[pnm]

Someone wrote to me that puzzle 126 of the new "Colossal Killer Sudoku" book was especially difficult.

I started solving it, and got a start in it pretty quickly. I'll describe the steps below, and am inviting other
puzzlers to add suggestions

Here's the puzzle:



Some initial steps:
- the sum of all cages in columns a, b, c, and d = 184, so e4 = 4
- the sum of all cages in rows 1, 2, and 3 = 146, so a4 + h4 = 11: doesn't help me much,
other than determine that a4 cannot be 1
- using similar steps, d6 + f6 = 12 (doesn't help me yet)
- the "17" cage at d8d9 must hold 8 and 9
- so the "11" cage at e7e8 must hold 5 and 6
- so the "17" cage at e1e2e3 must hold 1, 7, and 9
- so e5e6 must have 3 and 8, so f5 = 1
- so e9 = 2 and f9 = 3

After these steps (more or less):

[pnm]

Some further steps:

- from the numbers in the centre 3x3 box we see that d4d5 have to hold 2 and 6 (in some order)
- so f4 = 9
- so f1f2f3 have to hold 2,6,8 (in some order)
- so d1d2d3 have to hold 3,4,5 (in some order)
- so d1 = 5 (otherwise we can't make 8 in the cage c3d2d3)
- so c3 = 1
- so b2b3 hold 2,5 (in some order)(not 3,4 because then the puzzle solution
would no longer be unique)
- the cages in the top left add up to: 19 + 16 + 7 + 8 + 10 = 60
extending from there are d1d2d3 and a4. The sum of the numbers in d1d2d3: 12
so a4 = 60 - 45 - 12 = 3
- so a3 = 7
- from adding all cages in the top 3 boxes and a4 and h4 we find that h4 = 8
- so h3 + i3 = 8, so they hold 2,6 or 3,5
- so i1i2 holds 1,4 (and not 2,3)

The situation after these steps:

[nailhouse]

...because then the puzzle solution would no longer be unique...

What do you mean with this?

[nicow]

...because then the puzzle solution would no longer be unique...

What do you mean with this?

This is a fundamental strategy for sudoku, called 'unique rectangles'.
At this moment d2d3 holds {3,4} so, if b2b3 also should hold {3,4} two solutions would be found, because the four cells form a rectangle.
But trust the developer: only one solution exists.

[nailhouse]

Thank you for the explanation

[nailhouse]

It helped me today, solving the 'difficult' 8x8 calcudoku.

[marblevolcano]

b2b3 hold 2,5 (in some order)(not 3,4 because then the puzzle solution
would no longer be unique)

Same as nailhouse - I've used this a few times lately in the Holiday book.