How do they figure out how many possible Sudoku combinations there are?

September 16th, 2008

I've heard there trillions, but how is that figured?
Is it 81^81?

The number of ways 9 numbers can be arranged in one 3×3 square is: 9P9 or 362880.

If you consider the three large squares (each containing 9 numbers) on a diagonal, the arrangement of numbers in each of them is independent of the arrangement of numbers in the other two as they share no common rows or columns.

Therefore the number of ways those three large squares can have their numbers arranged is:
362880 × 362880 × 362880 = 47,784,725,839,872,000

This is 47784 trillion or 4.7784×10^16 and that's just those three squares - so there's more combinations than that. The rest of the calculations get rather complicated, but that's a start for you.

So yes, there are trillions - but the number is definitely not 81^81 = 3.87×10^154

Even if all 9 large squares were independent, the number of arrangements would be: 362880^9 which is 1.09×10^50. So the answer must be less than this.

As a complete guess, I'd say the total number of options is around 10^22 (a million arrangements for each option above).

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One Response to “How do they figure out how many possible Sudoku combinations there are?”

  1. Comment by Valithor
    January 6th, 2008

    The number of ways 9 numbers can be arranged in one 3×3 square is: 9P9 or 362880.

    If you consider the three large squares (each containing 9 numbers) on a diagonal, the arrangement of numbers in each of them is independent of the arrangement of numbers in the other two as they share no common rows or columns.

    Therefore the number of ways those three large squares can have their numbers arranged is:
    362880 × 362880 × 362880 = 47,784,725,839,872,000

    This is 47784 trillion or 4.7784×10^16 and that's just those three squares - so there's more combinations than that. The rest of the calculations get rather complicated, but that's a start for you.

    So yes, there are trillions - but the number is definitely not 81^81 = 3.87×10^154

    Even if all 9 large squares were independent, the number of arrangements would be: 362880^9 which is 1.09×10^50. So the answer must be less than this.

    As a complete guess, I'd say the total number of options is around 10^22 (a million arrangements for each option above).
    References :