/30. Close to 1, but not equal, as we are seeking. You can get even closer of course, but you can’t get arbitrarily close; there is a maximum.