Ken's POTW


A Counter-Intuitive Trip, and a Product Square
  1. A Counter-Intuitive Trip

    One summer day, Jack, deciding to paint his house, placed his ladder against the side of the house and began to climb. However, having placed the ladder perpendicular to the ground, he began to tip over backwards when just half-way up. It's easy to see that while Jack clung hopelessly to the middle rung, his trip to the ground (Point A on the side of the house to Point B on the ground) followed a path EXACTLY ONE QUADRANT OF A CIRCLE WHOSE CENTER WAS ON THE GROUND NEAR THE FOUNDATION OF HIS HOUSE AND WHOSE RADIUS WAS HALF THE LENGTH OF THE LADDER.

    Six months later, Jack had recovered from his broken bones and decided to finish the job he started. Even though it was now middle of winter, Jack hauled out his ladder and placed it just as he had before. Being a slow learner, Jack repeated his original error concerning the angle of the ladder, and again had some bad luck when he reached the half-way point. But now the ground was a frozen sheet of ice and instead of tipping backwards, now the base of his ladder slipped out and the top slid down the side of the house. Jack clinging to the center rung, wound up moving from Point A on the side of his house to Point B on the ground as before. Question: DESCRIBE THE PATH HE TRAVELED THE SECOND TIME!

  2. Magic Squares 
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    1. The Lo Shu is found in Chinese legends as early as the fifth century B.C. [See The Penguin book of Curious and Interesting Puzzles, p.21.] The figure contains each of the digits 1 through 9 in the 3x3 grid, such that the sum of the digits in each row, column, and diagonal is the same. Can you reproduce the Lo Shu?
    2. Instead of a sum, can you find nine unique positive integers, such that the product in each direction is the same? What is the lowest product that can be made?

    Source: 1. Submitted by Bill Coyle. 2. Many, many places.

    Solution
    Mail to Ken