Tommy and cards
There are 8 cards. Tommy wrote on them each of the following numbers: { -1, 3, 4, -5, 7, -9, -10, 11}. Then he flipped them down and shuffled. After that he writes on the other side of each card each of the following numbers: { -1, 3, 4, -5, 7, -9, -10, 11}. After all Tommy add numbers on each card and multiply all 8 sums that he got.
What is the smallest non-negative number he could've got.
The answer is 16.
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1 solution
If the numbers on the cards are in order:
Then the product will be ( 3 − 1 ) 2 ∗ ( 4 − 5 ) 2 ∗ ( 7 − 9 ) 2 ∗ ( 1 1 − 1 0 ) 2 = 4 ∗ 1 ∗ 4 ∗ 1 = 1 6
If we try to change the order of numbers, each sum will be greater in absolute value since its components will be further from each other or will have the same sign.
P.S I don't think it counts as strict mathematical proof. So hope you can strictly justify your answer:)