Let's discuss a problem from the AMC 2021 Middle Primary Category: Problem 26 which revolves around basic algebra.

Question

This is a magic square, so that all rows, columns and diagonals add up to the same sum. Some numbers are already filled in.

When we complete it and multiply the numbers in the three shaded squares, what do we get?

Solution

We know that the sum of the each row, columns and the diagonals add to the same sum.

From the picture we can see apart from of the diagonals all the rows and culomns are empty in two boxes. If we add the numbers diagonally we get: $16 + 10 + 7 + 1 = 34$. Thus all the rows, columns and the diagonals will add to $34$.

Thus we check the $4th$ column we get, $x + 8 + 12 + 1 = 34$.

So, $x = 34 - (8+12+1) = 34 - 21 = 13$

If $x = 13$ then $a = 34 - (16 + 2 + 13) = 3$.

If $a = 3$ then $b = 34 - (3 + 10 + 15) = 6$.

If $b = 6$ then $z = 34 - (6 + 7 + 12) = 9$.

If $z = 9$ then $d = 34 - (16 + 9 + 4) = 5$.

If $d = 5$ then $c = 34 - (5 + 10 + 8) = 11$.

Thus the product of $a \times b \times c = 3 \times 6 \times 11 = 198$

What is AMC (Australian Mathematics Competition)?

The Australian Mathematics Competition (AMC) is one of Australia's largest and oldest annual mathematics competitions, aimed at fostering interest and excellence in mathematics among students.

Watch the video to get the refernce book idea.