Welcome to our community

Be a part of something great, join today!

Find the smallest possible value of abc + def + ghi

  • Thread starter
  • Admin
  • #1

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,755
Let a, b, c, d, e, f, g, h, i be a permutation of (1, 2, 3, 4, 5, 6, 7, 8, 9).

Find the smallest possible value of abc + def + ghi.
 

Bacterius

Well-known member
MHB Math Helper
Jan 26, 2012
644
Let a, b, c, d, e, f, g, h, i be a permutation of (1, 2, 3, 4, 5, 6, 7, 8, 9).

Find the smallest possible value of abc + def + ghi.
Is that a product or concatenating digits? E.g. if a = 2, b = 3, c = 5 is abc equal to 30 or 235? Just making sure.
 
  • Thread starter
  • Admin
  • #3

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,755
Is that a product or concatenating digits? E.g. if a = 2, b = 3, c = 5 is abc equal to 30 or 235? Just making sure.
I'm sorry I wasn't clear on that part...:eek:

abc, def and ghi are all products of three numbers.
 

Bacterius

Well-known member
MHB Math Helper
Jan 26, 2012
644
I'm sorry I wasn't clear on that part...:eek:

abc, def and ghi are all products of three numbers.
Well normally it would be clear you meant a product but since the variables were between 1 and 9 it was at least conceivable you could have meant something else. All good now though, and an interesting puzzle :p
 

chisigma

Well-known member
Feb 13, 2012
1,704
Let a, b, c, d, e, f, g, h, i be a permutation of (1, 2, 3, 4, 5, 6, 7, 8, 9).

Find the smallest possible value of abc + def + ghi.
Empirically it seems to be $8 \cdot 9 \cdot 1 + 6 \cdot 7 \cdot 2 + 3 \cdot 4 \cdot 5 = 216$... but of course we have to demonstrate that any other sum is greater than 216...

Kind regards

$\chi$ $\sigma$
 
  • Thread starter
  • Admin
  • #6

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,755
Empirically it seems to be $8 \cdot 9 \cdot 1 + 6 \cdot 7 \cdot 2 + 3 \cdot 4 \cdot 5 = 216$... but of course we have to demonstrate that any other sum is greater than 216...

Kind regards

$\chi$ $\sigma$
Hmm...the smallest value that I've gotten is 214...
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
  • Thread starter
  • Admin
  • #8

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,755
Let a, b, c, d, e, f, g, h, i be a permutation of (1, 2, 3, 4, 5, 6, 7, 8, 9).

Find the smallest possible value of abc + def + ghi.
I should have said the answer that I've found by taking a peek into the work of other is 214 and that I didn't solve this question on my own.

In the solution that I'm referring to, the idea of AM-GM inequality was used to get 214 as the lowest possible value for abc+def+ghi.

For starters, AM-GM inequality states that

\(\displaystyle \frac{abc+def+ghi}{3} \ge \sqrt[3]{abc\cdot def \cdot def}\)

\(\displaystyle abc+def+ghi \ge 3\cdot\sqrt[3]{9!}\)

\(\displaystyle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ge 213.98\)

\(\displaystyle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge 214\)

He then guessed the minimum of the sum of abc, def and ghi is achieved when they're near 70 and 72 because \(\displaystyle \frac{214}{3}\approx 71.333\).

By some guesswork we see that

\(\displaystyle abc=9\cdot8\cdot1=72\)

\(\displaystyle def=3\cdot4\cdot6=72\)

\(\displaystyle gjo=2\cdot5\cdot7=70\)

Hence, the smallest possible value of abc+def+ghi=72+72+70=214, which agrees with the result that we found from AM-GM inequality.