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

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The discussion focuses on finding the smallest possible value of the expression abc + def + ghi, where a, b, c, d, e, f, g, h, i are permutations of the digits 1 through 9. Participants clarify that abc, def, and ghi represent products of three numbers rather than concatenated digits. Through empirical testing and application of the AM-GM inequality, the consensus emerges that the minimum value achievable is 214. This value is derived from specific groupings of the digits, with abc equating to 72, def also to 72, and ghi to 70. The findings confirm that 214 is the lowest possible sum for the given expression.
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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.
 
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anemone said:
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.
 
Bacterius said:
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...:o

abc, def and ghi are all products of three numbers.
 
anemone said:
I'm sorry I wasn't clear on that part...:o

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
 
anemone said:
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

$\chi$ $\sigma$
 
chisigma said:
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...
 
anemone said:
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.
If the natural generalization of the rearrangement inequality holds (see Rearrangement inequality - Wikipedia, the free encyclopedia) then the answer seems to be 216.

EDIT: I made a mistake. Please ignore this post.
 
anemone said:
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

$$\frac{abc+def+ghi}{3} \ge \sqrt[3]{abc\cdot def \cdot def}$$

$$ abc+def+ghi \ge 3\cdot\sqrt[3]{9!}$$

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ge 213.98$$

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge 214$$

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

By some guesswork we see that

$$abc=9\cdot8\cdot1=72$$

$$def=3\cdot4\cdot6=72$$

$$gjo=2\cdot5\cdot7=70$$

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.
 
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