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## A Path That Always Ends?

Stu Dent has red, gray, and blue flagstones for making a walk. He wants no two consecutive stones to be the same color, plus no consecutive pair of stones to have the same two colors in the same order, not repetition of three consecutive colors, etc.

Stu starts out by first laying a red stone, then a gray, and continues until he finishes laying the seventh stone.

Stu then finds himself stymied and unable to use any stone for the eighth without repetition of some color pattern. What were the colors of the first seven stones?

My Editorial Note: This situation suggests others questions as well...

• Under the same rules, could Stu ever lay the stones and get past the seventh stone?
• If Stu gets past the seventh stone, will there always be a stone that prevents continuation of the path?
• Is seven stones the least number of stones possible that will halt the process?
• What happens if there are four different colors for stones? n different colors?

Source: J.L.'s lesson plan (a former student)

Hint: Use some colored squares as manipulatives....try out Stu's process. Does a tree diagram help?

Solution Commentary: J.L. writes: "I like this problem because its main requirement is insight, ingenuity, and mental gymnastics, rather than formal mathematical training..... Red, gray, red, blue, red, gray, red. Stu now cannot use red, for then there would be two consecutive reds. He cannot use gray because there would be two consecutive red-grays. He cannot use blue, for then there would be two consecutive red-gray-red-blues. Any other pattern for the first seven stones would have allowed a choice for the eighth not involving a repetition."

Do you agree with her last claim? And how did you do in your explorations of the other questions posed?