# PSE Advent Calendar 2021 (Day 20): Candy Cane Sudoku

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2021. The accepted answer to this question will be awarded a bounty worth 50 reputation.

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So I was planning on doing a holiday themed Sudoku: I had my grid laid out, and I was cleaning off the table. But my wife said it was time to walk the dogs, and they went crazy like they normally do; Nutmeg ran into me and I dropped a basket of candy canes right on the grid, with some of them breaking in the process:

When we got back I went to clean up, but fortunately, I looked first...there was my puzzle! The puzzle is a classic Sudoku, with Renban lines (purple) and German whisper lines (green). Renban lines must contain a set of consecutive digits, though the digits may be placed in any order on the line. German whisper lines must have the difference between two successive cells on the line be at least 5. I hope you enjoy!

### Solver Notes

PLEASE, if you are providing a solution, be prepared to present the key logic points, ideally with diagrams of partial progress. And no computers, please.

• That's fun. I thought I was on the way too finish, and I reached a self-contradiction. Will be hard to find by solving mistake now... Dec 20, 2021 at 14:10

Before we start solving, there's a few logical deductions we can make at the very start:

- For Green, a 6 or a 4 can only ever be on the ends (assuming the two adjacent cells 'see' each other), or the two adjacent cells will have the same value
- Green can also never contain a 5 as no number is 5+ away in sudoku

So with this basic information, we can start trying to narrow down the candidates, and hopefully placing some numbers.

For all images, click/open link for a larger, clearer version - otherwise images are very large

### 1. Placing candidates

The obvious first step is a 9 must be next to the 4. After this, there are no obvious numbers to place, but the majority of the green lines only can have certain candidates in each cell, or other cells along the line will be impossible to fill. Not too much to show here.

### 2. A breakthrough!

Consider the 7th column. The 8 in this column can only go in the 2 highlighted cells, as it can not be part of the purple line. Now looking at the green line, with all 8s removed, the top of the green must be 2/3 as a 9 would need another 9 in the square. This places quite a few numbers, including the 8!

### 3. Moving on to purple

Look now at the purple line bottom right. It is only 3 numbers, but it is actually has to be 5-6-7 or the green above in the same column won't have any candidates. There is obviously now a triple, and this means there must be an 8 in the aforementioned green. This leaves only a 4 in the column, top right. The last bit step here, is there now must be a 8 in the bottom right green, as its the only place an 8 can go in that column:

### 4. The big cane gives one

There is one obviously larger cane - a purple of length 9. This must contain all the numbers from 1-9 in some order. There is already only one place an 8 can go, and this solves the green in the same column, and this in turn places some numbers in the big green cane - a nice flow of placements! The final bit of logic is to see that a 1 must be in the bottom left of the big cane, which removes some candidates to the right and places a 6.

### 5. The flow of candy

This puzzle is beautifully made in the such that you place one number and it opens the door to the next, and the solution flows very nicely. The last 6 placed solves a couple of numbers in the green cane in the same row. As there is a 1-5-7 in row 7, this places the 9 in column 1. Now consider the purple middle right. It cannot be 2-3-4 or R9C7 has no candidates. So the purple must be 3-4-5 or 4-5-6. Either way, there is a 5, which fixes a 1 in the column and places multiple other numbers.

### 6. An unused purple

Consider the purple top left. It must be 3-4-5-6 as it cannot contain 2 and 3 or the green to the right has no candidates. 4 can then only go in one place, and a 2 and 8 is fixed to the right. This dominoes around the grid placing a few other numbers as a result. A considerable portion of the board is now filled and we can almost solve.

### 7. Major progress!

This stage can now be reached with some normal sudoku logic. The next breakthrough is to look at the small 3-purple top left. It can be a 2/8 in one cell so this reduces the triplets it can be. However if it has a 2, it must be a 1-2-3 or a 2-3-4, neither of which is possible. It must be an 8, and a 6-7-8 or 7-8-9. Either way, a 7 is used created a pair of doubles in row 1 and 2, placing a 7 at the bottom of row 3. There is also only one place for 8 and 1 to go middle left.

### 8. The last breakthrough

The final breakthrough here is to look at row 9 column 6. If it is 4, or 5, then there is no where for 4 or 5 on the large purple cane. So it must be a 9. The rest of the numbers all fall into place with normal sudoku logic.

# The solution:

Great puzzle! Had a lovely flow to it while still being very difficult :)

• Correct, and very well presented...glad you enjoyed! Dec 20, 2021 at 18:29
• @JeremyDover very enjoyable! Might have taken me 2 and half hours but time well spent :P Dec 20, 2021 at 19:06