A man is going on a thru-hike of the Appalachian Trail. The trail is 2160 miles. He has a disposable camera containing 40 photographs for him to take. Approximately how many miles apart should each photograph be taken in order for him to capture a glimpse of the entire trail, rather than running out of photos before the trail ends or having photos left over after the trail ends?
(made it up myself, because my friend is embarking on an appalachian thru-hike and I'm giving him a disposable camera to send to me when he (hopefully) successfully finishes his journey! and I want to see some shots from beginning to middle to end! boy I hope he makes it :D)
(made it up myself, because my friend is embarking on an appalachian thru-hike and I'm giving him a disposable camera to send to me when he (hopefully) successfully finishes his journey! and I want to see some shots from beginning to middle to end! boy I hope he makes it :D)
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A-ha! This is a great problem for mathematicians AND computer scientists. It's called a "fencepost problem", and the reason it's called that is actually explained quite easily in your version of the problem. Instead of pictures, imagine you were making a 2160 mile fence, and you had 40 fenceposts to be placed equally.
You might think that the answer is 2160/40 miles apart, but it's not quite right! Think of a simpler version of the problem. If you had 2 fenceposts, would you place them 2160/2 miles apart? Nope! That's because you'd place one fencepost at the 0 mile marker, one at the 1080 mile marker, and....you're out of fenceposts! You would place them 2160 miles apart.
If you had THREE fenceposts, you'd place them 2160 / 2 = 1080 miles apart. You have to subtract 1 fencepost because you place one at the start!
So, for your problem, you'd take pictures every 2160 / 39 = 55.38 miles apart. But notice that that's pretty close to 2160 / 40 = 54 miles.
Fencepost problems are important in math and computer science because sometimes that one "extra" post makes a difference! You might be tempted to say that there are 50 integers between 50 and 100 inclusive...but there are 51! Notice that the numbers between 50 and 100 would be the same as the numbers between (50-49) and (100-49), which is just the numbers between 1 and 51, inclusive. Then it just becomes a counting problem.
You might think that the answer is 2160/40 miles apart, but it's not quite right! Think of a simpler version of the problem. If you had 2 fenceposts, would you place them 2160/2 miles apart? Nope! That's because you'd place one fencepost at the 0 mile marker, one at the 1080 mile marker, and....you're out of fenceposts! You would place them 2160 miles apart.
If you had THREE fenceposts, you'd place them 2160 / 2 = 1080 miles apart. You have to subtract 1 fencepost because you place one at the start!
So, for your problem, you'd take pictures every 2160 / 39 = 55.38 miles apart. But notice that that's pretty close to 2160 / 40 = 54 miles.
Fencepost problems are important in math and computer science because sometimes that one "extra" post makes a difference! You might be tempted to say that there are 50 integers between 50 and 100 inclusive...but there are 51! Notice that the numbers between 50 and 100 would be the same as the numbers between (50-49) and (100-49), which is just the numbers between 1 and 51, inclusive. Then it just becomes a counting problem.