Razonamiento matemático

Kenken For Teachers
Tom Davis
tomrdavis@earthlink.net http://www.geometer.org/mathcircles June 27, 2010

Abstract Kenken is a puzzle whose solution requires a combination of logic and simple arithmetic skills. The puzzles range in difficulty from very simple to incredibly difficult. Students who get hooked on the puzzle will be forced to drill their simple addition, subtraction, multiplicationand division facts.

1 Kenken Rules
On the left in the figure below is a simple Kenken puzzle and on the right is the solved version of that puzzle.
3 8+ 3− 3 8+ 3−

3
7+ 7+

2 4
8+ 8+

1 2 3
1

4 3 2 1

1
8+ 8+

4
1

1 3

2

4

Kenken puzzles are square (this one happens to be 4 × 4, but they can be of any size). Like Sudoku, the solution requires that an n × n puzzlecontain the digits 1 through n exactly once in each row and column, but in any order. In addition, the puzzle board is divided into possibly irregular “cages”, each with an indication of a goal and an operation1. If there is no operation indicated in a 1 × 1 cage, that number is simply to be inserted into the cage. If the operation is + (or ×), then the sum (or product) of all the numbers in thatcage have to yield the goal number. If the operation is − (or ÷) then the cage must consist of two squares and the difference (or quotient) of the two numbers must yield the goal number. The numbers can be in any order in the cages. Check to see that the board on the right above is a solution to the puzzle on the left. A well-constructed Kenken puzzle, of course, has a unique solution (and all thepuzzles in this document are well-constructed). This document contains a number of puzzles suitable for classroom use in the sections at the end, but there are lots of websites that contain them as well. The author’s favorite, at this time, is: http://www.nytimes.com/ref/crosswords/kenken.html
1 Some

advanced puzzles do not even include the operation, but in this paper we will always includeit.

1

where a new set of puzzles of differing size and difficulty is posted every day. The following Kenken puzzles are not at all easy, but they are beautiful: http://www.stanford.edu/∼tsnyder/kenken.htm

2 Classroom Use
Kids can learn a lot just trying to solve a few simple puzzles either by themselves or better, in small groups knowing nothing other than the goal. Have the kids work onan easy puzzle, and then have members of each group explain the sorts of logic they used to solve the puzzle. Make a list on the board of the strategies (even the easiest ones, like “If you know all but one of the numbers in a row/column, then the last number is the one that’s missing from the set {1, 2, · · · , n}.”, where n will probably be 4 for the first puzzles you give them). In Section 10is a page of six 4 × 4 puzzles that are fairly easy. The solutions, in case you need them, are in Section 16. All the other puzzles are 6 × 6 (and one very easy 9 × 9). Often students are much more comfortable with addition/subtraction than multiplication/division, and there is an entire page of 6 × 6 puzzles that use addition and subtraction goals only. The author has found it useful inpresentations to make a transparency of the puzzles to be presented and to project them onto a whiteboard so that the solution can be filled in and discussed using normal whiteboard markers. For advanced kids, after some practice with puzzle solutions, you can have them try to generate their own puzzles. This is discussed in Section 9. You can use Kenken puzzles as a gentle introduction to proof. If you lookat the descriptions of he solutions to the puzzles in Sections 3, 4 or 8, you will see that each is basically a mathematical proof that the solution is correct. The statements involve possible numbers to fill in the squares, and each requires a reason. You could assign puzzles (probably of size 4 × 4 to keep the number of steps in the proof manageable) and require a step-by-step explanation of...