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© Copyright 2001, Jim Loy
This famous puzzle looks like it should be relatively easy. We
have two buildings with a gap between them. Two ladders lean from one building
to the other, as seen in the diagram. We know the length of each ladder (105
and 87 in the diagram) and the height at which the two ladders cross (35 in the
diagram). What is the gap x between the two buildings?
Other dimensions are possible. A reader sent me these: 30, 20, 10.
Exact solution: It turns out that finding an exact solution is very difficult. Two books tell me that if the lengths of the ladders are a and b, and the height at which they cross is c, then the height of the top of ladder b is k in the following equation (k^4 means k to the 4th power):
k^4-2ck^3+k^2(a^2-b^2)-2ck(a^2-b^2)+c^2(a^2-b^2)=0
Solving a 4th degree equation is a little difficult. If you can
find k, then you can easily deduce x using
The Pythagorean Theorem. If we keep the
lengths of the ladders constant, and vary the gap between buildings, the
intersection point follows an interesting curve (diagram on the right using
Cinderella). I have continued the curve
below street level, and allowed x to go negative, to show the entire curve. It
is certainly a fourth degree equation (related to the above equation).
Computer estimate: But there is an easy way to estimate x, without dealing with any fourth degree equations. We can write a computer program which keeps the lengths of the ladders constant, and varies x, just as in my graphing routine. This program can calculate the height of the intersection (using proportions and the Pythagorean Theorem). See the next paragraph for the equations. As the height of the intersection goes from a number lower than that given to one higher than that given, we get upper and lower bounds for x. The program then uses those bounds, and a smaller increment for varying x. For the reader's problem (30, 20, 10), it rapidly zeros in on x=12.31185724.
Without a computer: From the dimensions that I gave originally (105, 87, 35), we might guess that there is an integer solution. In that case, we can do without a computer. As in the previous paragraph, we try different integer x values, and see if one of them gives a height of 35. We only have quadratic equations to solve. The heights of the two ladders are h=sqrt(a^2-x^2) and k=sqrt(b^2-x^2) by the Pythagorean theorem. And the height of the intersection is c=hk/(h+k) by similar triangles. We can try many x's, and we will find that x=63 is the answer. There is only one solution.
The Crossed Ladder Problem is sometimes stated as "Find smallest values for the various lengths (with the ladders being of unequal length) so that all of these lengths are integers." And we don't want the shorter ladder to be lying flat on the ground. Which lengths are specified, differs from puzzle to puzzle. It is not entirely clear what an ideal integer solution is, since we have several interacting variables. Let's say that a ideal solution is where we minimize the longest ladder. Here is a table that I get:
short ladder long ladder height of crossing x 87 105 35 63 100 116 35 80 70 119 30 56 119 175 40 105 74 182 21 70 174 210 70 126 182 210 45 168 156 219 44 144 200 232 70 160 113 238 14 112 140 238 60 112 175 273 90 105 104 296 35 96
And my diagram at the top was our ideal puzzle.
Also see Another Ladder Problem and Yet Another Ladder Problem.