Open Middle®
Directions: Using the digits 1 to 9 at most one time each, place a digit in each box to create a true statement.
Hint
How does the numerator change when the value increases/decreases? How does the denominator change when the value increases/decreases?
Answer
There are many possible answers, but all should have the fraction with the least value on the left and increase in value to the right. One example is 1/9 < 2/8 < 3/7
Source: Robert Kaplinsky
If you use all the digits….
54/3 < 76/2 < 98/1 OR
82/91 < 6/7 < 5/4
If you only use 6 digits…
6/7<8/9<5/3
It called improper fraction when numerator is bigger than the denominator. And when your numerator is smaller than the denominator , that means your final answer is always going to be less than one because, numerator is smaller.
We aren’t using the term improper fraction in our district anymore, but fraction greater than one. Improper implies something is wrong with the fraction.
2/5 <1/2< 3/4
1/9 < 2/8 < 5/6
4/9 < 5/8 < 6/7
If we allow improper fractions, there are 9,684 solutions out of the (9 choose 6)*6! = 60,480 ways to choose 6 digits and map them to the 6 boxes.
If we don’t allow improper fractions, there are 1,162 solutions, 52 of which start with a fraction of value greater than 1/2 and 69 of which have a fraction with a value of less than 1/2 at the end. The last fraction ranges between 3/9 and 8/9, the first fraction ranges between 1/9 and 4/5 and the middle fraction ranges between 2/9 and 6/7.
The one where the distance between the first and last number is smallest is 2/5 < 3/7 < 4/9 and the ones where the distance between the first and last numbers is highest are 1/9 < 2/3 < 7/8, 1/9 < 2/6 < 7/8, 1/9 < 4/6 < 7/8 and 1/9 < 5/6 < 7/8.
55 solutions (when using one-line notation) have all the digits in order, including 4/5 < 6/7 < 8/9 and 1/2 < 3/4 < 5/6.
1/2 < 3/4 < 5/6
1/2 < 3/4 < 6/7
1/2 < 3/4 < 7/8
1/2 < 3/4 < 7/9
1/2 < 3/4 < 8/9
1/2 < 3/5 < 4/6
1/2 < 3/5 < 6/7
1/2 < 3/5 < 6/8
1/2 < 3/5 < 6/9
1/2 < 3/5 < 7/8
1/2 < 3/5 < 7/9
1/2 < 3/5 < 8/9
1/2 < 4/5 < 6/7
1/2 < 4/5 < 7/8
1/2 < 4/5 < 8/9
1/2 < 4/6 < 5/7
1/2 < 4/6 < 7/8
1/2 < 4/6 < 7/9
1/2 < 4/6 < 8/9
1/2 < 4/7 < 3/5
1/2 < 4/7 < 5/6
1/2 < 4/7 < 5/8
1/2 < 4/7 < 6/8
1/2 < 4/7 < 6/9
1/2 < 4/7 < 8/9
1/2 < 5/6 < 7/8
1/2 < 5/6 < 8/9
1/2 < 5/7 < 3/4
1/2 < 5/7 < 6/8
1/2 < 5/7 < 8/9
1/2 < 5/8 < 3/4
1/2 < 5/8 < 4/6
1/2 < 5/8 < 6/7
1/2 < 5/8 < 6/9
1/2 < 5/8 < 7/9
1/2 < 5/9 < 3/4
1/2 < 5/9 < 4/6
1/2 < 5/9 < 4/7
1/2 < 5/9 < 6/7
1/2 < 5/9 < 6/8
1/2 < 5/9 < 7/8
1/2 < 6/7 < 8/9
1/2 < 6/8 < 4/5
1/2 < 6/8 < 7/9
