An Introduction to Sudoku Logic - W.H. Freeman

Taalman/Kohn • Calculus

An Introduction to Sudoku Logic Laura Taalman and Peter Kohn

In our Calculus textbook, we build up a tower of ideas, starting with a base of definitions and building up fact by fact with theorems and applications. Along the way we use logic and reasoning to make sure that our tower of ideas is strong at each level. Sudoku offers a fun example of how one can build such a tower of ideas, as well as good practice for explaining mathematical reasoning.

Sudoku Logic This puzzle follows usual Sudoku rules, which means that we must fill in the cells in the grid so that each row, each column, and each 3-by-3 block contains the integers 1-9 exactly once. One way to start filling in this particular Sudoku grid is to look at the patterns of 2’s given in the clues. The lower-left block must contain a 2 in some cell, and the 2’s already given in the seventh and eighth columns mean that the 2 in that block must be in cell A. Once we know this, the 2 at A and the 2’s given in the second column and the sixth row allow us to conclude that there is a 2 in cell B. This allows us to deduce that there is a 2 in cell C, and from that, that there is a 2 in cell D. This completes all of the 2’s in the puzzle!

More Sudoku, More Logic Every Sudoku puzzle has a unique solution, and each time we fill in a cell we must know that it is something that we have to do, not just something that we can do. Describing the reasoning process that leads us to fill in a Sudoku puzzle with certainty is good practice for describing any kind of mathematical reasoning. One way to start here is to fill in cells A, B, and C the same way that we filled in the 2’s in the previous puzzle, then fill in D using the 8 in the rightmiddle block, and then fill in E, F, and G, in that order. Determine what numbers go in these cells, and explain your reasoning.

Four-Square Sudoku In calculus, after we have established one set of rules and strategies, we look for ways to extend our knowledge to new situations. This Sudoku variation is an extension of Sudoku that requires new solution strategies. The usual rules of Sudoku apply, but in addition, each of the four shaded squares must contain 1-9 exactly once. The additional squares in this puzzle allow us to make new deductions. Consider the lowerleft green-shaded square. This green square must contain a 3, but since there is already a 3 in the very lower-left corner of the Sudoku grid, we know that the only location in which a 3 can appear in that green square is cell A. From this fact we can also deduce that a 6 must appear in cell B. Why?

Tile Sudoku Taking our extensions one step further, we can consider Sudoku puzzles of different sizes and with different sets of rules. In this variation we must fill in the cells so that each shaded region contains the integers 1-5 exactly once, and each row and column contains the integers 1-5 exactly twice. Since numbers appear more than once in each row and column of this Sudoku variation, we must develop new solving strategies. For example, in the second row of this puzzle we see that two 3’s are already given. Explain why this fact implies that we must have a 3 in cell A. Then explain why the 3 in cell A, together with the two 5’s in column 8, allow us to conclude that there must be a 5 in cell B. (Hint: Along the way, argue that a 5 must appear in the open oval in the second row of the puzzle.)

Solutions

Sudoku Logic

More Sudoku, More Logic

Four-Square Sudoku

Tile Sudoku

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