There are two Russian Army motorcyclists. In the section from a map given in our illustration below we are shown three long straight roads,
forming a right-angled triangle at C.
The General asked the two men how far it was from A to B. Pipipoff replied that all he knew was that in riding right round the triangle,
from A to B, from there to C and home to A,
his cyclometer registered exactly sixty miles, while Sliponsky could only say that he happened to know that C was exactly twelve miles from the road
A to B — that is, to the point D,
as shown by the orange line.
Whereupon the General made a very simple calculation in his head and declared that the distance from A to B must be ______. Can the reader
discover so easily how far it was?
Solution to the Problem:
The distance from A to B is 25 miles.
I don't see how the general was able to determine the distances in his head.
I needed to use several theorems from geometry to solve the problem.
Let a, b, c, d, and h be the lengths of the segments in the triangle where h = 12, a + b + c + d = 60, and you are trying to find the length (c + d).
See the diagram below:
We know that a + b + c + d = 60 and that h = 12.
We know that a2 + b2 = (c + d)2 (Pythagorean theorem).
We know that the altitude drawn to the hypotenuse of a right triangle divides it into two right triangles, each similar to the larger one.
Therefore, h is the geometric mean of c and d. So, h2 = (c)(d) or in other words, c d = 144.
From the similar triangles, we can show that a2= d (c + d) and b2 = c (c + d).
From the perimeter formula above, we know that c + d = 60 - a - b.
We can rewrite it as c + d = 60 - (a + b)
Then squaring, we obtain (c + d)2 = 3600 - 2(60)(a + b) + (a + b)2
After putting everything together and solving, we obtain (c + d) = 25 miles.
a = 20, b = 15, c = 9, and d = 16.
The diagram below gives all the correct distances.
Everything checks. The three right triangles are 9 - 12 - 15, and 12 - 16 - 20, and 15 - 20 - 25.
The perimeter is (a + b + c + d) = 20 + 15 + 9 + 16 = 60 miles.
Jacob Branson first sent in a correct solution using the geometric mean to set up his proportion. But then he sent in the following solution:
"Again another method that was much quicker was since the perimeter was a whole number then you know its a special right triangle like a 3,4,5 but
you would eventually times those in your head my 5 to come to be the numbers 15,20,25 and they add to be 60 and the hypotenuse is AB or as we now know it to be, 25."
I believe that this must be the solution that Dudeney was referring to in the problem (the general solved it in his head). So, Jacob gets
extra credit for figuring out two correct solutions.
Correctly solved by:
| 1. Brijesh Dave | Mumbai City, Maharashtra, India |
| 2. James Alarie | Flint, Michigan |
| 3. Jacob Branson * |
Mountain View High School, Mountain View, Wyoming |
| 4. Ashlee Rudy |
Mountain View High School, Mountain View, Wyoming |
| 5. Frankie Jenkins |
Mountain View High School, Mountain View, Wyoming |
| 6. Ivy Joseph | Pune, Maharashtra, India |