The Prisoner’s Dilemma

In 2015, Sudoku made the news in Britain. A couple of prisoners in The Victorian Jail wrote a letter of complaint to a local newspaper asserting that the paper had published a Sudoku that was IMPOSSIBLE to complete (emphasis theirs). They and 84 other prisoners were unable to complete the Hard Sudoku printed in the paper and they sent a copy of their work along with their letter. They did not have access to the solution; as that was in the next issue which they never received. It was, after all, a prison. Their work was in pen with several entries in broad marker; making for a rather messy display. There were at least six violations of the Prime directive – meaning there were at least six instances in which there were two of the same Digit in some Box, Row or Column. It is impossible to determine which error came first, but the publisher found the Sudoku to be legitimate and the prisoners’ complaint was unfounded.

When I examined the Sudoku, the errors were immediately obvious – nothing hidden or obtuse. My initial take was that all of this could have been avoided with one simple trick – counting. It had become clear to me by this time that counting is the core of every solution strategy. In a row with solutions 3, 6, 9 already in place; the trick is to account for: one, two, four, five, seven and eight. When the Matrix is complete, one may, and I do, check my answers by counting the Digits in each Row, Column and Box – rather than flipping back and forth between the Sudoku at hand and the solution printed in the back of the book. Whenever I discover a violation in my own Sudoku, I get into counting mode to try to figure out if the error is correctable; and as often as not; it is. Three, six, nine (Solutions) – one, two, four, five, seven, eight – the cadence often supersedes the order. After taking into account the existing solutions the remainder reveal themselves by counting; or more appropriately, accounting. It is the most common activity in Sudoku solving.

But counting takes on a completely new meaning than the traditional “One two, buckle my shoe; three four, shut the door.” How do you count within a set? Suppose we have a set of three Digits and three Cells. Assume further that the ‘numbers’ assigned to the three ‘elements’ of the Set are 2, 5 and 8. So, counting, in this set, is two, five and eight. So, if the remaining unknown cells, are 1, 2, 4, 5, 7 and 8 and 258 represent an independent set, then 1, 4 and 7, must constitute another set of three Digits. This is what we refer to as Partitioning. The six unassigned Digits, after accounting for the 369 solutions divide themselves into two independent sets of three Digits each; 258 and 147. Partitioning allows you to divide the missing Digits into smaller groups that facilitate memory. It is the key to a No-Notes approach to solving Sudoku