The highest possible number by multiply three 3-digit numbers, no repeat numbers allowed.

Question

The highest I got was 611 721 516. Solution: 941 × 852 × 763.

A) Is there anyway to prove that this is the best answer?

First the obvious:

• The first number of each of the 3- digits must be either: 9, 8, 7.

  The second number: 6, 5, 4

  And third number: 3, 2, 1.

Given that this must be the case, the next challenge is in how to arrange these numbers.

My reasoning:

• Getting the numbers as close to each other as possible, meaning that if you subtracted the numbers, you would get a small value.

  Illustrated with this simple example: 4 x 4 = 16. Which is higher than 5 x 3. And higher than 2 x 6. 4 - 4 = 0

  Yet another example: 91 x 65 is higher than 95 x 61. This I reason is because 91 - 65 = 26 is less than 95 - 61 = 34.

B) Is this a correct observation?

C) Anyway to think about this more mathematically?

Answer 1

For six distinct digits, we want to maximize the following

$$ \left(a\cdot 10^{2}+b\cdot 10^{1}+c\right)\times\left(d\cdot 10^{2}+e\cdot 10^{1}+f\right) $$

Let’s examine the terms containing $b$ and $e$

$$ be\cdot 10^{2}+b\cdot 10^{1}\left(d\cdot 10^{2}+f\right)+e\cdot 10^{1}\left(a\cdot 10^{2}+c\right) $$

Greedy theorem: multiply big number with big number and small number with small number. If $a>d$ then we select $e>b$ to maximize. Similarly we can show that we need $f>c$ to maximize. Therefore to maximize $ABC\times DEF\times GHI$ we arrange the digits such that:

$$ A>D>G\\ B<E<H\\ C<F<I $$

Answer 2

If you decide that your three numbers start with 9,8 and 7, you want to increase each by the largest percentage possible. If you add 63 to 700 instead of 900, you achieve a higher increase in percent.

That’s not very mathematical but could have led you to your result. To prove that your result is optimal: Each number must be three digits in descending order. Then you try systematically. One number starts with 9. The eight goes with the 9 or on its own, giving less than 987 x 765 x 654, too small. So the 8 is the largest digit of a number. 7 goes with 9 (at most 976 x 865 x 654, too small) or with 8 (at most 965 x 876 x 654, too small) so it goes on its own. Then you try the three numbers 6 could go with and so on.