1/2 < 6/9 < 3/4
1/2 < 6/9 < 4/5
1/2 < 6/9 < 5/7
1/2 < 6/9 < 7/8
1/2 < 7/9 < 4/5
1/2 < 7/9 < 5/6
1/3 < 2/4 < 5/6
1/3 < 2/4 < 5/7
1/3 < 2/4 < 5/8
1/3 < 2/4 < 5/9
1/3 < 2/4 < 6/7
1/3 < 2/4 < 6/8
1/3 < 2/4 < 6/9
1/3 < 2/4 < 7/8
1/3 < 2/4 < 7/9
1/3 < 2/4 < 8/9
1/3 < 2/5 < 4/6
1/3 < 2/5 < 4/7
1/3 < 2/5 < 4/8
1/3 < 2/5 < 4/9
1/3 < 2/5 < 6/7
1/3 < 2/5 < 6/8
1/3 < 2/5 < 6/9
1/3 < 2/5 < 7/8
1/3 < 2/5 < 7/9
1/3 < 2/5 < 8/9
1/3 < 4/5 < 6/7
1/3 < 4/5 < 7/8
1/3 < 4/5 < 8/9
1/3 < 4/6 < 5/7
1/3 < 4/6 < 7/8
1/3 < 4/6 < 7/9
1/3 < 4/6 < 8/9
1/3 < 4/7 < 5/6
1/3 < 4/7 < 5/8
1/3 < 4/7 < 6/8
1/3 < 4/7 < 6/9
1/3 < 4/7 < 8/9
1/3 < 4/8 < 5/6
1/3 < 4/8 < 5/7
1/3 < 4/8 < 5/9
1/3 < 4/8 < 6/7
1/3 < 4/8 < 6/9
1/3 < 4/8 < 7/9
1/3 < 4/9 < 5/6
1/3 < 4/9 < 5/7
1/3 < 4/9 < 5/8
1/3 < 4/9 < 6/7
1/3 < 4/9 < 6/8
1/3 < 4/9 < 7/8
1/3 < 5/6 < 7/8
1/3 < 5/6 < 8/9
1/3 < 5/7 < 6/8
1/3 < 5/7 < 8/9
1/3 < 5/8 < 4/6
1/3 < 5/8 < 6/7
1/3 < 5/8 < 6/9
1/3 < 5/8 < 7/9
1/3 < 5/9 < 4/6
1/3 < 5/9 < 4/7
1/3 < 5/9 < 6/7
1/3 < 5/9 < 6/8
1/3 < 5/9 < 7/8
1/3 < 6/7 < 8/9
1/3 < 6/8 < 4/5
1/3 < 6/8 < 7/9
1/3 < 6/9 < 4/5
1/3 < 6/9 < 5/7
1/3 < 6/9 < 7/8
1/3 < 7/9 < 4/5
1/3 < 7/9 < 5/6
1/4 < 2/3 < 5/6
1/4 < 2/3 < 5/7
1/4 < 2/3 < 6/7
1/4 < 2/3 < 6/8
1/4 < 2/3 < 7/8
1/4 < 2/3 < 7/9
1/4 < 2/3 < 8/9
1/4 < 2/5 < 3/6
1/4 < 2/5 < 3/7
1/4 < 2/5 < 6/7
1/4 < 2/5 < 6/8
1/4 < 2/5 < 6/9
1/4 < 2/5 < 7/8
1/4 < 2/5 < 7/9
1/4 < 2/5 < 8/9
1/4 < 2/6 < 3/5
1/4 < 2/6 < 3/7
1/4 < 2/6 < 3/8
1/4 < 2/6 < 5/7
1/4 < 2/6 < 5/8
1/4 < 2/6 < 5/9
1/4 < 2/6 < 7/8
1/4 < 2/6 < 7/9
1/4 < 2/6 < 8/9
1/4 < 2/7 < 3/5
1/4 < 2/7 < 3/6
1/4 < 2/7 < 3/8
1/4 < 2/7 < 3/9
1/4 < 2/7 < 5/6
1/4 < 2/7 < 5/8
1/4 < 2/7 < 5/9
1/4 < 2/7 < 6/8
1/4 < 2/7 < 6/9
1/4 < 2/7 < 8/9
1/4 < 3/5 < 6/7
1/4 < 3/5 < 6/8
1/4 < 3/5 < 6/9
1/4 < 3/5 < 7/8
1/4 < 3/5 < 7/9
1/4 < 3/5 < 8/9
1/4 < 3/6 < 5/7
1/4 < 3/6 < 5/8
1/4 < 3/6 < 5/9
1/4 < 3/6 < 7/8
1/4 < 3/6 < 7/9
1/4 < 3/6 < 8/9
1/4 < 3/7 < 5/6
1/4 < 3/7 < 5/8
1/4 < 3/7 < 5/9
1/4 < 3/7 < 6/8
1/4 < 3/7 < 6/9
1/4 < 3/7 < 8/9
1/4 < 3/8 < 2/5
1/4 < 3/8 < 5/6
1/4 < 3/8 < 5/7
1/4 < 3/8 < 5/9
1/4 < 3/8 < 6/7
1/4 < 3/8 < 6/9
1/4 < 3/8 < 7/9
1/4 < 3/9 < 2/5
1/4 < 3/9 < 5/6
1/4 < 3/9 < 5/7
1/4 < 3/9 < 5/8
1/4 < 3/9 < 6/7
1/4 < 3/9 < 6/8
1/4 < 3/9 < 7/8
1/4 < 5/6 < 7/8
1/4 < 5/6 < 8/9
1/4 < 5/7 < 6/8
1/4 < 5/7 < 8/9
1/4 < 5/8 < 2/3
1/4 < 5/8 < 6/7
1/4 < 5/8 < 6/9
1/4 < 5/8 < 7/9
1/4 < 5/9 < 2/3
1/4 < 5/9 < 6/7
1/4 < 5/9 < 6/8
1/4 < 5/9 < 7/8
1/4 < 6/7 < 8/9
1/4 < 6/8 < 7/9
1/4 < 6/9 < 5/7
1/4 < 6/9 < 7/8
1/4 < 7/9 < 5/6
1/5 < 2/3 < 6/7
1/5 < 2/3 < 6/8
1/5 < 2/3 < 7/8
1/5 < 2/3 < 7/9
1/5 < 2/3 < 8/9
1/5 < 2/4 < 6/7
