### Brain Teasers

# Highest Value

Using the digits 1 to 9 only once each, create three 3 digit numbers. Your goal is to create three numbers which, when multiplied together, give the highest value.

### Answer

763 x 852 x 941 = 611721516Hide Answer Show Answer

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## Comments

this one is kinda hard. the first person 2 read this comment then put ur own up... this one needs more comments up. by the way, if the person who made this teaser reads this then please put up if u got this from a book or in ur head... if u got it from a book then please make a comment and tell me the name of it

Apr 17, 2003

Um...good...but....i dno.

Apr 18, 2003

I really enjoyed this teaser. My first instinct was that 963 x 852 x 741 would produce the highest value. It wasn't difficult to see that I was wrong, but I can't find a simple, logical explanation for the solution. Perhaps one of the math whizzes that voted this teaser 'easy' could take some time to enlighten me? *feels dumb*

A fair question, Anonymous. I trust it's

not too hard to see that you want 7, 8, and 9

to be the hundreds, 4, 5, and 6 to be the

tens, and 1, 2, and 3 to be the ones, in some

order. The question then is what is the best

order? If you group 6 with 9 and 4 with 7, then

you are multiplying 60 by (700+800) and 40 by (800+900);

if you reverse them, it goes the other way.

Essentially, the question is would you rather

have 6*15+4*17 or 4*15+6*17? By grouping the

6 with the 7, you get two extra "17"s at the

expense of two "15"s, so this gives you the

biggest total. Similar considerations apply

to the location of the other digits. Basically,

you want to group the biggest tens and ones digits

with the smallest hundreds digit, because this gives

you the biggest bang for the buck when you

multiply them by the bigger hundred's digits.

I thought this was fairly easy, but it is

certainly not trivial to *prove* that the

answer is correct, and there is no reason

to feel dumb about not getting it immediately.

not too hard to see that you want 7, 8, and 9

to be the hundreds, 4, 5, and 6 to be the

tens, and 1, 2, and 3 to be the ones, in some

order. The question then is what is the best

order? If you group 6 with 9 and 4 with 7, then

you are multiplying 60 by (700+800) and 40 by (800+900);

if you reverse them, it goes the other way.

Essentially, the question is would you rather

have 6*15+4*17 or 4*15+6*17? By grouping the

6 with the 7, you get two extra "17"s at the

expense of two "15"s, so this gives you the

biggest total. Similar considerations apply

to the location of the other digits. Basically,

you want to group the biggest tens and ones digits

with the smallest hundreds digit, because this gives

you the biggest bang for the buck when you

multiply them by the bigger hundred's digits.

I thought this was fairly easy, but it is

certainly not trivial to *prove* that the

answer is correct, and there is no reason

to feel dumb about not getting it immediately.

It is not that hard, to solve such riddles: First digits:7, 8, 9-second:4,5,6-third:1,2,3. First we get 7--, 8--, 9--.Now put the lowest and highest number as close together as possible: 76- and 94-. The 85- comes automatically. The same procedure for the third digit:763, 941 and automatically the 852. Between 763 and 941 is the lowest possible difference. To get the lowest product of the 9 digits, you must do the opposite of the procedure: 1--, 2--, 3--. Put the 4, 5, 6 so that you get the highest difference between the lowest and highest number: 14- and 36- (middle number 25-). And for the third digit: 147 and 369 (middle number 25. Now we have the highest possible difference between the lowest and highest number. 147 x 258 x 369 = 13.994.694

Very easy.

It's obvious that the 7,8,9 are the hundreds, 4,5,6 the tens and 1,2,3 the ones. All you need to know beyond that is that 3x5 is less than 4x4 to realize that you want to assign the digits in a way to make the numbers as close as possible. This same principle also tells you that you want three 3-digit numbers.

It's obvious that the 7,8,9 are the hundreds, 4,5,6 the tens and 1,2,3 the ones. All you need to know beyond that is that 3x5 is less than 4x4 to realize that you want to assign the digits in a way to make the numbers as close as possible. This same principle also tells you that you want three 3-digit numbers.

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