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'The Mathemagician's Seven Spells' printed from http://nrich.maths.org/

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The great spell has to be broken! It has seven parts. Two children, Anna and David, must break it, but you can help...
The Mathemagician stands by his immense cauldron, flames dancing round. He stares hard at the cauldron and then at the two children.
mathemagician and his cauldron
"Tell me the next two numbers in each of these seven minor spells," chanted the Mathemagician, "and the great spell will crumble away! Watch the smoke to see the spells."
Spell the first: sequence: 1, 2, 4, 8, ?, ?
Spell the second: sequence: 6, 13, 20, 27, ?, ?
Spell the third: sequence: 127, 63, 31, 15, ?, ?
Spell the fourth: sequence: 1, 3, 6, 10, 15, ?, ?
Spell the fifth: sequence: 2, 3, 6, 11, 18, ?, ?
Spell the sixth: sequence: 1, 8, 27, 64, ?, ?
Spell the seventh: sequence: 216, 168, 126, 90, 60, 36, ?, ?
Anna and David managed to break the spell, did you?