Posts: 187
 
Location: New Zealand | I guess some people might not have come across optimising puzzles before, so I thought I'd give a walkthrough for how I approached the Tetra-Sudoku puzzle. If someone else wants to discuss the other two, feel free.
I tried placing tetraminos in the grid, then filling them with small digits, but that did not go well. So then I chose a different approach. Also I reread the instructions, and noticed that I didn't have to use all 7 shapes!
It was recently proven that a solveable sudoku must have at least 17 clues, but there are thousands of ways to achieve that. So I started with a 17-clue sudoku like this:
..5 ... 46.
1.. 27. ...
8.. ... ...
... ... ...
... ... .12
..3 .45 ...
... ... ..1
..4 ... ...
.56 ..4 ...
The clues are spread out. I'm allowed to swap around rows and columns, as long as each 'group of 3' stays together. So I do this to try to bring the clues closer together, hopefully into groups of size 4.
... ... ...
.3. .45 ...
... ... .12
... ... ..1
.4. ... ...
56. ..4 ...
.5. ... 46.
..1 27. ...
..8 ... ...
Now I have the clues grouped into two full tetraminoes, and two groups of three, a pair, and a single. So 6 tetraminoes would cover all these clues. But I can't find a way to cover them all with only 5 tetraminoes.
Next I throw away that single 3. So I have a sudoku with 16 clues, and it has 1000+ solutions, but I can cover the remaining clues with 5 tetraminoes. And I can choose which 4 extra digits to add (two next to the 45, one beside the 121, and one joining the 4 to the 46)...
Now is when I also relabel the digits so that the most common (the 4s) become 1s, etc, to minimize the sum. So I'd like to add 4 more 'small' digits if I can.
... ... ...
... .12 ...
... ... .34
... ... ..3
.1. ... ...
25. ..1 ...
.2. ... 15.
..3 47. ...
..6 ... ...
If I add an 8 to R2C4 and an 8 to R7C6 then the grid is solveable except for a 2x2 square. This leads to a score of 69+4=73.
Or if I add a 6 to R2C4 and a 7 to R6C7 then again there is 2 solutions: all of the digits 1-7 can be placed, but there are no 8 or 9 clues to resolve where those go.
... ... ...
... 612 ...
... ... .34
... ... ..3
.1. ... ...
25. ..1 7..
.2. ... 15.
..3 47. ...
..6 ... ...
I could add an 8-clue now, but going from 2 solutions to 1 solution only saves me 3 points, so I'd rather add a an 'unused' 1-clue and a 2-clue and stay with 2 solutions. After a final switch of columns 4 and 5 (so each shape is different), and another relabeling, this gives a score of 66+4=70.
... 2.. ...
... 163 ...
... ... .24
... ... .12
.1. ... ...
35. ..1 7..
.3. ... 15.
..2 74. ...
..6 ... ...
(I hope those are readable, it's the only way I could think of to format a 9x9 sudoku grid)
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