Impossible  Problems


These pages are dedicated to seemingly impossible problems.  I personally do not believe in impossible problems so that is not a real obstacle to me.   I will eventually solve the puzzle, given enough time.   Some of these really do not look as though there is enough data to solve it but if you are creative you see that you can figure out something.   Like that temperature problem on the posers page, it only sounds strange to you if your mind does not open as wide as it could.  Although there is insufficient data to calculate most of these problems, you still can do it with a little ingenuity and an open mind.  A computer would help a lot.  Mine saved me weeks of calculations using a pencil.

 
 

Goat in the field
Wrapped up flag pole
 Expanding Rail Road Track
Ladder and the box
   
 


The Ladder & the Box

Click here to zoom in closer
   This problem is a real fooler.  It looks simple enough but i have had a man with two degrees in mathematics tell me it is impossible to solve.  When i solved it, he refused to believe i did it, even though the answer checked out mathematically to be correct.
   At left is a model of a house.  On the left side of the house is a ten foot long ladder, leaning against the wall.   On the ground at the base of the ladder is a wooden box that measures 1 foot in all directions.  The ladder is touching the nearest corner of this box and the box is up against the wall as well.   So then, the ladder is touching the ground, the box, and the side of the house all at the same time.   How high off the ground is the point of contact with the house?


The Goat in the Field

Picture a circular field of grass.   This field is 100 feet in diameter and a steel stake is driven into a point on the perimeter of this field.   Attached to the stake is a 50 foot chain and on the end of the chain is attached a Goat.  The question is this:  What is the amount the field's total area will the goat be free to graze?  (answer in square feet)

    Assume that the distance from the stake to the Goat's mouth is exactly 50 feet, just to keep clutter out of the problem in the form of goat physiology.    This is another of those, "Gee, that doesn't sound too tough to figure..." category posers.   This looks so very easy to draw, that one fails to see how tough it is to calculate.  Calculate  your answer to three decimal accuracy.
 
 



The Wrapped up Flag Pole

    In this problem we have a big flag pole and a reel of one half inch diameter rope.   The Flag pole is tapered so that the base is much bigger than the very top of the pole.   The height of the pole is 40 feet and the diameter at the bottom of the pole is 24 inches.  The diameter at the top of the pole is only 3 inches.  You can see that there is a taper to the pole.   Now take the rope, and starting at the bottom of the pole, begin wrapping the flag pole with the rope.  Keep each successive coil touching the previous one so there are no gaps between the coils of rope.  Continue until the entire pole has been wrapped, from the very bottom to the very top of the pole.   The problem is: What is the length of the rope it took to completely wrap the flag pole, top to bottom?
    This problem sounds tough doesn't it?   Well it is tough.   It is so tough i almost gave up on it when it was presented to me.  However i seldom ever give up 'completely' so i just "filed the flanges off my wheels" and did some creative thinking with the right side of my brain instead of Mr. Mathematician on the left.  Pretty soon i figured out a simple solution that worked out just fine.
 


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The Expanding Rail Road Track

 
 

     At left is a rather poor and out of scale sketch of the problem.   What we have in the top picture is a rail road track that measures exactly one mile long.   At both ends of the track are "super stoppers" that will prevent the track from expanding if it warms up.    
     Now the sun comes out and shines on the track, causing it to grow by one foot (12 inches).   Since the track can not expand, it bulges up in the middle as shown in the bottom picture.  Imagine this to be a true radius.   Now the track is one foot longer than it used to be and it is raised up off of the ground by an unknown distance.   The question in this problem is:


How high off the ground is the highest point under the rail road track?


 

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