Open Middle®
Directions: Using the digits 0 to 9 at most one time each, fill in the boxes to make the value of this expression is as large as possible. Challenge: Try to make this expression as close to 30 as possible (for an added challenge, try to make this expression as close to 30 as possible using just the digits 1 through 9).
Hint
How can we figure out which part of the expression has the biggest impact on the answer? The smallest impact?
Answer
The largest (using 0-9) value is 8 x 7.6 + 9/0.1 = 150.8
You can make exactly 30 several ways (using 0-9) including:
4 x 7.5 + 0/1.2 = 30
7 x 4.0 + 3/1.5 = 30
6 x 4.5 + 9/3.0 = 30
If you only use digits 1-9 this is harder but you can still get exactly 30: 3 x 7.5 + 9/1.2 = 30
You can make exactly 30 several ways (using 0-9) including:
4 x 7.5 + 0/1.2 = 30
7 x 4.0 + 3/1.5 = 30
6 x 4.5 + 9/3.0 = 30
If you only use digits 1-9 this is harder but you can still get exactly 30: 3 x 7.5 + 9/1.2 = 30
Source: Erick Lee
Another solution: 4 x 6.5 + 8/2.0
8 x 0.3 + 2 / 0.9
All 65 solutions to get exactly 30 (with digits 0-9)
1 * 7.5 + 9 / 0.4
2 * 7.5 + 6 / 0.4
2 * 7.5 + 9 / 0.6
3 * 2.5 + 9 / 0.4
3 * 7.5 + 6 / 0.8
3 * 7.5 + 9 / 1.2
3 * 8.0 + 9 / 1.5
3 * 8.5 + 9 / 2.0
3 * 9.5 + 6 / 4.0
4 * 2.5 + 6 / 0.3
4 * 6.0 + 9 / 1.5
4 * 6.5 + 8 / 2.0
4 * 6.7 + 8 / 2.5
4 * 6.8 + 7 / 2.5
4 * 7.0 + 3 / 1.5
4 * 7.1 + 8 / 5.0
4 * 7.2 + 6 / 5.0
4 * 7.5 + 0 / 1.2
4 * 7.5 + 0 / 1.3
4 * 7.5 + 0 / 1.6
4 * 7.5 + 0 / 1.8
4 * 7.5 + 0 / 1.9
4 * 7.5 + 0 / 2.1
4 * 7.5 + 0 / 2.3
4 * 7.5 + 0 / 2.6
4 * 7.5 + 0 / 2.8
4 * 7.5 + 0 / 2.9
4 * 7.5 + 0 / 3.1
4 * 7.5 + 0 / 3.2
4 * 7.5 + 0 / 3.6
4 * 7.5 + 0 / 3.8
4 * 7.5 + 0 / 3.9
4 * 7.5 + 0 / 6.1
4 * 7.5 + 0 / 6.2
4 * 7.5 + 0 / 6.3
4 * 7.5 + 0 / 6.8
4 * 7.5 + 0 / 6.9
4 * 7.5 + 0 / 8.1
4 * 7.5 + 0 / 8.2
4 * 7.5 + 0 / 8.3
4 * 7.5 + 0 / 8.6
4 * 7.5 + 0 / 8.9
4 * 7.5 + 0 / 9.1
4 * 7.5 + 0 / 9.2
4 * 7.5 + 0 / 9.3
4 * 7.5 + 0 / 9.6
4 * 7.5 + 0 / 9.8
5 * 4.2 + 9 / 1.0
5 * 4.6 + 7 / 1.0
5 * 4.8 + 6 / 1.0
6 * 3.5 + 9 / 1.0
6 * 4.0 + 9 / 1.5
6 * 4.5 + 3 / 1.0
6 * 4.5 + 9 / 3.0
7 * 4.0 + 3 / 1.5
8 * 3.0 + 9 / 1.5
8 * 3.4 + 7 / 2.5
8 * 3.5 + 2 / 1.0
8 * 3.5 + 4 / 2.0
8 * 3.6 + 9 / 7.5
8 * 3.7 + 1 / 2.5
8 * 3.7 + 2 / 5.0
9 * 2.5 + 3 / 0.4
9 * 2.5 + 6 / 0.8
9 * 3.2 + 6 / 5.0
As you can see, the following 6 of the 65 expressions resulting in 30 don’t contain the digit 0:
3 * 7.5 + 9 / 1.2
4 * 6.7 + 8 / 2.5
4 * 6.8 + 7 / 2.5
8 * 3.4 + 7 / 2.5
8 * 3.6 + 9 / 7.5
8 * 3.7 + 1 / 2.5
The greatest result possible with digits 0-9 is 8 * 7.6 + 9 / 0.1 = 150.8 and with digits 1-9 is 9 * 8.7 + 6 / 1.2 = 83.3
The lowest result possible with digits 0-9 is 0 * 7.6 + 1 / 9.8 = 0.1020408163265306 and with digits 1-9 is 1 * 2.4 + 3 / 9.8 = 2.706122448979592