Sudoku Phantoms

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|| | | || 2 | 5 | || | 1 | ||

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|| | | 7 || | 8 | || 9 | | 3 ||

||---|---|---||---|---|---||---|---|---||

|| | | || | | || 6 | | ||

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|| | 3 | || | 1 | || | 4 | : ||

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|| | | 8 || 6 | | 9 || 5 | | || Phantom 8's are shown by :

||---|---|---||---|---|---||---|---|---|| in B8 and B6. Only one 8

|| | | || | 7 | || | 9 | : || will truly exist in each box.

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|| | | 3 || : | | : || | | ||

||---|---|---||---|---|---||---|---|---||

|| | | 4 || | 3 | || 8 | | ||

||---|---|---||---|---|---||---|---|---||

|| | 7 | || 9 | 6 | 1 || | | ||

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The puzzle above demonstrates the concept of "Phantoms".

My definition of phantoms is a row or column in a single box where a number MUST appear,

but you can't determine exactly where it goes. Let me illustrate with the puzzle above.

Notice the lone "8" in R8 of B9. It is the only 8 in the bottom Row-Boxes (B7,B8,B9).

B7 and B8 both need an 8. The "Sudoko Wall" in R9 of B8 means that 8 can only go in R7 of B8.

But you don't know where exactly. HOWEVER, that means the other 8 must go in R9 of B7.

That we can determine because of the 7 at (R9,C2), and the 8 at (R5,C3). So (R9,C1) = 8.

There's another just like it. Again, start with the "8" in B9. The "8" in R5 blocks the center

row of B6. The 8 at (R8,C7) blocks C7. So the only places left in B6 for 8 are in C9. But,

again we don't know exactly where. HOWEVER, that means the "8" in B3 must be in C8,

and there's only one place left for it: (R3,C8), because of the "8" in R2 and "1" in C8.

We still haven't determined where to place the 8's in B8 and B6, because they are "phantoms".

But now, with 8's in B7 and B3, we can find more, like (R1,C2) = 8.

Many times you'll discover a pair of numbers that MUST occur in a row or column.

Such pairs are called "Naked Pairs", and they signal that those two numbers CAN NOT

occurs anywhere else in that row or column. The requirement is that there are

no other numbers that can go in the pair of cells holding the naked pair. Also, if the

naked pair is in a single box, those numbers can't go anywhere else in that box.

Naked Pairs are another kind of phantom find, forcing the pair to be in other rows or

columns in companion boxes.

In the B6 box shown above, the number 6 can only be in the same two places as the number 8.

So those two places hold naked pairs consisting of 6 and 8. Find one, and you find the other.