2001 AMC 10 Problems/Problem 22
In the magic square shown, the sums of the numbers in each row, column, and diagonal are the same. Five of these numbers are represented by , , , , and . Find .
We know that , so we could find one variable rather than two.
The sum per row is .
Since we needed and we know , .
The magic sum is determined by the bottom row. .
Solving for :
To find our answer, we need to find . .
Really Easy Solution
A nice thing to know is that any numbers that go through the middle form an arithmetic sequence.
Using this, we know that , or because would be the average.
We also know that because is the average the magic sum would be , so we can also write the equation using the bottom row.
Solving for x in this system we get , so now using the arithmetic sequence knowledge we find that and .
Adding these we get .
Systems of Equations
Create an equation for every row, column, and diagonal. Let be the sum of the rows, columns, and diagonals. .
Notice that and both have . Equate them and you get that . Using that same strategy, we use instead. is good for our purposes. It turns out that . Since we already know those numbers, and , We can say that will be . We are now able to solve: , , , and . Respectively, , , , , and . We only require The sum of , which is . We get that the sum of and respectively is
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