# Deficits and Surpluses (UK Sudoku Championship, 2020)

Last weekend I participated in the UK Sudoku Championship 2020. Participants had two hours to score points, by solving as many of 16 puzzles as they could – the top contestant finished everything correctly in 56 minutes 34 seconds, and sixteen people cleared everything within the time limit. It’s pretty impressive how long the tail of the distribution is. I finished thirteen out of sixteen puzzles, scoring 621 of 720 points (puzzles are worth different numbers of points depending on difficulty, and the ones I left behind were generally on the lower-valued side of things), and would probably have taken another 15 minutes to finish out the last three puzzles.

Nonetheless, 621 points was enough to log a 28th place finish. I finished on rank 42 two years ago, and although I missed last year’s I did it offline and ranked somewhere in the 40s as well, so this looks like an improvement. Of course, it could also be the case that the field got weaker.

Most of the puzzles in this contest were fairly common sudoku variants – while I normally scan through the instruction booklet before starting the contest I don’t usually prep explicitly, and here it’s not really needed. The UK championship seems to have had three standard 9×9 Sudokus, a Killer (extra regions are added with target sums; numbers in a region can’t repeat, and must add up to the target), a Diagonal (numbers along the two major diagonals must be unique) and a Thermo (thermometers are added, which introduce inequality constraints: along these numbers must increase). I’d usually go through all of these – based on past data I’m usually reasonably fast at these.

There are also a couple of more peculiar puzzles. There’s “OEBS” (Odd-Even-Big-Small), where a couple of kanji characters (in this case they’re the same as the Chinese words) are placed next to the grid, introducing constraints on the first two numbers seen in that direction. This is usually an 8×8 grid, and is a type I like; it often admits logic around pairs and sets. Then there’s a Surplus and Deficit Sudoku. These puzzles tend to make my head spin a lot, and so I didn’t plan on going for them at all.

• Surplus Sudoku: Place 1-N in each row and column once each. Some M-cell (M > N) regions are marked – these regions must contain each number at least once. In the contest, N = 7 and M = 8.
• Deficit Sudoku: Place 1-N in each row and column once each. Some M-cell (M < N) regions are marked – these regions must not contain any duplicates. In the contest, N = 7 and M = 6.

I already find the geometry of an Irregular Sudoku (which has the standard rules, except the 9-cell boxes are weirdly shaped) tough – spatial reasoning is not one of my strong suits. Working with these is even harder, especially because some rudimentary Sudoku techniques become invalid in the regions (but are still valid, and often needed, in the rows and columns!):

• Naked Single: If all possibilities other than X are present in the same row, column or region as this cell, then X must be the value of this cell. This doesn’t work in Surplus Sudoku regions, because X could be a duplicated digit.
• Hidden Single: If X can only be placed in one location in this row, column or region, then X must go there. This doesn’t work in Deficit Sudoku regions, because X might simply not appear in the region.

Applying these two techniques is pretty much automatic for me, and these deductions remain valid in almost all Sudoku variants, but are no longer always applicable in these puzzles. The additional checking and context switching slows things down by a lot.

It’s probably a much more involved technical discussion, but to me at least Surplus Sudoku feels like a pretty weird sudoku variant (probably because of having repeated numbers in a region). I don’t find it symmetric: Deficit feels less strange, maybe because “numbers cannot repeat in a specifically marked region” is not too far off from many of the other constraints in common variants (e.g. a Killer with size eight cages – though you’re told which number is missing, or a Thermo with thermometers of length seven or eight – strictly increasing means numbers can’t repeat). I probably don’t have that much experience puzzle-solving, but I can only think of one variant which ends up forcing a lot of equality in sub-regions of the grid: Anti-Diagonal Sudoku (each of the major diagonals has only three distinct digits), though that’s not a super common variant and I find they solve kind of weirdly as well.

Writing those two puzzles off probably turned out to be a mistake, as I had about four minutes at the end with the Surplus, Deficit and Irregular left. I don’t think this was intentional, but the spatial theme here is kind of amusing. I opted to try the 20-point Deficit Sudoku instead of the 30-point Irregular which was unlikely to have been solvable that quickly, but barely got anywhere. Given how close I was to otherwise finishing the set, I might need to be careful about writing off more than one puzzle (and at some point, even about writing off one puzzle).

Nonetheless, I think I performed well, and enjoyed the contest – maybe next time around I’ll actually give these puzzles a shot! (Though probably at the end.)