Extreme Products – 1
THE CHALLENGE
- Using the digits from 1 to 9, each at most once, make two 2-digit numbers whose product is as large as possible.
- Also, using the digits from 1 to 9, each at most once, make two 2-digit numbers whose product is as small as possible.
EXPLORATION
Can you apply what you learned for multiplying two numbers to do this with multiplying three two-digit numbers? Can you think of other interesting variations?
Notes
THE CHALLENGE
This is small enough and obvious enough, that not much experimenting or analysis is needed.
To make the product large we want to have tens digits that are 8 and 9. The only remaining question can be settled by multiplying: should it be 97 x 86 = 8342 or 96 x 87 = 8352.
Similar logic for making the product small produces a choice of 13 x 24 = 312 or 14 x 23 = 322.
EXPLORATION
For three numbers, the analysis is similar.
To make the product large, we have first digits of 7, 8, and 9, and second digits of 4, 5, and 6. The only question is how to combine them. Some experimentation verifies what you would expect from the two number case: the largest value comes from 94 x 85 x 76.
Similarly, the smallest value comes from 14 x 25 x 36.
A natural variation of this problem is to look at three-digit numbers. We will do this in the “Extreme Products – 2” puzzle.