Submissions and Requests

As a puzzle-site owner, I get an occasional request for help on one problem or another. When they're especially interesting, I usually turn them into a POTW quickly. Though some may not be completely applicable to a POTW, they're still interesting. Here are a few questions I've received, followed by a few puzzle submissions I've received.

Requests for help

  1. An interior designer has an intriguing dilemma. She has to move an amazingly heavy armchair, but the only possible movement is to rotate it through ninety degrees about any of its corners. Can it be moved so it is exactly beside its starting position and facing the same way? [Update 9/18: One reader suggested this problem should be attacked without requiring the angle of rotation always be 90 degrees. Consider this as part (b) of this problem.]
  2. An angus is anchored to a 6 meter rope. The rope is attached to the outside corner of a shed measuring 4 meters by 5 meters in a rolling grassy meadow with varying elavations. What area of grass can the cow graze in 1.5 hrs?
  3. Prove that: Within a set of triangles having a constant base and constant perimeter, the isosceles triangle has the maximum area.
  4. What is the missing number ?
        26  34
      22   ?  26
    11  18  27  31
  5. You have 6 pool balls, all identical in size though differently numbered. 5 balls are of the same weight, the 6th is different (i.e. heavier or lighter). You have a set of balancing scales and, using the scales twice only, you have to identify the rogue ball.
Small Submissions
  1. S - - - - S
    Enter a 4 letter word, divisible by 2, to get odd numbers.
  2. 14 people are eating at their hotel's restaurant. They are seated at two tables which hold 8 and 6 people. If they are seated at these tables for every meal, how many meals are needed to assure that every pair of people sit at the same table at least once? What if the tables instead hold 9 and 5 people?
  3. a/b + b/c + c/a = 1
    Does a solution exist for a, b, and c belonging to the set of negative and positive integers? Does any solution exist? If so, show a simple example.
  4. Simplify the infinite product (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^16)..., given |x| < 1.

Source: Many sources. Submissions from Denis Borris and Ravi Subramanian.

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