Maximum product

[tuanl]

You have nine digits 1,2,..9. Make two numbers using these digits exactly once such that product of the two numbers is maximum. Eg. 4321 x 95768.

My guess was 97531*8642, but it's not correct. I haven't had any ideas to do this systematically yet

 

[SirWankAlot]

I still don't quite understand it but the answer seems to be available.

Spoiler

http://puzzle.dse.nl/math/big_numbers_us.html

 

[tuanl]

I still don't quite understand it but the answer seems to be available.

Spoiler

http://puzzle.dse.nl/math/big_numbers_us.html

Thanks a lot for the link! Beautiful solution indeed. The key insight of this solution: "If you have two pairs of numbers with equal sums, the pair of which the numbers have the smallest absolute difference, is the one of which the numbers have the largest product" comes from the identity: (a-b)^2=(a+b)^2-4ab.

 

[Kangxi]

The answer should be 97521*8643. Assume only 1, 2, 3 to be assigned, we know that the maximum is 21*3. If 1, 2, 3, 4, 5 available, we know it should be 521*43, add 5 to 21 and add 4 to 3, say, add a largest one to more digital number and add a second largest to the smaller digital number. Therefore, for the case of 1, 2, 3, 4, 5, 6, 7, we should have 7521 * 643. And hence for 1, 2, 3, 4, 5, 6, 7, 8, 9, we should have 97521 * 8643.