Logic Masters Deutschland e.V.

As a math(s) teacher, I've learned a lot of 'divisibility' tricks over my lifetime. You probably have learned some -- like numbers divisible by 5 end in 5 or 0. Or numbers that are divisible by 2 are even. You might even know that numbers divisible by 3 or 9 are divisible when the sum of their digits is divisible by 3 or 9.

There is a divisibility trick for 11 too, which lends itself to providing some sudoku clues. To check if a number is divisible by 11:

1. Find the 'alternating sum' of the number:
   • Subtract the 2nd digit from the first...
   • Add the 3rd digit.
   • Subtract the 4th digit.
   • Add the 5th digit...
2. If this alternating sum is divisible by 11 (0, 11, 22, or -11, -22...) then the original number is.

• An example: 84359 has an alternating sum of 8-4+3-5+9 = 11, which is divisible by 11, so 84359 is divisible by 11 (84359/11 = 7669).
• Another example: 176 has an alternating sum of 1-7+6 = 0, which is divisible by 11, so 176 is divisible by 11 (176/11 = 16.).
• Another strategic example: 379X has an alternating sum of 3 - 7 + 9 - X, or 5-X. To be divisible by 11, 5-x must equal 0, 11, 22, etc... If X is a sudoku digit, it would have to be 5. The number 3795 is divisible by 11 (3795/11 = 345)

Some observations that might prove helpful (the proofs are left as exercises for the reader):

• If a number is divisible by 11, it will be in reverse order too.
• A shortcut for three digit numbers: if the outside digits add to the middle digit, it's divisible by 11. e.g. 253 is divisible because 2+3 = 5
  • Note, there are some numbers like 319 that are divisible where this shortcut wouldn't work. You could still use this shortcut if you 'cast off 11's -- meaning subtract 11 away if the total is more than 11, e.g. 3 + 9 (-11) = 1

Short summary of the puzzle:

• Normal sudoku rules apply.
• Little Killer diagonals are shown when the cells taken as one number are divisible by 11. Numbers can repeat along these diagonals. Sorry/not sorry: only some sums are given.
• Sandwich sudoku totals are given when the cells taken as one number are divisible by 11 (and not zero).
• Many killer sudoku cages are shown, some overlapping. Every killer sudoku cage, when considering the cells inside as one number, is divisible by 11.Sorry/not sorry: only some sums are given.

Exercise for the reader (and helpful hint, so highlight text to show):

If you know the digits in a three digit number that is divisible by 11 sums to X, what is the middle digit?