# Next Sudoku Number

Here is a sudoku puzzle coming from this magazine

I found that B5 and B6 can only contain 6 and 7. 8 cannot be in B5 because D1 would be a 8 and neither B3 not C1 could be a 8.

I thought it would help me to find out the next number, but I'm still stuck at this step.

What is the next step?

• Welcome to Puzzling, take our tour! Could you please provide proper attribution for this question?
– Jafe
Commented Aug 2, 2023 at 23:17

## 1 Answer

This is a formidable puzzle. Fortunately OP's request is to prove the next digit. The grid with all candidates is needed for this. I use chess notation: rows 1 to 9 (bottom to top) and columns a to i (left to right).

.---------------------------------------------------------------------.
| 567    2      356    | 4579   1      4567   | 37      379    8      | 9
| 9      1367   368    | 278    678    267    | 1237    5      4      | 8
| 1578   157    4      | 25789  578    3      | 6       1279   19     | 7
|----------------------+----------------------+-----------------------|
| 5678   567    2      | 1      567    9      | 4       3678   356    | 5
| 3      5679   568    | 457    4567   4567   | 1578    16789  2      | 5
| 4567   45679  1      | 3      2      8      | 57      679    569    | 4
|----------------------+----------------------+-----------------------|
| 145    1345   7      | 6      3458   145+2  | 9       12(3)8 1(3)5  | 3
| 1456   8      9      | 245    345    1245   | 12(3)5  12(3)6 7      | 2
| 2      1356   356    | 578    9      157    | 1(3)58  4      1(3)56 | 1
'---------------------------------------------------------------------'
a      b      c        d       e     f        g       h      i


Claim: Every possible position for the digit 3 at box ghi123 implies that h3 ≠ 2.

Proof:

. h3=3 => h3 ≠ 2
. g2=3 or h2=3 => e2 ≠ 3 => e3=3 => e3 ≠ 8 => h3=8
. g1=3 => g1 ≠ 8 => h3=8
. i1=3 or i3=3 => i6 ≠ 3 => h6=3 =>
h6 ≠ 8 => a6=8 => c5 ≠ 8 => c8=8
=> def8 has no 8 => (672) locked at def8
=> d7 ≠ 2 => h7=2


Having proved the claim, it follows that f3=2.