Science Focus (Issue 23)

7 you do the division, you start with the total length of the fence (eight meters) and divide by the length of a segment. That will give you an answer of the number of segments of fence. But the question was about the fenceposts, not the fence segments. That means that when dealing with these kinds of problems, the important question is whether you need to count the numbers or the spans [1]. Once you can distinguish between fence segments and fenceposts, it’s quite easy to see the oddities in everyday counting conventions. In music theory, a third denotes an interval of three notes: C-D-E is a third, and so is E-F-G [2]. When you put them together you get C-G, a fifth. In other words, two thirds make a fifth. This is the same as making a longer fence out of two existing fences: You’ll find that there is an extra post left over since the “post” E has been counted twice. The issue is that thirds and fifths refer to the spans between notes but are named for the number of notes they contain: They count the posts instead of the segments [3]. Simi lar ly, fence segments can be disguised as fenceposts. When you celebrate your birthday what you’re actually celebrating is the number of years you were alive [2, 3]. You can even see it in the wording that most people use: On the f i r st bi r thday you celebrate, you turn one year old. The years are the fence segments; the birthdays are the fenceposts. In our original problems, you have four assignments to complete and what you need to do is match them with four time spans in which you can complete them. That means you have to consider the fence segments in the problem. In the wording of the problem, the times 1 p.m., 2 p.m., … , 4 p.m. are markers of time (fenceposts) whi le the days May 1, May 2, … , May 4 are time spans (fence segments) [4]. One gives a span of three hours, the other a span of four days. That’s why your cramming session can fit into one schedule but not the other. A related problem that thi s al so rai ses i s whether your count starts at zero or one [3]. We all know that the first floor can refer either to the ground floor or the floor above it depending on what country you are in. You have ten fingers, but it is possible to count 11 numbers with your fingers: Everyone forgets to include zero fingers [3]. But these are mostly linguistic conventions. More interestingly, think about labeling a series of fence segments #1, #2 … and so on and consider the question of how to label the fenceposts: It naturally requires a fencepost somewhere marked #0. (If you think of the fence segments as timespans spent alive, in years, and the fenceposts as birthdays then this all becomes a version of our birthday discussion above, in which the day of birth can be considered as the zeroth birthday.) This is where many of the problems of confusing fenceposts and fence segments come into play. For this reason, if you reread our or iginal problems about the eight-meter fence, you can see that the question of whether to start counting from zero is tied to whether you want to count fence segments or fenceposts. Counting fence segments is easy. Counting fenceposts requires you to remember that extra zero – zero fingers, the zeroth floor, the zeroth birthday – and add one accordingly [1]. So once again it’s very impor tant to know the context of your quest ion: Should you count the numbers or the spans; fenceposts or fence segments? What kind of fence are you dealing with? Just to leave you with something to think about, suppose the eightmeter preschool fence actually stretches between two walls and doesn’t need posts at either end. Or maybe it closes up to form an enclosure. How many posts are needed now?