Twelve-tone Breakfast
(Published on 2. March 2024, 19:19 by Teal)
Twelve-tone
Place the digits 0 through 11 in each row and column exactly once (T = 10, E = 11). In the case of symmetry and inversion, numbers are modular (in base 12), and can be visualized on a 'clock' where the straight line between 0 and 6 is the axis of reflection.
The topmost row, when read left to right, is the Prime Row.
Each row is a simple transposition of the Prime Row (if your prime row is 1, 2, 3, 4... and row 7 starts on a 6, then row 7 is 6, 7, 8, 9...).
Each column is an inversion of the Prime Row, and is transposed relative to its starting digit. (Inversion is like mirroring the number on the modulo 'clock,' 0 -> 0, 6 -> 6, 3 -> 9, 2 -> t, etc. If your prime row is 1, 2, 3, 4... the leftmost column would be, from top to bottom, 1, 0, E, T...)
Symmetrical Lines
Digits on dotted lines are Symmetrical (The difference between each subsequent digit by value is the same, e.g., 1, 7 is symmetrical, as is 0, 4, 8 and 2, 5, 8, e. Digits are unordered on the line.)
'Typical' Variant Rules
Digits in a cage sum to the small clue in the top left of the cage. Digits along an arrow sum to the digit in that arrow's circle. Digits along a thermometer increase starting from the bulb.
Solve in SudokuPad
Solve in Penpa
Solution code: Row 12
Comments
on 7. March 2024, 21:42 by Teal
@tobymgk sorry for the late reply! and yes, 'transposition' is referring to musical transposition (Twelve-tone is a musical concept and the origin of the idea of this puzzle), so you maintain the same "distance of movement" between each digit while changing the starting note. your description is accurate, if the prime row was 1234 each row would have to start with 4 consecutive ascending digits.
inversion is trickier, and i take it the confusion is because it's a technical term in music and differs from the math term. the simplest way to explain it is it's like a negative transposition, so you maintain the distance of movement but change its direction. if the prime row is 1234, then a transposition must start with 4 consecutive *increasing* digits, and an inversion must start with 4 consecutive *decreasing* digits. i hope this helps, and i assure you the next twelve-tone puzzle i publish will have much clearer rules and examples!
on 4. March 2024, 18:45 by tobymgk
I know you said you’re working on updating the rules/explanations but I have a question on the transposition example. If row 7 starts with 6789, is that assuming that row 1 (prime row) is 123456789TE0? Or is it more in line with musical transposition where, since row 1 starts with 4 consecutive numbers in ascending order, then all the other rows must start with 4 consecutive numbers in ascending order?
Also for the inversion example, I’m not following how you got those values for the left-hand column, unless again we’re assuming row 1 is 123456789TE0.
on 3. March 2024, 19:48 by Teal
@topoi thank you for the feedback! my biggest concern with the puzzle was the confusing rules, so i'll try and workshop a clearer explanation for the twelve-tone concept! i'm at least glad you were able to solve it. i hadn't considered the possible confusion with mathematical transposition and inversion so i'll try and disambiguate
on 3. March 2024, 13:47 by topoi
@Piatato
Oh I see. I don't know anything about music. I am familiar with these terms only from math and was very confused
on 3. March 2024, 12:37 by Piatato
@topoi Transposition and inversion are used as musical terms here, for which everything works exactly as expected. However, I understand how it could be confusing, since these terms also exist in mathematics.
on 3. March 2024, 02:45 by topoi
This was an interesting puzzle with good flow.
But the rules are written in a very confusing way, and I had to spend a long time figuring out what you meant.
"Transposition" and "inversion" don't have their expected meanings for sequence of digits.