1/5 < 2/4 < 6/8
1/5 < 2/4 < 6/9
1/5 < 2/4 < 7/8
1/5 < 2/4 < 7/9
1/5 < 2/4 < 8/9
1/5 < 2/6 < 3/4
1/5 < 2/6 < 3/7
1/5 < 2/6 < 3/8
1/5 < 2/6 < 4/7
1/5 < 2/6 < 4/8
1/5 < 2/6 < 4/9
1/5 < 2/6 < 7/8
1/5 < 2/6 < 7/9
1/5 < 2/6 < 8/9
1/5 < 2/7 < 3/4
1/5 < 2/7 < 3/6
1/5 < 2/7 < 3/8
1/5 < 2/7 < 3/9
1/5 < 2/7 < 4/6
1/5 < 2/7 < 4/8
1/5 < 2/7 < 4/9
1/5 < 2/7 < 6/8
1/5 < 2/7 < 6/9
1/5 < 2/7 < 8/9
1/5 < 2/8 < 3/4
1/5 < 2/8 < 3/6
1/5 < 2/8 < 3/7
1/5 < 2/8 < 3/9
1/5 < 2/8 < 4/6
1/5 < 2/8 < 4/7
1/5 < 2/8 < 4/9
1/5 < 2/8 < 6/7
1/5 < 2/8 < 6/9
1/5 < 2/8 < 7/9
1/5 < 2/9 < 3/4
1/5 < 2/9 < 3/6
1/5 < 2/9 < 3/7
1/5 < 2/9 < 3/8
1/5 < 2/9 < 4/6
1/5 < 2/9 < 4/7
1/5 < 2/9 < 4/8
1/5 < 2/9 < 6/7
1/5 < 2/9 < 6/8
1/5 < 2/9 < 7/8
1/5 < 3/4 < 6/7
1/5 < 3/4 < 7/8
1/5 < 3/4 < 7/9
1/5 < 3/4 < 8/9
1/5 < 3/6 < 4/7
1/5 < 3/6 < 7/8
1/5 < 3/6 < 7/9
1/5 < 3/6 < 8/9
1/5 < 3/7 < 2/4
1/5 < 3/7 < 4/6
1/5 < 3/7 < 4/8
1/5 < 3/7 < 4/9
1/5 < 3/7 < 6/8
1/5 < 3/7 < 6/9
1/5 < 3/7 < 8/9
1/5 < 3/8 < 2/4
1/5 < 3/8 < 4/6
1/5 < 3/8 < 4/7
1/5 < 3/8 < 4/9
1/5 < 3/8 < 6/7
1/5 < 3/8 < 6/9
1/5 < 3/8 < 7/9
1/5 < 3/9 < 2/4
1/5 < 3/9 < 4/6
1/5 < 3/9 < 4/7
1/5 < 3/9 < 4/8
1/5 < 3/9 < 6/7
1/5 < 3/9 < 6/8
1/5 < 3/9 < 7/8
1/5 < 4/6 < 7/8
1/5 < 4/6 < 7/9
1/5 < 4/6 < 8/9
1/5 < 4/7 < 2/3
1/5 < 4/7 < 6/8
1/5 < 4/7 < 6/9
1/5 < 4/7 < 8/9
1/5 < 4/8 < 2/3
1/5 < 4/8 < 6/7
1/5 < 4/8 < 6/9
1/5 < 4/8 < 7/9
1/5 < 4/9 < 2/3
1/5 < 4/9 < 3/6
1/5 < 4/9 < 6/7
1/5 < 4/9 < 6/8
1/5 < 4/9 < 7/8
1/5 < 6/7 < 8/9
1/5 < 6/8 < 7/9
1/5 < 6/9 < 3/4
1/5 < 6/9 < 7/8
1/6 < 2/3 < 4/5
1/6 < 2/3 < 5/7
1/6 < 2/3 < 7/8
1/6 < 2/3 < 7/9
1/6 < 2/3 < 8/9
1/6 < 2/4 < 3/5
1/6 < 2/4 < 5/7
1/6 < 2/4 < 5/8
1/6 < 2/4 < 5/9
1/6 < 2/4 < 7/8
1/6 < 2/4 < 7/9
1/6 < 2/4 < 8/9
1/6 < 2/5 < 3/4
1/6 < 2/5 < 3/7
1/6 < 2/5 < 4/7
1/6 < 2/5 < 4/8
1/6 < 2/5 < 4/9
1/6 < 2/5 < 7/8
1/6 < 2/5 < 7/9
1/6 < 2/5 < 8/9
1/6 < 2/7 < 3/4
1/6 < 2/7 < 3/5
1/6 < 2/7 < 3/8
1/6 < 2/7 < 3/9
1/6 < 2/7 < 4/5
1/6 < 2/7 < 4/8
1/6 < 2/7 < 4/9
1/6 < 2/7 < 5/8
1/6 < 2/7 < 5/9
1/6 < 2/7 < 8/9
1/6 < 2/8 < 3/4
1/6 < 2/8 < 3/5
1/6 < 2/8 < 3/7
1/6 < 2/8 < 3/9
1/6 < 2/8 < 4/5
1/6 < 2/8 < 4/7
1/6 < 2/8 < 4/9
1/6 < 2/8 < 5/7
1/6 < 2/8 < 5/9
1/6 < 2/8 < 7/9
1/6 < 2/9 < 3/4
1/6 < 2/9 < 3/5
1/6 < 2/9 < 3/7
1/6 < 2/9 < 3/8
1/6 < 2/9 < 4/5
1/6 < 2/9 < 4/7
1/6 < 2/9 < 4/8
1/6 < 2/9 < 5/7
1/6 < 2/9 < 5/8
1/6 < 2/9 < 7/8
1/6 < 3/4 < 7/8
1/6 < 3/4 < 7/9
1/6 < 3/4 < 8/9
1/6 < 3/5 < 7/8
1/6 < 3/5 < 7/9
1/6 < 3/5 < 8/9
1/6 < 3/7 < 2/4
1/6 < 3/7 < 4/5
1/6 < 3/7 < 4/8
1/6 < 3/7 < 4/9
1/6 < 3/7 < 5/8
1/6 < 3/7 < 5/9
1/6 < 3/7 < 8/9
1/6 < 3/8 < 2/4
1/6 < 3/8 < 2/5
1/6 < 3/8 < 4/5
1/6 < 3/8 < 4/7
