double false position I have this question for a capstone class and im not sure how to even go about it or what the question is saying necessarily. I cannot seem to find it anywhere else on the internet where it is explained.
Use the method of double false position to solve the following problem: suppose that a man went into an orchard and picked some apples. But the orchard has 3 gates and each is protected by a watchman. The man then (equally) shares the apples with the first watchman and gives him two extras. Then he shares his packing with the second watchman and gives him two extra. Finally, he shares what is left with the third watchman, also giving him an extra two, and leaves the orchard with one apple. How many apples had he picked? try $H_1=100$ and $H_2=40$
I know how to use false position normally, where you make a "convenient" guess that eliminates fractions, then alter the input value based on how off it is based on the original example. I'm not really sure how to even format this question to answer it, or what it means by double false position. Do I just use false position twice? Or is there some way to use false position on two variables? ($H_1$ and $H_2$ being these two variables maybe). Even if either of these are the case im not sure why you would try using $100$ and $40$ as numbers.
Anyone with any ideas about how this works would be much appreciated as I am thoroughly stumped...
Thanks!
 A: Before talking about the method of double false position, let's talk in modern algebra terms one of the steps involved in the use of this method. 
Exercise
Given a line that passes through the points $(x_1, y_1)$ and $(x_2, y_2)$, find its $x$-intercept. 
Solution
The slope of the line is $m = (y_2 - y_1)/(x_2 - x_1)$, so the $x$-intercept is found to be
$$ x_0 = x_1 -  y_1 / m = \frac{x_1 y_2 - x_2 y_1 }{y_2 - y_1} \tag{*}$$

Now onto the method of the double false position. 
Double false position
Given an affine function $x \mapsto y$, for which we wish to solve for the value $x$ such that $y = 0$. The method asks you to 


*

*First make two "false guesses" $x_1, x_2$. 

*Use the affine function to evaluate $y_1, y_2$. 

*Use equation (*) to find $x_0$. 


(Historically, the affine function is often given as an "algorithm" for arriving at $y$ given $x$. Before the advent of modern algebraic notation, the notion of reversing the algorithm may be difficult to conceive. Hence the method of the double false positions, which only involve applying the algorithm in the "forward" direction.)
Applied to your story
If you let $f(x) = x/2 - 2$, your story can be written given by finding $x$ such that $f(f(f(x))) = 1$, or $f(f(f(x))) - 1 = 0$. 
By the method of double false position, you would first make two random guesses, say $x_1 = 100$ and $x_2 = 40$ as in the hint. 
You will then find
$$ f(f(f(x_1))) - 1 = f(f(48)) - 1 = f(22) - 1 = 9 - 1 = 8 $$
and
$$ f(f(f(40))) - 1 = f(f(18)) - 1 = f(7) - 1 = 7/2 - 3 = 0.5 $$
So we conclude
$$ x_0 = \frac{100 \times 0.5 - 40 \times 8}{0.5 - 8} = \frac{270}{7.5} = 36$$
(Check: after the first gate the men has 16, after the second gate 6, after the third gate 1.)
