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Multiply the Addition Square

Age 11 to 14
Challenge Level

Take an addition table from $1$ to $10$ (or any other that you like better!):

addition square from 1 to 10
Now take a $3\times 3$ square of numbers on it, such as this red one:
addition square with two three by three squares highlighted
Now multiply the two diagonally opposite corner numbers together:
three by three square with top row 5, 6, 7
And then find the difference between the two answers:
$5 \times 9 = 45$
$7 \times 7 = 49$
$49 - 45 = 4$
Now try the numbers in the green square:
three by three square with 8, 9, 10 on top row
What is the answer?
Investigate other $3\times 3$ squares.
What do you notice?
Can you explain it?
Now try with $2\times 2$ squares and $4\times 4$ squares.
Can you predict your answers? What is happening?