1/6 < 3/8 < 4/9
1/6 < 3/8 < 5/7
1/6 < 3/8 < 5/9
1/6 < 3/8 < 7/9
1/6 < 3/9 < 2/4
1/6 < 3/9 < 2/5
1/6 < 3/9 < 4/5
1/6 < 3/9 < 4/7
1/6 < 3/9 < 4/8
1/6 < 3/9 < 5/7
1/6 < 3/9 < 5/8
1/6 < 3/9 < 7/8
1/6 < 4/5 < 7/8
1/6 < 4/5 < 8/9
1/6 < 4/7 < 2/3
1/6 < 4/7 < 3/5
1/6 < 4/7 < 5/8
1/6 < 4/7 < 8/9
1/6 < 4/8 < 2/3
1/6 < 4/8 < 3/5
1/6 < 4/8 < 5/7
1/6 < 4/8 < 5/9
1/6 < 4/8 < 7/9
1/6 < 4/9 < 2/3
1/6 < 4/9 < 3/5
1/6 < 4/9 < 5/7
1/6 < 4/9 < 5/8
1/6 < 4/9 < 7/8
1/6 < 5/7 < 3/4
1/6 < 5/7 < 8/9
1/6 < 5/8 < 2/3
1/6 < 5/8 < 3/4
1/6 < 5/8 < 7/9
1/6 < 5/9 < 2/3
1/6 < 5/9 < 3/4
1/6 < 5/9 < 4/7
1/6 < 5/9 < 7/8
1/6 < 7/9 < 4/5
1/7 < 2/3 < 4/5
1/7 < 2/3 < 5/6
1/7 < 2/3 < 6/8
1/7 < 2/3 < 8/9
1/7 < 2/4 < 3/5
1/7 < 2/4 < 5/6
1/7 < 2/4 < 5/8
1/7 < 2/4 < 5/9
1/7 < 2/4 < 6/8
1/7 < 2/4 < 6/9
1/7 < 2/4 < 8/9
1/7 < 2/5 < 3/4
1/7 < 2/5 < 3/6
1/7 < 2/5 < 4/6
1/7 < 2/5 < 4/8
1/7 < 2/5 < 4/9
1/7 < 2/5 < 6/8
1/7 < 2/5 < 6/9
1/7 < 2/5 < 8/9
1/7 < 2/6 < 3/4
1/7 < 2/6 < 3/5
1/7 < 2/6 < 3/8
1/7 < 2/6 < 4/5
1/7 < 2/6 < 4/8
1/7 < 2/6 < 4/9
1/7 < 2/6 < 5/8
1/7 < 2/6 < 5/9
1/7 < 2/6 < 8/9
1/7 < 2/8 < 3/4
1/7 < 2/8 < 3/5
1/7 < 2/8 < 3/6
1/7 < 2/8 < 3/9
1/7 < 2/8 < 4/5
1/7 < 2/8 < 4/6
1/7 < 2/8 < 4/9
1/7 < 2/8 < 5/6
1/7 < 2/8 < 5/9
1/7 < 2/8 < 6/9
1/7 < 2/9 < 3/4
1/7 < 2/9 < 3/5
1/7 < 2/9 < 3/6
1/7 < 2/9 < 3/8
1/7 < 2/9 < 4/5
1/7 < 2/9 < 4/6
1/7 < 2/9 < 4/8
1/7 < 2/9 < 5/6
1/7 < 2/9 < 5/8
1/7 < 2/9 < 6/8
1/7 < 3/4 < 5/6
1/7 < 3/4 < 8/9
1/7 < 3/5 < 4/6
1/7 < 3/5 < 6/8
1/7 < 3/5 < 6/9
1/7 < 3/5 < 8/9
1/7 < 3/6 < 4/5
1/7 < 3/6 < 5/8
1/7 < 3/6 < 5/9
1/7 < 3/6 < 8/9
1/7 < 3/8 < 2/4
1/7 < 3/8 < 2/5
1/7 < 3/8 < 4/5
1/7 < 3/8 < 4/6
1/7 < 3/8 < 4/9
1/7 < 3/8 < 5/6
1/7 < 3/8 < 5/9
1/7 < 3/8 < 6/9
1/7 < 3/9 < 2/4
1/7 < 3/9 < 2/5
1/7 < 3/9 < 4/5
1/7 < 3/9 < 4/6
1/7 < 3/9 < 4/8
1/7 < 3/9 < 5/6
1/7 < 3/9 < 5/8
1/7 < 3/9 < 6/8
1/7 < 4/5 < 8/9
1/7 < 4/6 < 8/9
1/7 < 4/8 < 2/3
1/7 < 4/8 < 3/5
1/7 < 4/8 < 5/6
1/7 < 4/8 < 5/9
1/7 < 4/8 < 6/9
1/7 < 4/9 < 2/3
1/7 < 4/9 < 3/5
1/7 < 4/9 < 3/6
1/7 < 4/9 < 5/6
1/7 < 4/9 < 5/8
1/7 < 4/9 < 6/8
1/7 < 5/6 < 8/9
1/7 < 5/8 < 2/3
1/7 < 5/8 < 3/4
1/7 < 5/8 < 4/6
1/7 < 5/8 < 6/9
1/7 < 5/9 < 2/3
1/7 < 5/9 < 3/4
1/7 < 5/9 < 4/6
1/7 < 5/9 < 6/8
1/7 < 6/8 < 4/5
1/7 < 6/9 < 3/4
1/7 < 6/9 < 4/5
1/8 < 2/3 < 4/5
1/8 < 2/3 < 5/6
1/8 < 2/3 < 5/7
1/8 < 2/3 < 6/7
1/8 < 2/3 < 7/9
1/8 < 2/4 < 3/5
1/8 < 2/4 < 5/6
1/8 < 2/4 < 5/7
1/8 < 2/4 < 5/9
1/8 < 2/4 < 6/7
1/8 < 2/4 < 6/9
1/8 < 2/4 < 7/9
1/8 < 2/5 < 3/4
1/8 < 2/5 < 3/6
1/8 < 2/5 < 3/7
1/8 < 2/5 < 4/6
1/8 < 2/5 < 4/7
1/8 < 2/5 < 4/9
1/8 < 2/5 < 6/7
1/8 < 2/5 < 6/9
1/8 < 2/5 < 7/9
1/8 < 2/6 < 3/4
1/8 < 2/6 < 3/5
