You forgot to mention this important bit of the story:

Arthur, initially dumbfounded by the senile babbling of Merlin, sought help from Merlin's colleague from a distant land named Oz. Below is an extract of the reply from the other wizard:

Merlin wrote:Rotating the puzzle I could restore the givens using 4 nontrivial and 3 trivial digit pair swaps.

This is the original puzzle and its 180-degree rotation:

- Code: Select all
`........3`

..1..56..

.9..4..7.

.....9.5.

7.......8

.5.4.2...

.8..2..9.

..35..1..

6........

........6

..1..53..

.9..2..8.

...2.4.5.

8.......7

.5.9.....

.7..4..9.

..65..1..

3........

To restore the rotated grid to the original puzzle, we need 2 different stages of operations:

Stage 1 - all blocks except b5:

3 trivial digit swaps: 1-1, 5-5, 9-9.

3 non-trivial digit swaps: 2-4, 3-6, 7-8.

Stage 2 - b5:

1 non-trivial digit swap: 4-9.

1 clue movement: the 2-clue @ r4c4 is moved to r6c6. (Merlin must have forgotten about this important bit.)

So the 4 non-trivial digit swaps Merlin talked about were 2-4, 3-6, 7-8, 4-9, and his 3 trivial digit swaps were 1-1, 5-5, 9-9.

Merlin wrote:Outside the centre cell just one of them was valid for the completed grid as well.

At this stage we can place a hidden single of 5 @ r5c5.

According to Merlin one of the 7 digit swaps above (except the one involving the centre cell, i.e. 5-5) must also be applied globally in the solution grid (i.e. all cells involving those digit(s) must be 180-degree symmetrical to each other).

Since r5c5=5, 1-1 & 9-9 can't be possibly applied globally (considering b5).

Also, r37c5=[42] & r4c6+r6c4=[94], so 2-4 & 4-9 can't be global swapping pairs.

This leaves us only 2 possible swaps: 3-6 & 7-8. One of these two must be a global 180-degree symmetrical swapping pair in the final solution grid.

Merlin wrote:For row 2 just one more of the ones I could see was true leaving me 4 digits to account for.

For this row these 4 digits also paired up on nontrivially on rotation and their pairings weren't observable in the givens.

Now we look at r28: from the given clues, 1-1 & 5-5 are already established as local 180-degree symmetrical swapping pairs in these 2 rows.

Also, the global swapping pair (3-6 or 7-8, we don't know which one yet) must also be applied in these 2 rows.

This leaves us 5 unaccounted digits. But Merlin told us that there was one more local swapping pair leaving him 4 unaccounted digits, so this swapping pair must only involve 1 digit, i.e. the remaining trivial swap, 9-9.

The 4 unaccounted digits must pair up in manners different to the 4 non-trivial swaps specified above (2-4, 3-6, 7-8, 4-9).

However, since r2c7+r8c3=[63], we know 3 & 6 are not one of these 4 digits, so 3-6 is the global swapping pair specified above.

And the 4 unaccounted digits must be {2,4,7,8}, and their pairings must be [2-7|4-8] or [2-8|4-7].

With these bits of information in mind, we can set out to solve the puzzle:

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`+-------------------------+-------------------------+-------------------------+`

| 2458 2467 245678 | 126789 16789 1678 | 24589 1248 3 |

| 2348 2347 1 | 23789 3789 5 | 6 248 249 |

| 2358 9 2568 | 12368 4 1368 | 258 7 125 |

+-------------------------+-------------------------+-------------------------+

| 12348 12346 2468 | 13678 13678 9 | 2347 5 12467 |

| 7 12346 2469 | 136 5 136 | 2349 12346 8 |

| 1389 5 689 | 4 13678 2 | 379 136 1679 |

+-------------------------+-------------------------+-------------------------+

| 145 8 457 | 1367 2 13467 | 3457 9 4567 |

| 249 247 3 | 5 6789 4678 | 1 2468 2467 |

| 6 1247 24579 | 13789 13789 13478 | 234578 2348 2457 |

+-------------------------+-------------------------+-------------------------+

We know 9-9 must be a local swap in r28, and the only possible way is r2c9=r8c1=9.

Also, since 7-8 can't be a swap in r28, r28c5 can't be {78}, must be [36].

Since 3-6 is a global swapping pair, it must also be applied in b5.

The only possible way is r5c46={36} (naked pair @ r5,b5).

After 10 hidden singles and a box-line intersection of 8:

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`+----------------------+----------------------+----------------------+`

| 245 6 24578 | 12789 1789 178 | 2458 1248 3 |

| 24 247 1 | 278 3 5 | 6 248 9 |

| 3 9 258 | 1268 4 168 | 258 7 125 |

+----------------------+----------------------+----------------------+

| 1248 3 6 | 178 178 9 | 247 5 1247 |

| 7 124 24 | 36 5 36 | 9 124 8 |

| 18 5 9 | 4 178 2 | 3 6 17 |

+----------------------+----------------------+----------------------+

| 145 8 457 | 137 2 1347 | 457 9 6 |

| 9 247 3 | 5 6 478 | 1 248 247 |

| 6 1247 2457 | 1789 1789 1478 | 24578 3 2457 |

+----------------------+----------------------+----------------------+

We know the remaining 8 cells of r28 (r2c1248+r8c2689) must have 4 swapping pairs with the pairings [2-7|4-8] or [2-8|4-7].

Now r2c1 is from {24}, so r8c9 can't be from {24}, must be 7.

Now r8c2 is from {24}, so r2c8 can't be from {24}, must be 8.

Now r8c8 is from {24}, so r2c2 can't be from {24}, must be 7.

Thus r2c14=[42], and we can confirm the pairings are [2-8|4-7].

Hence r8c268=[284].

All naked singles from here.