My Problem with Sudoku.

Well admittedly, the title of this post is somewhat tongue in cheek.

In and of itself, Sudoku is a great puzzle.

Sudoku is nothing more than a glorified Latin square with the added restriction of numbers being unique within each box (3x3 in a conventional 9x9 Sudoku). The thing is, Sudoku has nothing to do with math/arithmetic or more generally- The main objective of Sudoku has nothing to do with the actual entries. Don’t get me wrong, I am fully aware of the math (mostly combinatorics and graph theory) BEHIND Sudoku, but the Sudoku-solving process itself doesn’t oblige the solver in any way to use arithmetic or any other quantitative considerations. The digits 1-9 are treated purely as symbols, which means one could easily create a mapping between the digits 1-9 and some other set of symbols or pictures. The deductions required are solely related to the spatial organization of said symbols in relation to one another. One could even imagine a tactile- or auditory-based Sudoku.

One isn’t necessarily limited to expanding the required skills to the arithmetical domain, we can imagine a purely visual puzzle (as I’ll demonstrate in this post) that forces the solver to apply visual transformation on the entries and work through the logic of the given clues.

As a full 9x9 walkthrough solve would be quite tedious, and as I am sure the vast majority, if not all, of you are fully capable of solving your average Sudoku puzzle on your own, I’ll only do a quick demonstration of how one would fill in a “latin square”.

From now on, whenever I write “Standrd Latin Square Rules” or “SLSR” I am referring to the following:

Given a set of N different symbols, each symbol in the NXN array:

1. appears exactly once in each row

2. appears exactly once in each column

3. the only symbols used are those from the given set

So a classic 9X9 Sudoku for example would have the set {1,2,3,4,5,6,7,8,9} and obey SLSR plus the extra rule:

4. each 3X3 box contains the digits 1-9 exactly once

The basic approach for attempting a standard Sudoku is the same (even though the extra constraint on the 3x3 boxes provides more restrictions on the possible digits). If we have the following 5x5 Latin square:

We can quickly recognize that the two empty digits in the left-most column have to be a 4 and a 5 from-top-to-bottom. Otherwise, it would have contradicted the 5 in the third-from-left column and the 4 in the second-from-left-column. So we have:

It is also easy to see that neither the middle nor the bottom-most digit in the middle column can be a 3 since it would contradict the 3 in the bottom left-most column and the 3 in the middle of the second column from-the-right. Same logic goes for the the bottom of the middle column with respect to the digit 4. After filling in the digits we get:

Now, since the only missing digit in the middle column is 1, we know this has to be correct digit. We can now also clearly see that 2 is the only missing digit in the middle row (after filling in the 1 of course), and by the same logic we also can fill in the only missing digit in the right-most column, which is a 3.

From here, it is pretty much self-evident which digits should go where. So the solution to our Latin Square is:

Other more “mathy” Sudoku-variants DO exist, such as “mathdoku/KenKen”, “Killer Sudoku”, “neighbours”, Kropki Sudoku and “Futoshiki”, the first and third of which I deeply enjoy.

As you probably know by now, I love math and I especially enjoy numbers, computation and arithmetic so in this post I wanted to create some Sudoku-variants of my own that would put an emphasis on using quantitative and arithmetical considerations as well as the usual spatial ones.

Let us first solve one small (4x4) example together and then I’ll leave you with a somewhat bigger challenge you can attempt on your own. The rules are SLSR+the Sudoku constraint plus the following:

1.The dotted area (with a number and an operation) shows the result obtained after applying the operation to the digits contained within the region.
For Example: The three digits in the dotted area with "+9" should all sum to 9.

2."gpf" stands for "greatest prime factor". When the digits within the region are read as a single number (from left-to-right/top-to-bottom).
For Example: a 2-cell dotted region with "gpf 7" would mean the only possibilities would have been (1,4) and (2,1) since 14=2*7 and 21=3*7. 

3.The inequalities >,< tell you if the digit is bigger/smaller than the one in an adjacent cell.

Let's start by looking and the region with gpf 67:
we know that we are looking for a three digit number with all of its digits between 1 and 4 so since floor(500/67)=7 ("floor function" means to round down to the nearest integer) and since 67*1=067, we know we only need to check multiples of 67 between 3*67 and 7*67. So:
2*67=134
3*67=201 (X-contains a 0)
4*67=268 (X-contains a 6 and an 8)
5*67=335 (X- contains a repeated digit and a 5)
6*67=402 (X-contains a 0)
7*67=469 (X-contains a 6 and a 9)

And so clearly the only possible option is "134" from top-to-bottom.

Now, since the only 3 digits smaller than 4 that can multiply together to a 6 are 1,2 and 3 - the 4 in the right-most column can't be in the top cell. But since 4 is the greatest digit in a 1-4 grid and the "<" are strictly increasing from bottom-to-top, the only place for the 4 in the right-most column is the second cell from the top:

I will leave the rest of this puzzle to you, so you can prepare for the more challenging version at the end.

Now for a non-arithmetic variant of my own making.

The rules are SLSR plus the following explanation of the given iconography:

1. H- horizontal flip; meaning the 2 adjacent cells sharing this symbol are mirror images of one another around the Y-axis

2. V- vertical flip; meaning the 2 adjacent cells sharing this symbol are mirror images of one another around the X-axis

3. (2)- 180 turn; meaning the 2 adjacent cells sharing this symbol are a double counter-/clockwise turn from one another

4. twisted arrow- 90 turn; meaning the 2 adjacent cells sharing this symbol are (when read left-to-right/top-to-bottom) a one either counter-clockwise or clockwise turn from one another, depending on the direction of the arrow.

and as for the more challenging (6X6) arithmetic variant (SLSR+SUdoku constraint+ the already mentioned clarifications): 

And that's all for now! A huge huge thank you to everyone who tested the puzzles and gave feedback (Michael, Sebastien, Jakob and Hamza), I am truly sorry it took me so long to publish this one, it was actually already mostly done a month ago but I have a lot going on right now. I hope you liked it, I hope you learned something- I promise to keep posting semi-regularly even while having other stuff to take care of. Please be safe and watch out during this whole COVID-19 outbreak (wherever you may be), it is probably also a good idea to store some food, meds and other supplies in advance. Stay tuned for next time and keep that cortex active!