1/8 < 2/6 < 3/7
1/8 < 2/6 < 4/5
1/8 < 2/6 < 4/7
1/8 < 2/6 < 4/9
1/8 < 2/6 < 5/7
1/8 < 2/6 < 5/9
1/8 < 2/6 < 7/9
1/8 < 2/7 < 3/4
1/8 < 2/7 < 3/5
1/8 < 2/7 < 3/6
1/8 < 2/7 < 3/9
1/8 < 2/7 < 4/5
1/8 < 2/7 < 4/6
1/8 < 2/7 < 4/9
1/8 < 2/7 < 5/6
1/8 < 2/7 < 5/9
1/8 < 2/7 < 6/9
1/8 < 2/9 < 3/4
1/8 < 2/9 < 3/5
1/8 < 2/9 < 3/6
1/8 < 2/9 < 3/7
1/8 < 2/9 < 4/5
1/8 < 2/9 < 4/6
1/8 < 2/9 < 4/7
1/8 < 2/9 < 5/6
1/8 < 2/9 < 5/7
1/8 < 2/9 < 6/7
1/8 < 3/4 < 5/6
1/8 < 3/4 < 6/7
1/8 < 3/4 < 7/9
1/8 < 3/5 < 4/6
1/8 < 3/5 < 6/7
1/8 < 3/5 < 6/9
1/8 < 3/5 < 7/9
1/8 < 3/6 < 4/5
1/8 < 3/6 < 4/7
1/8 < 3/6 < 5/7
1/8 < 3/6 < 5/9
1/8 < 3/6 < 7/9
1/8 < 3/7 < 2/4
1/8 < 3/7 < 4/5
1/8 < 3/7 < 4/6
1/8 < 3/7 < 4/9
1/8 < 3/7 < 5/6
1/8 < 3/7 < 5/9
1/8 < 3/7 < 6/9
1/8 < 3/9 < 2/4
1/8 < 3/9 < 2/5
1/8 < 3/9 < 4/5
1/8 < 3/9 < 4/6
1/8 < 3/9 < 4/7
1/8 < 3/9 < 5/6
1/8 < 3/9 < 5/7
1/8 < 3/9 < 6/7
1/8 < 4/5 < 6/7
1/8 < 4/6 < 5/7
1/8 < 4/6 < 7/9
1/8 < 4/7 < 2/3
1/8 < 4/7 < 3/5
1/8 < 4/7 < 5/6
1/8 < 4/7 < 6/9
1/8 < 4/9 < 2/3
1/8 < 4/9 < 3/5
1/8 < 4/9 < 3/6
1/8 < 4/9 < 5/6
1/8 < 4/9 < 5/7
1/8 < 4/9 < 6/7
1/8 < 5/7 < 3/4
1/8 < 5/9 < 2/3
1/8 < 5/9 < 3/4
1/8 < 5/9 < 4/6
1/8 < 5/9 < 4/7
1/8 < 5/9 < 6/7
1/8 < 6/9 < 3/4
1/8 < 6/9 < 4/5
1/8 < 6/9 < 5/7
1/8 < 7/9 < 4/5
1/8 < 7/9 < 5/6
1/9 < 2/3 < 4/5
1/9 < 2/3 < 5/6
1/9 < 2/3 < 5/7
1/9 < 2/3 < 6/7
1/9 < 2/3 < 6/8
1/9 < 2/3 < 7/8
1/9 < 2/4 < 3/5
1/9 < 2/4 < 5/6
1/9 < 2/4 < 5/7
1/9 < 2/4 < 5/8
1/9 < 2/4 < 6/7
1/9 < 2/4 < 6/8
1/9 < 2/4 < 7/8
1/9 < 2/5 < 3/4
1/9 < 2/5 < 3/6
1/9 < 2/5 < 3/7
1/9 < 2/5 < 4/6
1/9 < 2/5 < 4/7
1/9 < 2/5 < 4/8
1/9 < 2/5 < 6/7
1/9 < 2/5 < 6/8
1/9 < 2/5 < 7/8
1/9 < 2/6 < 3/4
1/9 < 2/6 < 3/5
1/9 < 2/6 < 3/7
1/9 < 2/6 < 3/8
1/9 < 2/6 < 4/5
1/9 < 2/6 < 4/7
1/9 < 2/6 < 4/8
1/9 < 2/6 < 5/7
1/9 < 2/6 < 5/8
1/9 < 2/6 < 7/8
1/9 < 2/7 < 3/4
1/9 < 2/7 < 3/5
1/9 < 2/7 < 3/6
1/9 < 2/7 < 3/8
1/9 < 2/7 < 4/5
1/9 < 2/7 < 4/6
1/9 < 2/7 < 4/8
1/9 < 2/7 < 5/6
1/9 < 2/7 < 5/8
1/9 < 2/7 < 6/8
1/9 < 2/8 < 3/4
1/9 < 2/8 < 3/5
1/9 < 2/8 < 3/6
1/9 < 2/8 < 3/7
1/9 < 2/8 < 4/5
1/9 < 2/8 < 4/6
1/9 < 2/8 < 4/7
1/9 < 2/8 < 5/6
1/9 < 2/8 < 5/7
1/9 < 2/8 < 6/7
1/9 < 3/4 < 5/6
1/9 < 3/4 < 6/7
1/9 < 3/4 < 7/8
1/9 < 3/5 < 4/6
1/9 < 3/5 < 6/7
1/9 < 3/5 < 6/8
1/9 < 3/5 < 7/8
1/9 < 3/6 < 4/5
1/9 < 3/6 < 4/7
1/9 < 3/6 < 5/7
1/9 < 3/6 < 5/8
1/9 < 3/6 < 7/8
1/9 < 3/7 < 2/4
1/9 < 3/7 < 4/5
1/9 < 3/7 < 4/6
1/9 < 3/7 < 4/8
1/9 < 3/7 < 5/6
1/9 < 3/7 < 5/8
1/9 < 3/7 < 6/8
1/9 < 3/8 < 2/4
1/9 < 3/8 < 2/5
1/9 < 3/8 < 4/5
1/9 < 3/8 < 4/6
1/9 < 3/8 < 4/7
1/9 < 3/8 < 5/6
1/9 < 3/8 < 5/7
1/9 < 3/8 < 6/7
1/9 < 4/5 < 6/7
1/9 < 4/5 < 7/8
1/9 < 4/6 < 5/7
