Nine digits => two numbers => largest product

[bonanova]

Here are your digits: 1 2 3 4 5 6 7 8 9.

Using each of them exactly once,  form two numbers: for example, 314879 and 652.

What's the largest product two such numbers can have?

 

You don't need a computer

[phil1882]

in general the largest number produced by multiplication is achieved when the two numbers are close together.

so we want a 4 digit and a 5 digit number.

a b c d e 

* f g h i

----------

 

the largest first digit occurs when a and f are 9 and 8. since we want them to be close, its better for a to be 8 and f to be 9.

the second largest digit will be 2 + 9*b +8*g. the largest value occurs when b is 7 and g is 6. 

this puts the max value at 8300000 aprox. the next largest digit is 7*6 +9*c +8*h. the max value occurs when h is 4 and c is 5.

you can continue. the final answer is: 9642 *87531.

[wolfgang]

we make:

98765432 as first number
and 1 as second number
and we will get the largest product

[bonanova]

we make:

98765432 as first number

and 1 as second number

and we will get the largest product

 

98765431 x 2 = 197530862 is bigger.

[kukupai]

9 x 87654321

[bonanova]

9 x 87654321

 

91 x 87654321 is bigger

[kukupai]

maybe 911 x 87654321 then?

 

Anyway, back to work.

Edited August 3, 2014 by kukupai

[kukupai]

9652 x 87431

(and running out of ideas )

[bonanova]

Sooooo close. Swap a single digit from each number and you have it.

[kukupai]

9642 x 87531 ???

Edited August 4, 2014 by kukupai

[Quantum.Mechanic]

Toying with a script to spit out the answers quickly, it seems this is the general pattern:

 

If the digits are d_m..d_n (contiguous) the answer looks like:

 

    d _n d_n-3d_n-5.. . * d_n-1d_n-2d_n-4...

 

regardless of m and n. This works regardless of whether there is an even or odd number of digits.

[bonanova]

Toying with a script to spit out the answers quickly, it seems this is the general pattern:

 

If the digits are d_m..d_n (contiguous) the answer looks like:

 

    d _n d_n-3d_n-5.. . * d_n-1d_n-2d_n-4...

 

regardless of m and n. This works regardless of whether there is an even or odd number of digits.

 

Nice!