Patterned 9 x 9, 31 May 2016

The patterned 9 x 9 on Tuesday, 31 May 2016 seemed like too much of a challenge at first but then actually turned out to be a rewarding experience! I struggled for a time and then resorted to trial and error and went wrong somewhere, but later on I suddenly found the clues to a comparatively simple "analytical" solution. To celebrate, I'm now posting them for the benefit of other puzzlers who were similarly challenged

Screenshot note: I took the liberty of entering some numbers as a single "candidate" or "pencil" number

in the cells where they can go within their column, row or section. That's more or less self-explanatory, I hope.

As usual, multiple candidates in one cell are meant as an exhaustive list of the numbers that cell can contain.

The first clue and the second clue both refer to this screenshot.**First clue – "dressed to the 9s"**The first, trivial steps in solving the puzzle include noticing that the 18x cage in the rightmost column must contain [3,6] because of the pair [1,9] in the 9x cage. After that, closer inspection reveals that there are only four cages in the four lower rows that can contain the number 9, and that each of those cages can contain that number only once. This means that each of those cages (10+, 72x, 2- and 1080x) must contain 9 exactly once.

The set of numbers in the 2- cage (where the cell indicating the 2- requirement is highlighted in orange) must then be [1,6,9] or [2,5,9] or [3,4,9]. [2,5,9] can be immediately eliminated because there are already two 5s in the rows that contain the cage. [1,6,9] in the 2- cage means, together with [1,9] in the 10+ cage, that the 72x cage can't contain two 1s – as in [1,1,8,9] – so its combination must be [1,2,4,9]. The 1s in those three cages mean that the 40x cage can't contain a 1 and must have the combination [2,4,5]. But wait, that means that only five different numbers, [1,2,4,5,9], are allowed in the six leftmost cells of the fourth row from the bottom, which is of course impossible! With [1,6,9] thus also eliminated, only the set [3,4,9] remains. And that, my friends, leads to a whole lot of other steps

**Second clue – about 5s and 7s in the second and third columns**The 7 in the leftmost column must be in one of its two uppermost cells, that is, in the 3024x cage. And since 3024 only contains 7 as a factor once, the part of the cage that is in the second and third columns from left can't contain any 7s. This very much limits where the 7s, along with the 5s, can go in the second and third columns. In the third column they have to share the third and fourth cells from top between them, and in the second column they must both be in the 25+ cage. This means that the 25+ cage must contain either [5,5,7,8] or [5,6,7,7]. Looking at possible distributions of "single and double threes" (the numbers 3, 6 and 9) in the first three columns may give you a – well, a clue – to the right combination

**Third clue – placing powers of 2**Many of the steps that follow from the first and second clues are "sudoku-like" in that they don't require any calculation, just attention to where a given number can go within its row or column. They lead to an "endgame" where the remaining cells need to be filled in with mostly 2s, 4s and 8s. The 960x cage in the upper right corner must contain [1,4,5,6,8], of which the 8 can only be placed in the highlighted cell, and the rest is "sudoku". Of course this clue isn't as important as the first and second ones since it merely illustrates a rather trivial step, but I included it nonetheless because I initially overlooked some of the trivial steps that led to the stage shown in the screenshot, which meant I still needed to figure out the arrangement of

*all* the powers of 2, including 2^0 = 1, in the 960x cage. I did manage to do that, using a rule that often comes in handy when dealing with difficult puzzles, but I won't go into that here since it actually wasn't necessary. Rather than being a needed clue, the screenshot illustrates a fond memory

And

using Imgur is fun and easy! Hope you enjoyed this, thanks for reading