1/9 < 4/6 < 7/8
1/9 < 4/7 < 2/3
1/9 < 4/7 < 3/5
1/9 < 4/7 < 5/6
1/9 < 4/7 < 5/8
1/9 < 4/7 < 6/8
1/9 < 4/8 < 2/3
1/9 < 4/8 < 3/5
1/9 < 4/8 < 5/6
1/9 < 4/8 < 5/7
1/9 < 4/8 < 6/7
1/9 < 5/6 < 7/8
1/9 < 5/7 < 3/4
1/9 < 5/7 < 6/8
1/9 < 5/8 < 2/3
1/9 < 5/8 < 3/4
1/9 < 5/8 < 4/6
1/9 < 5/8 < 6/7
1/9 < 6/8 < 4/5
2/3 < 4/5 < 6/7
2/3 < 4/5 < 7/8
2/3 < 4/5 < 8/9
2/3 < 5/6 < 7/8
2/3 < 5/6 < 8/9
2/3 < 5/7 < 6/8
2/3 < 5/7 < 8/9
2/3 < 6/7 < 8/9
2/3 < 6/8 < 4/5
2/3 < 6/8 < 7/9
2/3 < 7/9 < 4/5
2/3 < 7/9 < 5/6
2/4 < 3/5 < 6/7
2/4 < 3/5 < 6/8
2/4 < 3/5 < 6/9
2/4 < 3/5 < 7/8
2/4 < 3/5 < 7/9
2/4 < 3/5 < 8/9
2/4 < 5/6 < 7/8
2/4 < 5/6 < 8/9
2/4 < 5/7 < 6/8
2/4 < 5/7 < 8/9
2/4 < 5/8 < 6/7
2/4 < 5/8 < 6/9
2/4 < 5/8 < 7/9
2/4 < 5/9 < 6/7
2/4 < 5/9 < 6/8
2/4 < 5/9 < 7/8
2/4 < 6/7 < 8/9
2/4 < 6/8 < 7/9
2/4 < 6/9 < 5/7
2/4 < 6/9 < 7/8
2/4 < 7/9 < 5/6
2/5 < 3/4 < 6/7
2/5 < 3/4 < 7/8
2/5 < 3/4 < 7/9
2/5 < 3/4 < 8/9
2/5 < 3/6 < 4/7
2/5 < 3/6 < 7/8
2/5 < 3/6 < 7/9
2/5 < 3/6 < 8/9
2/5 < 3/7 < 4/6
2/5 < 3/7 < 4/8
2/5 < 3/7 < 4/9
2/5 < 3/7 < 6/8
2/5 < 3/7 < 6/9
2/5 < 3/7 < 8/9
2/5 < 4/6 < 7/8
2/5 < 4/6 < 7/9
2/5 < 4/6 < 8/9
2/5 < 4/7 < 6/8
2/5 < 4/7 < 6/9
2/5 < 4/7 < 8/9
2/5 < 4/8 < 6/7
2/5 < 4/8 < 6/9
2/5 < 4/8 < 7/9
2/5 < 4/9 < 3/6
2/5 < 4/9 < 6/7
2/5 < 4/9 < 6/8
2/5 < 4/9 < 7/8
2/5 < 6/7 < 8/9
2/5 < 6/8 < 7/9
2/5 < 6/9 < 3/4
2/5 < 6/9 < 7/8
2/6 < 3/4 < 7/8
2/6 < 3/4 < 7/9
2/6 < 3/4 < 8/9
2/6 < 3/5 < 7/8
2/6 < 3/5 < 7/9
2/6 < 3/5 < 8/9
2/6 < 3/7 < 4/5
2/6 < 3/7 < 4/8
2/6 < 3/7 < 4/9
2/6 < 3/7 < 5/8
2/6 < 3/7 < 5/9
2/6 < 3/7 < 8/9
2/6 < 3/8 < 4/5
2/6 < 3/8 < 4/7
2/6 < 3/8 < 4/9
2/6 < 3/8 < 5/7
2/6 < 3/8 < 5/9
2/6 < 3/8 < 7/9
2/6 < 4/5 < 7/8
2/6 < 4/5 < 8/9
2/6 < 4/7 < 3/5
2/6 < 4/7 < 5/8
2/6 < 4/7 < 8/9
2/6 < 4/8 < 3/5
2/6 < 4/8 < 5/7
2/6 < 4/8 < 5/9
2/6 < 4/8 < 7/9
2/6 < 4/9 < 3/5
2/6 < 4/9 < 5/7
2/6 < 4/9 < 5/8
2/6 < 4/9 < 7/8
2/6 < 5/7 < 3/4
2/6 < 5/7 < 8/9
2/6 < 5/8 < 3/4
2/6 < 5/8 < 7/9
2/6 < 5/9 < 3/4
2/6 < 5/9 < 4/7
2/6 < 5/9 < 7/8
2/6 < 7/9 < 4/5
2/7 < 1/3 < 4/5
2/7 < 1/3 < 4/6
2/7 < 1/3 < 4/8
2/7 < 1/3 < 4/9
2/7 < 1/3 < 5/6
2/7 < 1/3 < 5/8
2/7 < 1/3 < 5/9
2/7 < 1/3 < 6/8
2/7 < 1/3 < 6/9
2/7 < 1/3 < 8/9
2/7 < 3/4 < 5/6
2/7 < 3/4 < 8/9
2/7 < 3/5 < 4/6
2/7 < 3/5 < 6/8
2/7 < 3/5 < 6/9
2/7 < 3/5 < 8/9
2/7 < 3/6 < 4/5
2/7 < 3/6 < 5/8
2/7 < 3/6 < 5/9
2/7 < 3/6 < 8/9
2/7 < 3/8 < 4/5
2/7 < 3/8 < 4/6
2/7 < 3/8 < 4/9
2/7 < 3/8 < 5/6
2/7 < 3/8 < 5/9
2/7 < 3/8 < 6/9
2/7 < 3/9 < 4/5
2/7 < 3/9 < 4/6
2/7 < 3/9 < 4/8
2/7 < 3/9 < 5/6
2/7 < 3/9 < 5/8
2/7 < 3/9 < 6/8
2/7 < 4/5 < 8/9
2/7 < 4/6 < 8/9
2/7 < 4/8 < 3/5
2/7 < 4/8 < 5/6
2/7 < 4/8 < 5/9
2/7 < 4/8 < 6/9
