A maths number puzzle

Question

We have these digits : $1,2,3,4,5,6,7,8,9$. Using these digits, we have to make two numbers such that their product must be as large as possible. We have to use all digits only once.

I did $97531\times 8642$ and got $842862902$

Answer 1

Note that if $a\gt b$ and $c\gt 0$ then $(10a+b)c\gt (10b+a)c$ - so we can organise the two numbers so that the largest digits are in the places of highest significance.

If the two numbers we are multiplying have $m$ and $n$ digits their product is less than $10^m\times 10^n=10^{m+n}=10^9$ so the product must have at most nine digits, and since nine digits is possible, the largest product will have nine digits.

Work down from the most significant digits.

$9\times 8=72$

Now we place $7$ and since $7\times 9\gt 7\times 8$ we choose $9\times 87$.

Next $6$ has to be allocated. It occupies a more significant place (second) if we have $96\times 87$ than $9\times 876$ where it is third - check $8352\gt 7884$.

Now $5$ goes in the most significant place it can, and since both are equal, it goes where it is multiplied by the highest number i.e. $96\times 875$

Continuing we get $9642\times87531=843,973,902$

@imakesmalltalk got there first, but this explanation was too long for a comment.

Answer 2

I got a bigger one : $87531\times 9642=843973902$

What I did was start with two blanks. Then I started putting the greater digits one by one, keeping the sum of digits of both the number somewhat balanced. The greater digits are given more preference, by putting them at a greater place value.

Answer 3

I did $963×875421 = 843030423 > 97531×8642 = 842862902$.

Answer 4

$87531 \cdot 9642 = 843973902$

I could confirm imakesmalltalk's answer by a JavaScript program.

Install node.js, execute node the-saved-script-name.js and the last line from the program's output should be the largest possible product.

The program should run reasonably fast, it needs 3.936 seconds on my PC.

/* Create all given permutations of an array
 * @author Oriol <http://stackoverflow.com/users/1529630/oriol
 * @link http://stackoverflow.com/a/11509565/603003
 * @license CC BY-SA 3.0 with attribution required
 */
function permute(input) {
    var permArr = [],
    usedChars = [];
    function main(){
        var i, ch;
        for (i = 0; i < input.length; i++) {
            ch = input.splice(i, 1)[0];
            usedChars.push(ch);
            if (input.length == 0) {
                permArr.push(usedChars.slice());
            }
            main();
            input.splice(i, 0, ch);
            usedChars.pop();
        }
        return permArr;
    }
    return main();
}

var numbers = [1,2,3,4,5,6,7,8,9];
var maxProduct = 0;

permute(numbers).forEach(function (permutation) {
    for (var i=1; i<9; i++) {
        var factor1 = parseInt(permutation.slice(0, i).join(""), 10);
        var factor2 = parseInt(permutation.slice(i, permutation.length).join(""), 10);
        var product = factor1 * factor2;
        if (product > maxProduct) {
            maxProduct = product;
            console.log("New maxProduct: " + maxProduct + " (factor1: " + factor1 + ", factor2: " + factor2 + ")");
        }
    }
});

Output:

...
the last line follows:
New maxProduct: 843973902 (factor1: 87531, factor2: 9642)