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- Feb 14, 2012

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

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- Thread starter
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- Feb 14, 2012

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

- Jan 26, 2012

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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.

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

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- Feb 14, 2012

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I'm sorry I wasn't clear on that part...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.

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

- Jan 26, 2012

- 644

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 puzzleI'm sorry I wasn't clear on that part...

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

- Feb 13, 2012

- 1,704

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...

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

Kind regards

$\chi$ $\sigma$

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- Feb 14, 2012

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Hmm...the smallest value that I've gotten is 214...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

$\chi$ $\sigma$

- Mar 10, 2012

- 835

If the natural generalization of the rearrangement inequality holds (see Rearrangement inequality - Wikipedia, the free encyclopedia) then the answer seems to be 216.

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

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

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- Feb 14, 2012

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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.

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

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.