2/7 < 4/9 < 3/5
2/7 < 4/9 < 3/6
2/7 < 4/9 < 5/6
2/7 < 4/9 < 5/8
2/7 < 4/9 < 6/8
2/7 < 5/6 < 8/9
2/7 < 5/8 < 3/4
2/7 < 5/8 < 4/6
2/7 < 5/8 < 6/9
2/7 < 5/9 < 3/4
2/7 < 5/9 < 4/6
2/7 < 5/9 < 6/8
2/7 < 6/8 < 4/5
2/7 < 6/9 < 3/4
2/7 < 6/9 < 4/5
2/8 < 1/3 < 4/5
2/8 < 1/3 < 4/6
2/8 < 1/3 < 4/7
2/8 < 1/3 < 4/9
2/8 < 1/3 < 5/6
2/8 < 1/3 < 5/7
2/8 < 1/3 < 5/9
2/8 < 1/3 < 6/7
2/8 < 1/3 < 6/9
2/8 < 1/3 < 7/9
2/8 < 3/4 < 5/6
2/8 < 3/4 < 6/7
2/8 < 3/4 < 7/9
2/8 < 3/5 < 4/6
2/8 < 3/5 < 6/7
2/8 < 3/5 < 6/9
2/8 < 3/5 < 7/9
2/8 < 3/6 < 4/5
2/8 < 3/6 < 4/7
2/8 < 3/6 < 5/7
2/8 < 3/6 < 5/9
2/8 < 3/6 < 7/9
2/8 < 3/7 < 4/5
2/8 < 3/7 < 4/6
2/8 < 3/7 < 4/9
2/8 < 3/7 < 5/6
2/8 < 3/7 < 5/9
2/8 < 3/7 < 6/9
2/8 < 3/9 < 4/5
2/8 < 3/9 < 4/6
2/8 < 3/9 < 4/7
2/8 < 3/9 < 5/6
2/8 < 3/9 < 5/7
2/8 < 3/9 < 6/7
2/8 < 4/5 < 6/7
2/8 < 4/6 < 5/7
2/8 < 4/6 < 7/9
2/8 < 4/7 < 3/5
2/8 < 4/7 < 5/6
2/8 < 4/7 < 6/9
2/8 < 4/9 < 3/5
2/8 < 4/9 < 3/6
2/8 < 4/9 < 5/6
2/8 < 4/9 < 5/7
2/8 < 4/9 < 6/7
2/8 < 5/7 < 3/4
2/8 < 5/9 < 3/4
2/8 < 5/9 < 4/6
2/8 < 5/9 < 4/7
2/8 < 5/9 < 6/7
2/8 < 6/9 < 3/4
2/8 < 6/9 < 4/5
2/8 < 6/9 < 5/7
2/8 < 7/9 < 4/5
2/8 < 7/9 < 5/6
2/9 < 1/3 < 4/5
2/9 < 1/3 < 4/6
2/9 < 1/3 < 4/7
2/9 < 1/3 < 4/8
2/9 < 1/3 < 5/6
2/9 < 1/3 < 5/7
2/9 < 1/3 < 5/8
2/9 < 1/3 < 6/7
2/9 < 1/3 < 6/8
2/9 < 1/3 < 7/8
2/9 < 1/4 < 3/5
2/9 < 1/4 < 3/6
2/9 < 1/4 < 3/7
2/9 < 1/4 < 3/8
2/9 < 1/4 < 5/6
2/9 < 1/4 < 5/7
2/9 < 1/4 < 5/8
2/9 < 1/4 < 6/7
2/9 < 1/4 < 6/8
2/9 < 1/4 < 7/8
2/9 < 3/4 < 5/6
2/9 < 3/4 < 6/7
2/9 < 3/4 < 7/8
2/9 < 3/5 < 4/6
2/9 < 3/5 < 6/7
2/9 < 3/5 < 6/8
2/9 < 3/5 < 7/8
2/9 < 3/6 < 4/5
2/9 < 3/6 < 4/7
2/9 < 3/6 < 5/7
2/9 < 3/6 < 5/8
2/9 < 3/6 < 7/8
2/9 < 3/7 < 4/5
2/9 < 3/7 < 4/6
2/9 < 3/7 < 4/8
2/9 < 3/7 < 5/6
2/9 < 3/7 < 5/8
2/9 < 3/7 < 6/8
2/9 < 3/8 < 4/5
2/9 < 3/8 < 4/6
2/9 < 3/8 < 4/7
2/9 < 3/8 < 5/6
2/9 < 3/8 < 5/7
2/9 < 3/8 < 6/7
2/9 < 4/5 < 6/7
2/9 < 4/5 < 7/8
2/9 < 4/6 < 5/7
2/9 < 4/6 < 7/8
2/9 < 4/7 < 3/5
2/9 < 4/7 < 5/6
2/9 < 4/7 < 5/8
2/9 < 4/7 < 6/8
2/9 < 4/8 < 3/5
2/9 < 4/8 < 5/6
2/9 < 4/8 < 5/7
2/9 < 4/8 < 6/7
2/9 < 5/6 < 7/8
2/9 < 5/7 < 3/4
2/9 < 5/7 < 6/8
2/9 < 5/8 < 3/4
2/9 < 5/8 < 4/6
2/9 < 5/8 < 6/7
2/9 < 6/8 < 4/5
3/4 < 5/6 < 7/8
3/4 < 5/6 < 8/9
3/4 < 6/7 < 8/9
3/4 < 7/9 < 5/6
3/5 < 4/6 < 7/8
3/5 < 4/6 < 7/9
3/5 < 4/6 < 8/9
3/5 < 6/7 < 8/9
3/5 < 6/8 < 7/9
3/5 < 6/9 < 7/8
3/6 < 4/5 < 7/8
3/6 < 4/5 < 8/9
3/6 < 4/7 < 5/8
3/6 < 4/7 < 8/9
3/6 < 5/7 < 8/9
3/6 < 5/8 < 7/9
3/6 < 5/9 < 4/7
3/6 < 5/9 < 7/8
3/6 < 7/9 < 4/5
3/7 < 1/2 < 4/5
3/7 < 1/2 < 4/6
3/7 < 1/2 < 5/6
3/7 < 1/2 < 5/8
3/7 < 1/2 < 5/9
3/7 < 1/2 < 6/8
3/7 < 1/2 < 6/9
3/7 < 1/2 < 8/9
