Open Middle®
Directions: Using the digits 1 to 9, at most one time each, place a digit in each box to make the equation true.
Hint
What digits would be challenging as denominators?
What digits would be challenging as numerators?
What digits would be challenging as numerators?
Answer
There are many answers. Here are two possibilities:
2/8 + 1/4 = 3/6
1/2 + 3/9 = 5/6
2/8 + 1/4 = 3/6
1/2 + 3/9 = 5/6
Source: Owen Kaplinsky
Another two answers discovered by my students: 1/2 + 9/6 = 8/4 and 5/3 + 2/6 = 8/4
2/4 + 1/3 = 12/10 simply it and get 5/6
6/8 + 1/2 = 5/4
9/3 + 4/2 = 5/1
2/1 + 6/9 = 8/3
7/2 + 3/6 = 4/1
8/2 + 9/3 = 7/1
8/4 + 6/2 = 5/1
8/4 + 9/3 = 5/1
4/2 + 6/2 = 5/1
2/8 + 6/3 = 9/4
54 unique solutions (times 2, if you count it as a different solution if you swap the two addends):
1/4 + 2/8 = 3/6 = 0.5
1/6 + 3/9 = 2/4 = 0.5
1/6 + 3/9 = 4/8 = 0.5
1/6 + 4/8 = 2/3 = 0.6667
1/2 + 3/9 = 5/6 = 0.8333
1/3 + 2/4 = 5/6 = 0.8333
1/3 + 4/8 = 5/6 = 0.8333
2/4 + 3/9 = 5/6 = 0.8333
3/9 + 4/8 = 5/6 = 0.8333
2/3 + 4/8 = 7/6 = 1.1667
1/2 + 6/8 = 5/4 = 1.25
1/2 + 5/6 = 4/3 = 1.3333
2/8 + 5/4 = 9/6 = 1.5
1/2 + 7/6 = 5/3 = 1.6667
2/4 + 7/6 = 5/3 = 1.6667
4/8 + 7/6 = 5/3 = 1.6667
2/8 + 9/6 = 7/4 = 1.75
1/2 + 9/6 = 8/4 = 2.0
2/6 + 5/3 = 8/4 = 2.0
2/8 + 7/4 = 6/3 = 2.0
4/3 + 6/9 = 2/1 = 2.0
4/8 + 9/6 = 2/1 = 2.0
5/4 + 6/8 = 2/1 = 2.0
2/8 + 6/3 = 9/4 = 2.25
3/2 + 6/8 = 9/4 = 2.25
2/6 + 8/4 = 7/3 = 2.3333
1/6 + 7/3 = 5/2 = 2.5
3/6 + 8/4 = 5/2 = 2.5
4/3 + 7/6 = 5/2 = 2.5
4/8 + 6/3 = 5/2 = 2.5
6/8 + 7/4 = 5/2 = 2.5
1/6 + 5/2 = 8/3 = 2.6667
2/1 + 4/6 = 8/3 = 2.6667
2/1 + 6/9 = 8/3 = 2.6667
4/2 + 6/9 = 8/3 = 2.6667
4/8 + 5/2 = 3/1 = 3.0
4/8 + 5/2 = 9/3 = 3.0
6/8 + 9/4 = 3/1 = 3.0
3/1 + 4/8 = 7/2 = 3.5
4/8 + 9/3 = 7/2 = 3.5
5/6 + 8/3 = 7/2 = 3.5
8/4 + 9/6 = 7/2 = 3.5
3/6 + 7/2 = 4/1 = 4.0
5/2 + 9/6 = 4/1 = 4.0
3/1 + 6/4 = 9/2 = 4.5
3/6 + 4/1 = 9/2 = 4.5
3/6 + 9/2 = 5/1 = 5.0
4/2 + 9/3 = 5/1 = 5.0
4/8 + 9/2 = 5/1 = 5.0
6/2 + 8/4 = 5/1 = 5.0
6/4 + 7/2 = 5/1 = 5.0
7/2 + 9/6 = 5/1 = 5.0
8/4 + 9/3 = 5/1 = 5.0
8/2 + 9/3 = 7/1 = 7.0
A student of mine came up with this one: 1/2 + 6/8 = 5/4
Another student just had 9/6 + 8/4 = 7/2 !!