The rules of Sudoku dictate that every row, column and 3×3 bolded box in the completed grid must contain the digits 1 through 9 exactly once. A surprising fact emerges from these simple rules.
In every completed Sudoku, the cells highlighted in orange will always contain the same digits as the cells highlighted in blue.
In this example from our recent “Expert Sudoku,” there are three 7’s in the orange ring and three 7’s in the blue 2×2 squares in the corners. Similarly, there are no 8’s in the orange ring and no 8’s in the blue squares, and so on. Explain why this happens in every Sudoku puzzle.
How do we know that in a completed Sudoku grid, the orange cells and blue cells below will contain the same digits?
We know that the full middle column and the full middle row of the grid must contain the same digits by the rules of Sudoku: every row and column has the digits 1 through 9. Furthermore, the middle cell of the puzzle is common to both of these sets, so if it is removed from both, they remain equal.
To solve this week’s puzzle, find two sets of cells that you know contain the same digits and remove the cells they have in common to create the desired pattern.
Every column in a completed Sudoku contains the digits 1 through 9. So the four columns highlighted in blue below altogether contain exactly four 1’s, four 2’s, four 3’s, and so on.
Similarly, the cells highlighted in orange below comprise two full rows and two full 3 x 3 boxes, so they also contain exactly four 1’s, four 2’s, four 3’s, and so on.
In other words, the orange and blue sets below contain the same composition of digits.
Removing cells that are common to both orange and blue gives us our desired pattern and maintains the equality of the sets.
The Sudoku community calls this pattern the Phistomefel Ring. It is named after the alias of a puzzle setter who popularized the observation and has designed variant Sudoku puzzles around it.
Notes from Readers
Our recent puzzle “Prime Time” asked you to find one million consecutive nonprime numbers. The problem did not ask for the smallest instance of one million consecutive nonprimes, but mathematical minds enjoy optimizing, and several readers wrote in ideas for finding smaller sequences that fit the bill.
Our original solution was:
1,000,001! + 2, 1,000,001! + 3, 1,000,001! + 4, ..., 1,000,001! + 1,000,001
Reader Mike Thwaites offered an elegant tweak by replacing the plus signs with minus signs:
1,000,001! – 2, 1,000,001! – 3, 1,000,001! – 4, ..., 1,000,001! – 1,000,001
Each of these terms is nonprime, just like their counterparts formed by addition, and we have indeed shifted things earlier in the number line, albeit slightly.
Another reader, Mike O’Connor, found a significantly smaller sequence. Instead of multiplying all of the numbers from 2 to 1,000,001, we only need to multiply the primes from 2 to 1,000,001. Call this product P. Then P + 2, P + 3, ..., P + 1,000,001 form one million consecutive nonprimes.
This one takes a bit of reflection to understand. If k is a number between 2 and 1,000,001, then why is P + k necessarily nonprime? There are two scenarios: If k is prime, P is divisible by k by definition, so P + k is divisible by k and therefore nonprime. If k is not prime, then there must be some smaller prime that divides it. That smaller prime will also divide P, so again, P + k can’t be prime.
Thanks to the Mikes for sharing their ideas.
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