3/7 < 2/4 < 5/6
3/7 < 2/4 < 5/8
3/7 < 2/4 < 5/9
3/7 < 2/4 < 6/8
3/7 < 2/4 < 6/9
3/7 < 2/4 < 8/9
3/7 < 4/5 < 8/9
3/7 < 4/6 < 8/9
3/7 < 4/8 < 5/6
3/7 < 4/8 < 5/9
3/7 < 4/8 < 6/9
3/7 < 4/9 < 1/2
3/7 < 4/9 < 5/6
3/7 < 4/9 < 5/8
3/7 < 4/9 < 6/8
3/7 < 5/6 < 8/9
3/7 < 5/8 < 4/6
3/7 < 5/8 < 6/9
3/7 < 5/9 < 4/6
3/7 < 5/9 < 6/8
3/7 < 6/8 < 4/5
3/7 < 6/9 < 4/5
3/8 < 1/2 < 4/5
3/8 < 1/2 < 4/6
3/8 < 1/2 < 4/7
3/8 < 1/2 < 5/6
3/8 < 1/2 < 5/7
3/8 < 1/2 < 5/9
3/8 < 1/2 < 6/7
3/8 < 1/2 < 6/9
3/8 < 1/2 < 7/9
3/8 < 2/4 < 5/6
3/8 < 2/4 < 5/7
3/8 < 2/4 < 5/9
3/8 < 2/4 < 6/7
3/8 < 2/4 < 6/9
3/8 < 2/4 < 7/9
3/8 < 2/5 < 4/6
3/8 < 2/5 < 4/7
3/8 < 2/5 < 4/9
3/8 < 2/5 < 6/7
3/8 < 2/5 < 6/9
3/8 < 2/5 < 7/9
3/8 < 4/5 < 6/7
3/8 < 4/6 < 5/7
3/8 < 4/6 < 7/9
3/8 < 4/7 < 5/6
3/8 < 4/7 < 6/9
3/8 < 4/9 < 1/2
3/8 < 4/9 < 5/6
3/8 < 4/9 < 5/7
3/8 < 4/9 < 6/7
3/8 < 5/9 < 4/6
3/8 < 5/9 < 4/7
3/8 < 5/9 < 6/7
3/8 < 6/9 < 4/5
3/8 < 6/9 < 5/7
3/8 < 7/9 < 4/5
3/8 < 7/9 < 5/6
3/9 < 1/2 < 4/5
3/9 < 1/2 < 4/6
3/9 < 1/2 < 4/7
3/9 < 1/2 < 5/6
3/9 < 1/2 < 5/7
3/9 < 1/2 < 5/8
3/9 < 1/2 < 6/7
3/9 < 1/2 < 6/8
3/9 < 1/2 < 7/8
3/9 < 2/4 < 5/6
3/9 < 2/4 < 5/7
3/9 < 2/4 < 5/8
3/9 < 2/4 < 6/7
3/9 < 2/4 < 6/8
3/9 < 2/4 < 7/8
3/9 < 2/5 < 4/6
3/9 < 2/5 < 4/7
3/9 < 2/5 < 4/8
3/9 < 2/5 < 6/7
3/9 < 2/5 < 6/8
3/9 < 2/5 < 7/8
3/9 < 4/5 < 6/7
3/9 < 4/5 < 7/8
3/9 < 4/6 < 5/7
3/9 < 4/6 < 7/8
3/9 < 4/7 < 5/6
3/9 < 4/7 < 5/8
3/9 < 4/7 < 6/8
3/9 < 4/8 < 5/6
3/9 < 4/8 < 5/7
3/9 < 4/8 < 6/7
3/9 < 5/6 < 7/8
3/9 < 5/7 < 6/8
3/9 < 5/8 < 4/6
3/9 < 5/8 < 6/7
3/9 < 6/8 < 4/5
4/5 < 6/7 < 8/9
4/6 < 5/7 < 8/9
4/7 < 2/3 < 5/6
4/7 < 2/3 < 6/8
4/7 < 2/3 < 8/9
4/7 < 3/5 < 6/8
4/7 < 3/5 < 6/9
4/7 < 3/5 < 8/9
4/7 < 5/6 < 8/9
4/7 < 5/8 < 2/3
4/7 < 5/8 < 6/9
4/8 < 2/3 < 5/6
4/8 < 2/3 < 5/7
4/8 < 2/3 < 6/7
4/8 < 2/3 < 7/9
4/8 < 3/5 < 6/7
4/8 < 3/5 < 6/9
4/8 < 3/5 < 7/9
4/8 < 5/9 < 2/3
4/8 < 5/9 < 6/7
4/8 < 6/9 < 5/7
4/8 < 7/9 < 5/6
4/9 < 1/2 < 3/5
4/9 < 1/2 < 5/6
4/9 < 1/2 < 5/7
4/9 < 1/2 < 5/8
4/9 < 1/2 < 6/7
4/9 < 1/2 < 6/8
4/9 < 1/2 < 7/8
4/9 < 2/3 < 5/6
4/9 < 2/3 < 5/7
4/9 < 2/3 < 6/7
4/9 < 2/3 < 6/8
4/9 < 2/3 < 7/8
4/9 < 3/5 < 6/7
4/9 < 3/5 < 6/8
4/9 < 3/5 < 7/8
4/9 < 3/6 < 5/7
4/9 < 3/6 < 5/8
4/9 < 3/6 < 7/8
4/9 < 5/6 < 7/8
4/9 < 5/7 < 6/8
4/9 < 5/8 < 2/3
4/9 < 5/8 < 6/7
5/7 < 3/4 < 8/9
5/8 < 2/3 < 6/7
5/8 < 2/3 < 7/9
5/8 < 3/4 < 6/7
5/8 < 3/4 < 7/9
5/8 < 4/6 < 7/9
5/8 < 6/9 < 3/4
5/9 < 2/3 < 6/7
5/9 < 2/3 < 6/8
5/9 < 2/3 < 7/8
5/9 < 3/4 < 6/7
5/9 < 3/4 < 7/8
5/9 < 4/6 < 7/8
5/9 < 4/7 < 2/3
5/9 < 4/7 < 6/8
6/8 < 7/9 < 4/5
6/9 < 3/4 < 7/8
6/9 < 4/5 < 7/8
6/9 < 5/7 < 3/4
3 7/8 plus 4 6/8 = 8 5/8