Directions: Using the digits 1 to 9 at most one time each, fill in the boxes to make the smallest (or largest) product.

### Hint

What number does each box represent?

### Answer

SS found the current smallest product with positive factors: 1.35 * 2.46. Kevin Crowl suggested 9.75 * 8.64 which led to 9.64 * 8.75 which is currently the largest product with positive factors. We are still looking for answers. If you have one, post it in the comments.

Note: This problem’s difficulty can be adjusted by altering the number of digits (boxes), picking smallest or largest, or by picking either a positive, negative, or both.

Source: Robert Kaplinsky

1.35 * 2.46 ?? smallest for both positive??

Thank you for that solution, it has been added to the problem.

I think largest both positive is 9.75 * 8.64

Thanks Kevin. Challenging, right?! Your solution reminded me how this is similar to another problem. https://www.openmiddle.com/rectangles-maximizing-area/. In this problem, you get the largest area when the side lengths are as close to the same amount as possible. Using that reasoning, I realized that 9.64 * 8.75 is a bit closer together and gives 84.35 as an answer as opposed to 84.24.

Might still be even larger ones!

As I was checking through my students’ work the other day, one of them had also found 9.64*8.75 as the largest possible. I appreciate the confirmation, and my whole class was very excited to hear that one of them had done “better” than me!

Excellent. That kid will remember that for the rest of his life. Those are good moments to have.

Wouldn’t the smallest value then be +9.64 * -8.75 ? Or -9.64 * +8.75 ? The negative value of the largest positive solution?

Does the +- in front of both confuse this problem?