Logic Masters Deutschland e.V.

Counting Circles are such a curious and clever constraint. I have wanted to set a Counting Circles puzzle for a while now, but at first, I struggled to find an interesting way to use them. Their global nature makes them tricky to utilize locally, because each Counting Circle you try to add to the puzzle completely alters the logic of all previous Counting Circles. Then I realized instead of fighting the global nature of the Counting Circles, I should embrace it! The complete idea for this puzzle then immediately followed, although it took some tinkering to make sure all of the details played out perfectly. :)

I hope you enjoy my latest puzzle, The Buddy System!

Update: This puzzle was featured on Cracking the Cryptic! Watch Simon's solve of the puzzle here!

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Normal Sudoku rules apply.

Counting Circles: A digit in a circle indicates how many circles contain that digit.

Palindrome Lines: Digits along a grey line form a palindrome, i.e. they read the same forwards and backwards.

Between Lines: Digits on a blue line between two circles have values between the digits in those circles.

Kropki Pairs: Digits separated by a black dot are in a 2:1 ratio.

Solve on SudokuPad

A clever puzzle can only be appreciated by those who discover the solution. I don't intend for this puzzle to be unapproachable for anyone, but the nature of its design means it may feel impossible to those who aren't struck by what my favorite math professor would call "divine inspiration" to see the path forward. If you are feeling hopelessly stuck, highlight the text below to reveal a few hints that should help give everyone a fighting chance to solve this puzzle. Each hint is more explicit than the last, so highlight one hint at a time and only move onto the next hint if needed.

1. This puzzle requires Set Equivalence Theory to solve. In other words, it's possible to identify two sets of cells that each contain exactly the same collection of digits, and this knowledge somehow helps in deducing the solution.

2. One set of cells is the union of rows 2, 4, 6, and 8. We know that these 36 cells contain exactly 4 copies of each digit 1-9. Now, find another set of cells that contains exactly the same collection of digits. Anytime you can prove that one cell in each set of cells contains the same digit, you can remove those cells from each set and it will still be true that the remaining cells in each set still contain exactly the same collection of digits.

3. The other set of cells is the union of columns 2, 4, 6, and 8. Color rows 2468 in one color and columns 2468 in a different color. Remove the coloring from any cell with both colors. Anytime a palindrome line forces a cell of one color to contain the same digit of a cell of the other color, remove the coloring from both cells. The remaining cells of one color must contain exactly the same collection of digits as the remaining cells of the other color. What does that tell you about the nature of the digits in the Counting Circles?

Good luck!