Fence Posts
Fine fancy formulas are ducky but I think you are a gentleman interested in going beneath the surface. The question is, how can you noodle up your own answer.
And, actually, as Subhotosh Khan has warned, it is not so simple as the final formula would have one believe.
You draw a diagram and count up the post holes, so really what you want is a formula that will do the counting for you.
Adding is shorthand counting. Being able to add circumvents the necessity of counting, no need to count two fingers, then three more to get five. Multiplying is shorthand adding. No need to add 5 + 5 =10 and then add five more to get 10+ 5 =15, you can just multiply 5 x 3 to get 15.
(Denis is probably pulling down on his lids to keep his eyeballs from popping out of his head in disbelief, but I am a plain thinker so let us proceed.)
With that understanding in order one need not walk around the perimeter with a ruler to measure it but rather, add together the lengths of the sides.
L + S + L +S to get 2L + 2S = 2(L +S) = 2(40+16) = 112, which, in fact is a MARVELOUS THING … and I am such a simpleton (dusting my lapels) as to be quite sincere in thinking so, as those who know me best will not contest.
Knowing the length of the fence one must now consider how that information can be used to determine the number of posts and here is where it gets tricky.
Consider a tape measure, the 1 meter mark, what does that refer too? I would suggest that it refers to the meter unit length between 0 and 1, the 2 meter mark refers to the length between 0 and 2, BUT ALSO, it reports the number of unit meter lengths, or, unit “intervals” between 0 and 2.
That is a fundamental point. By extension, 116 meters stands not only for a length of 116 meters but for 116 intervals of 1 meter each.
Translating length to intervals in our thinking allows us to ask, if there are 116 one meter intervals in our perimeter, how many 8 meters intervals are in our perimeter? Since the 8 meter interval is 8 times the unit meter interval it follows that there are only 112/8 = 14, eight meter intervals.
So now the question is, does the number of intervals equal the number of fence posts needed? Yes or no?
Consider a 3 meter fence?
0…..1…..2…..3, The fence is 3 meters long, 3 intervals, BUT, FOUR FENCE POSTS ARE NEEDED, A POST AT 0 !!!! Don’t tell me that I need to go that 100 miles back to the store! Do I need 15 rather then 14 fence posts?
No, why, because when you closed the perimeter you wrapped around to use your last fence post as the zero fence post. So Denis’s formula which we derived too, is correct. Whew (for him).
The points to remember in my view are:
1) A marking on a rule names both the length and the number of unit intervals preceding it.
2) One can change the unit of measure by calculating a ratio of the old unit length to the new unit length.
3) One must remember to add an additional 1 if you must enclose a length without wrapping the end back to the beginning.
Hope this makes sense, helps.
