Table 3 The left-hand column gives the number of puzzles that had the number of solutions listed in the middle column. Thus, the cumulative total of 58 at the end of the third row indicates the number of puzzles that had 3, 6, or 8 solutions. The right-hand column lists the cumulative number of puzzles accounted for. Thus, 21 puzzles had three solutions, 17 puzzles had nine solutions and 17 puzzles had 10 solutions, etc. The left-hand column lists the number of puzzles (out of 500) that had the number of solutions listed in the middle column. Table 2 The integers in this table represent the number of solutions per trial for 500 trials The sequence of integers in Table 2 is the sorted list of the numbers of solutions found in each of the 500 trials. Since I started from the "answer," there are no puzzles generated that will have no solution. I was amazed to discover that in a sequence of 500 puzzles, one puzzle had 190 solutions, and fourteen puzzles had exactly one solution. ![]() The puzzle shown above has eight solutions. To see the number of solutions, and the solutions themselves, click the button below. I then select the four digits to be shown, but write and solve the ten equations governing the other twelve numbers. Obviously, I start with a 4 × 4 array of sixteen numbers between one and nine, and form the row, column, and diagonal sums. It was designed to test the number of distinct solutions a puzzle can have. The puzzle generator I wrote deliberately does not accept a user-supplied puzzle. Then, if the icon below the puzzle is clicked, the number of solutions, and the actual solutions, will be printed. ![]() Each time the "New Puzzle" button is pressed, a new puzzle is generated. Well, I finally decided to look into this question, and the tool below gradually evolved. To what extent will these conditions determine the solution to the puzzle? I have always wondered about this puzzle - there are twelve numbers to be determined by some ten equations, and the condition that the numbers are integers between one and nine. Table 1 Directions for Challenger Puzzle from the Waterloo Region Record Vertical squares should add to totals on bottom.ĭiagonal squares though center should add to total in upper and lower right. Horizontal squares should add to totals on right. ![]() ![]() In fact, "There may be more than one solution" is explicitly stated below the directions, copyrighted by King Features Syndicate, Inc., that appear in my local newspaper, the Waterloo Region Record, and quoted in Table 1.įill each square with a number, one through nine. And unlike Sudoku, the puzzle can have multiple solutions. The object of the puzzle is to discover the missing twelve numbers. Indeed, on a 4 × 4 grid where sixteen integers would fit, four are given, along with the row, column, and diagonal sums of the numbers not shown. Do an internet search on "Challenger Puzzle" and you will find descriptions and solvers for a puzzle that involves sums of integers from one to nine.
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