Open Middle®
Directions: Using the digits 1 to 9 at most one time each, find the dimensions of three rectangular prisms so that their volumes are as close as possible. Note: diagram may not be drawn to scale.
Hint
All the volumes are under 100 cubic units.
Can you find 3 volumes that are within the range of 50 and 90?
Can you find 3 volumes that are within the range of 60 and 80?
The difference between all 3 volumes is less than 10.
Can you find 3 volumes that are within the range of 50 and 90?
Can you find 3 volumes that are within the range of 60 and 80?
The difference between all 3 volumes is less than 10.
Answer
You can’t get a perfect match, but you can get pretty close, with dimensions of (1,8,9), (2,5,7), (3,4,6)
Source: Daniel Walker
(1,8,9), (2,5,7), (3,4,6) is the best possible solution, no matter whether you define “as close as possible” as the “sum of the differences of the volumes, |volume1-volume2|+|volume1-volume3|+|volume2-volume3|, should be as small as possible” (which is equivalent to saying the difference between the smallest and greatest volume should be as small as possible) or as “the sum of the ratios of their volumes, volume1/volume2 + volume1/volume3 + volume2/volume3, should be as close to 3 as possible (with volume1<=volume2<=volume3)" (because three identical rectangular prisms would have the sum of the ratios of theirs volumes be 1+1+1=3)
In the case of (1,8,9), (2,5,7) and (3,4,6), this sum of the ratios of the volumes is 2.94444… and the sum of their differences is |1*8*9-2*5*7|+|1*8*9-3*4*6|+|3*4*6-2*5*7| = 4.
Now let's look at the opposite case: I want to find the solution where the volumes of the three rectangular prisms are as different as possible, so I'm looking for max(|volume1-volume2|+|volume1-volume3|+|volume2-volume3|) and min(volume1/volume2 + volume1/volume3 + volume2/volume3).
For (1,2,3), (4,5,6), (7,8,9), the sum of the (absolute value of the) differences of the three volumes is greatest:
|7*8*9-1*2*3|+|7*8*9-4*5*6|+|4*5*6-1*2*3| = 996.
However, the sum of the quotients of the three volumes is as far away from 3 as possible for (1,2,4), (3,5,6), (7,8,9) with
1*2*4/(3*5*6)+1*2*4/(7*8*9)+3*5*6/(7*8*9) = 0.28333…, while 1*2*3/(4*5*6)+1*2*3/(7*8*9)+4*5*6/(7*8*9) = 0.3