finding patterns - swaps

create a magic square using the first nine odd numbers: 1 , 3 , 5 , 7 , 9 , 11 , 13 , 15  and 17

then create a magic square using the first nine even numbers ( 2 up to 18 inclusive)

how is this easy after making one with the odd numbers?

magic square using the first nine odd numbers

















using the first nine even numbers

what happens when you half these?

in these magic squares, 2 of the numbers have been swapped over

without bothering about the middle number, students try to 'unswap' these magic squares

then they look at their squares, sort out what the central number will be and find patterns (there are many)

then they can try to create a different magic square (not a refection or rotation of these) with 10 in the middle:

they can do this by adjusting from the middle number moving outwards along both of the diagonals

what happens if a = 2b?

how can you avoid involving negative numbers?