Find the smallest possible value of abc + def + ghi

anemone

MHB POTW Director
Staff member
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
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.

anemone

MHB POTW Director
Staff member
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...

Bacterius

Well-known member
MHB Math Helper
I'm sorry I wasn't clear on that part...

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

chisigma

Well-known member
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

$\chi$ $\sigma$

anemone

MHB POTW Director
Staff member
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...

MHB Math Scholar

anemone

MHB POTW Director
Staff member
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.