# Puzzle and Game: Maximum Product

Puzzle:

Concatenate the digits: 4, 5, 6, 7, 8, and 9 and create numbers with as high a product as possible. Each digit must be used exactly once. (Why not create the same puzzle, but with maximum sum?? Well…)

[Subpuzzle: assume that a candidate is a pair of numbers of, with the digits in each number strictly descending from left to right. How many candidates of lengths 2 and 4 are there? Of lengths 3 and 3?]

Game:

Each player sets up 6 blanks: __ __ × __ __ __ __

A non-player generates a series of random integers, 0 – 9, and calls them out, one at a time. As each number is called, each player must place it into one of the remaining blanks. Maximum product wins.

This is similar to a game that Dave Marain posted at Math Notations last month. I have seen lots of variations; you could make up your own (quotient of a 4 digit number and a 2 digit number works. But so do lots of others.)

Do you want to exclude trivial “products” of one number? Otherwise it’s trivial for the same reason as maximum sum, no?

I may be missing something, I seem to be quite stupid at the moment.

Nice catch. Let’s exclude 987654.

A previous version said …create two numbers… Maybe I should have left that in?

First I figured that (obviously) large digits should be in higher place value places….

Then the question seemed to be did you want to make the numbers close to equal, or not. My guess was close to equal, but I figured I look at the opposite possibility.

Three (to me) most plausible guesses:

964*875 = 843,500

98764*5 = 493,820

97654*8 = 781,232

So I’m going with 964*875 = 843,500

That’s what I get, but I’d like to have a better grip on the placement of the 4, 5, 6, and 7.

My logic was…

— The 9 and the 8 go in the hundreds column

— The 7 and the 6 go in the tens column, placed so as to make the two numbers as close to equal as possible. So 96_ and 87_ rather than 97_ and 86_.

— The 5 and the 4 go in the ones column, again placed so as to make the two numbers as close to equal as possible.

One more thought — the between 964, 875 and 975, 864 is set by knowing that for a fixed sum, the product is maximized when the two numbers are as close equal as possible, since both pairs sum to 1839.

I find getting a grip on why 964*875 not 97654*8 question harder…

The kids loved the game, and did not want to be dismissed…

Thank you for the interesting game.