Question:

Here is a famous prize problem that Sam Loyd issued in 1882, offering $1000 as a prize for the best answer showing

How to arrange the seven figures and the eight “dots” .4.5.6.7.8.9.0. which would add up to 82.

Out of several million answers, only five were found to be correct.

Answer:

80+.5.+.97+.46 =82

It can also be written as 

80+5/99+97/99+46/99 =80+198/99=82

The dot over a number signifies that it is a repeater which would go on for ever, eg 1/6 decimally as 0.6666 . . . . (etc)
With a series of numbers we place the dot over the first and last, as with 0.97979797979 . . . (etc)
So If a proper fraction is divided by 9s e.g. 46/99 is exactly equal to the numerator with the repeater sign followed by the decimal.