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Why the centre of a 3 by 3 magic square is always 5

magic-squaresarithmeticproofclassic

The numbers 1 to 9 are arranged in a 3 by 3 grid so that every row, column and both diagonals add to the same total.

  1. What is that total?
  2. Prove that the centre square must contain 5.
  3. Which numbers can go in the corners?
Show a hint

Add up the four lines that pass through the centre: two diagonals, the middle row and the middle column. Every number is counted once, except the centre.

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Divisibility Tests and Digit SumsEstimating Square RootsApproximating Exponential GrowthFast Division and ReciprocalsFirst-Order Approximations in Your Head

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