PRIZE PUZZLE FOR SEPTEMBER 2004
LADDERS AGAINST THE WALL
The "Ladders Against The Wall"
problem shown here is well known. I first came across this
problem when studying mathematics as a young man. I spent hours
trying to solve the equation for x and eventually gave up.
Coming back to it many years later I resolved it into a fourth
order polynomial in x^{2} which is still difficult (but
not impossible) to work out analytically. The generally best way
to solve it is using a numerical method with the help of a
computer program.
But this month I have chosen a special case. I believe that
this is the smallest version of the problem that has an integer
answer with integer values for the ladders and the height of the
crossover point. This should make it easier because a few
inspired guesses will resolve the problem.
Please send me the value of x by 6th October to be entered
into the prize draw. If you can find values of the three
parameters that sum to a smaller value than 219 and still give an
integer answer you will get a special mention! Best of luck.
PS: The ladders should be of different lengths.
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Scroll down for the answer
SOLUTION
The answer is 56. John Stafford showed this to be the
smallest puzzle of this type in integers.
I received two correct answers: one from John Stafford and one
from Clem Robertson, but there was no draw this month as there
were only two competitors (a recent new rule requires that at
least three competitors must take part in the draw).
Given that the result was a whole number, you could make a
rough scale drawing and measure the width, round to the nearest
integer and check the answer using pythagoras.
Another method is to just make some guesses and test to see if
they fit the geometry of the diagram.
The mathematical method for noninteger cases involves finding
the solution to a fourth order polynomial equation. I would not
expect anyone to attempt this, but numerical methods exist for
solving equations that could be employed. I have even
implemented such a scheme on Microsoft's Works Spreadsheet that
gives an accurate answer to 14 decimal digits.
