The Basics of Popular Sudoku Variants
Last spring I published a walkthrough of the basics of killer sudoku, complete with a puzzle designed to highlight several different types of logic. In the same vein, this post will explore the basics of four popular variants—kropki dots, German whispers, renban lines, and arrows—using the puzzle below. Click one of the links under the rules to try it for yourself!
Fill each row, column, and 3x3 box with the digits 1-9 once each.
Digits in cells separated by a white dot are consecutive. Digits in cells separated by a black dot have a 1:2 ratio. (These clues are sometimes referred to collectively as “kropki dots”.)
Along green (“German whisper”) lines, digits must differ from their neighbors by at least 5.
Purple (“renban”) lines contain a non-repeating set of consecutive digits in any order.
The sum of the digits along an arrow is equal to the digit in the connected circle. Digits along an arrow can repeat if allowed by other rules, although in this puzzle none of the arrows are positioned in a way that would allow that.
Very rough difficulty estimate: 4/10
Play online: SudokuPad | F-Puzzles
Kropki Dots
There isn’t much to say about white dots—the digits on either side must simply be consecutive, and the order doesn’t matter—though it is sometimes helpful to note that any pair of consecutive digits will contain one odd and one even.
The properties of black dots are a bit more complex. To start with, there are only four possible pairs: 1-2, 2-4, 3-6, and 4-8. Some of the implications of that are shown below:
Three of the pairs consist of just four digits: 1, 2, 4, and 8 (the single-digit powers of 2), and 2 and 4 are the only digits that are in two different pairs. This means any sequence of three black dots that cannot contain a repeated digit must contain 2 and 4 in the central cells, with 1 and 8 at each end. (Similarly, a sequence of two dots with no repeats would be either 1-2-4 or 2-4-8.)
Since 1, 2, 4, and 8 have all been used, the other black dot in box one must contain 3 and 6. And the remaining cells in the box must contain the three digits that can never be in a 1:2 ratio with another sudoku digit: 5, 7, and 9.
The white dot in box one means the digits in r2c3 and r3c3 must be consecutive, and 1 cannot be consecutive with 5, 7, or 9, so there must be an 8 in r2c3, which determines the order of the rest of the sequence.
The black dot in box four now only has one possibility: the 4 in r2c2 rules out 2-4 and 4-8, and 3-6 is ruled out because that would leave no options for r1c2. So it must contain a 12-pair.
In fact, the order of that 12-pair is resolved by the white dot in row four: 1 can only be consecutive with 2, so a 1 in r4c2 would put a 2 in r4c3, and there would be two 2s in box four. So r4c2 must be 2 and r4c3 must be 3, which also resolves the 36-pair in box one.
Also, as mentioned earlier, a white dot must always contain an even digit and an odd digit, and with 2, 4, and 8 already placed in row two, there is only one even digit left for the dot between r2c4 and r2c5.
German Whispers
Along green (“German whisper”) lines, digits must differ from their neighbors by at least 5.
The most important properties of German whispers are:
- a whisper line can never contain a 5 (because 5 is not 5 apart from any sudoku digit)
- the digits along a whisper must alternate between low (1-4) and high (6-9)
- if 4 is on a whisper, it must be next to 9, and if 6 is on a whisper, it must be next to 1
With those properties in mind, a lot of work can be done in box three:
The only place for 5 in the box is r2c8. Now, every other digit must be on the whisper, and since 4 can only be next to 9, it must be on the end to avoid forcing two 9s into the box. There must be a 6 on the other end by the same logic, and the cells immediately next to the ends of the line must contain 1 and 9.
The order can be worked out by noting that there is already a 1 in row three, or by looking at where 3 can go in box three, and at this point it should be possible to fill in the entire box.
Digits along whispers alternate between low and high, which means any two-cell domino along a whisper must contain exactly one low digit. In row four, however, the 2 and 3 are already placed, and there are four cells in the row that are on a whisper, so the remaining low digits, 1 and 4, must be somewhere in those cells.
The 4 cannot go in r4c7 or r4c8 because that would put two 9s in the row. In theory it could go in r4c6, with 9s in r3c6 and r4c7, but the 9 in r3c7 rules that out. So 4 is in r4c9, which determines the low/high polarity of the line.
(Note: the pencil marks in column six above are just the available low or high digits for each cell, without considering the differ-by-at-least-five part of the rule, which would allow 6 to be removed as a candidate from r4c6.)
Renbans
Purple (“renban”) lines contain a non-repeating set of consecutive digits in any order.
Renban lines essentially represent a “slice” of a number line, which is especially powerful when a line is forced to contain an extreme digit.
Here, the 1 in column three must be on the four-cell renban, which means the other three digits are 2, 3, and 4. And digits can’t repeat on a renban line, which means the digit in r7c4 must be in a cell that is not on the line in box seven (of course, it could in theory be on one of the other lines, but that is not the case here).
Meanwhile, the five-cell renban contains 6, 7, 8, and 9 to fill out column two, which means its fifth digit must be a 5. In fact, any renban that contains five digits will always contain a 5.
(As lines get longer, that logic can be extended. A six-cell renban will always contain 4, 5, and 6. A seven-cell renban will always contain 3, 4, 5, 6, and 7. An eight-cell renban will contain every digit but 1 or 9.)
Finally, a two-cell renban oriented horizontally or vertically is exactly the same as a white dot. As noted earlier, a white dot always contains one even digit and one odd digit, and in this case, the only odd digit available is 7.
Arrows
The sum of the digits along an arrow is equal to the digit in the connected circle.
The logic around arrows has a lot in common with killer cages, with the important difference that digits can repeat along an arrow (although, as noted above, that is not possible for any of the arrows in this puzzle). And since arrow sums are usually single-digit numbers,* knowing a few properties of single-digit sums can be very helpful.
*Double- and triple-digit arrow sums are another beast entirely, and are outside the scope of this walkthrough.
When an arrow contains three digits that must all be different, like the arrow connected to the circle in r9c8, its sum must be 6 or greater (1+2+3 = 6).
A two-cell arrow could in theory have a sum as low as 2 if the digits are both 1, or as low as 3 if the digits must be different. But here, the two arrows that are entirely in box nine can be thought of collectively: the two digits in the circles have a sum equal to the five digits on the arrows, and the minimum sum of five different digits is 15 (1+2+3+4+5 = 15). So the digits in the circles also have a minimum sum of 15, which means the digit in r7c8 must also be at least 6.
Of course, by sudoku neither of those digits can be 9, so the whole arrangement is forced: the circles contain a 78-pair (the only way to reach of a sum of at least 15 without a 9), and the digits on the arrows are a 12345-quintuple.
This leaves a 69-pair in the remaining cells in box nine, but 9 can never be on a multi-cell arrow, so the digit in r9c7 must be 6.
A one-cell arrow, like the one in r7c6, simply means the digit in that cell is the same as the digit in the circle. Here, 1, 3, 4, 6, 8, and 9 can be ruled out because of already-placed digits that see one or both of those cells. And 2 can be ruled out because the two-cell arrow in box six cannot have a sum that low, leaving 5 and 7 as the only options. But 7 doesn’t work because 7s in both r6c7 and r7c6 would make it impossible to put a 7 on the two-cell renban in column one, so the digit on the one-cell arrow must be 5.
Ok, I’ll leave it there. I hope this has been helpful! I didn’t cover literally every way these rulesets can be used, but I think I got most of the basics in. If there’s anything I skipped over or could have explained more clearly, please feel free to leave a question in the comments.
Solved puzzle below:
