μ-Dots and Totient Sandwiches
(Published on 16. June 2022, 20:22 by Hrafnsvaengr)
What started out as a very silly idea turned into this, a very niche but fun little puzzle based on two functions from number theory. It shouldn't be too difficult, although the rules may seem more complicated at first than they actually are.
A link for the puzzle on f-puzzles can be found here and on CtC here.
Alternatively, the rules can be simplified as so, if you are familiar with the Möbius function, μ(n), and the Euler totient function, φ(n).
A link for the puzzle on f-puzzles can be found here and on CtC here.
- Normal sudoku rules apply.
- If the product of two orthogonally adjacent digits is the product of an even number of distinct primes, there is a blue dot between them.
- If the product of two orthogonally adjacent digits is the product of an odd number of distinct primes, there is an orange dot between them.
- If the product of two orthogonally adjacent digits is the product of a perfect square (other than 1) and any other number, there is no dot between them.
- The clues around the edge of the grid are totient sandwiches.
- Let N be the sum of the digits that appear between the 1 and 9 in a given row or column.
- The clue adjacent to that row or column is equal to the number of positive integers, up to N, which share no factors with N.
μ-Dots:
Totient Sandwiches:
Alternatively, the rules can be simplified as so, if you are familiar with the Möbius function, μ(n), and the Euler totient function, φ(n).
- Normal sudoku rules apply.
- Let a,b be two orthogonally adjacent cells. If there is a blue dot between them, μ(ab)=1; if there is an orange dot between them, μ(ab)=-1; if there is no dot, μ(ab)=0.
- Let si be the digits appearing between the 1 and 9 in a row or column. A clue outside the grid is equal to φ(Σ si) in the corresponding row or column.
Solution code: Row 4 and then column 3, without commas or spaces.
Last changed on on 16. June 2022, 20:23
Solved by Arashdeep Singh, kublai, RJBlarmo, baris, SKORP17, Xean, Leonard Hal, KevinTheMH, hibye1217, Woody03130
Comments
on 17. June 2022, 00:54 by RJBlarmo
Fun puzzle, nothing too difficult once you really consider how the Mobius function works. Needed to have a totient function table up though :)
on 16. June 2022, 22:13 by kublai
WOW! Once I got the mu and totient constraints sorted it wasn't so bad, but that part was tricky enough to bump it up a little on difficulty.
on 16. June 2022, 21:23 by Arashdeep Singh
Nice puzzle. Learned the functions through it. Thanks :)
on 16. June 2022, 20:23 by Hrafnsvaengr
Typo correction
