Brain Teaser #50

Numbers and Dots

This is a famous problem from 1882, to which a prize of $1000 was awarded for the best solution. The task is to arrange the seven numbers 4, 5, 6, 7, 8, 9, and 0, and eight dots in such a way that an addition approximates the number 82 as close as possible. Each of the numbers can be used only once. The dots can be used in two ways: as decimal point and as symbol for a recurring decimal. For example, the fraction 1/3 can be written as
.
.3

The dot on top of the three denotes that this number is repeated infinitely. If a group of numbers needs to be repeated, two dots are used: one to denote the beginning of the recurring part and one to denote the end of it. For example, the fraction 1/7 can be written as
. .
.142857

Note that '0.5' is written as '.5'.

The question: How close can you get to the number 82?

Solution

A possible solution is the following one:
    .
80.5

   . .
  .97

    . .
  .46
_____
82

Conclusion: The number 82 can be made exactly.