I think this would make a good puzzler, what do you think?
A man w/bow and arrow stands at the center of a regularly spaced rectangular grid, Cartesian x/y grid in other words, with trees located at each grid point. Assume the trees are narrow enough or the trees are far enough way that it doesn’t matter how close the shooter is to a particular tree. And that the arrow moves in a straight horizontal line indefinitely. The shooter turns (in a circle) a random number of degrees and fires. What is the probability the arrow will (eventually) hit the exact center of a tree?
the trees could be spaced in any distances. If you prefer, consider the trees spaced on a 100 foot grid in both directions. A tree exists only at each grid intersection, and those go on to infinity. So it’s an infinite forest of equally spaced trees. This is an imaginary puzzler, so the trees are infinitely thin. But it is still (theoretically) possible to hit one. For example if the shooter aimed directly along a grid line they’d hit the first tree.
I don’t actually know the answer, but my guess is that there’s zero probability of ever hitting a tree. The reason is that to hit a tree you’ll have to aim at an angle that corresponds to a direction that can be described with an x and y coordinate . But that sort of direction corresponds to a rational number, a number than can be expressed as a fraction of two integers) . Since there infinitely more irrational numbers than rational ones, a random aim isn’t going to hit any tree.
Since the archer is at position (0,0) and s/he fires an arrow in a straight horizontal line, s/he will hit the very first tree at either (0,1) or (0,-1)… The only thing puzzling about this is that you either did not understand the Cartesian x/y grid or you thought we would not understand it…
The fact that you state, “The shooter turns (in a circle) a random number of degrees and fires.” Is irrelevant since you prefaced it with “fires an arrow in a straight horizontal line”
If you were stumped with this, I think you should never play, “I’ve got your nose…”
The theoretical arrow doesn’t follow the typical parabolic arc physics requires, but instead flies at a constant distance from the ground. Horizontal is not related to the aim direction. I think the concept of the shooter turning a random number of degrees before firing is pretty clear.
I understand the two dimensional puzzle. Each tree has width and distance. You said assume the trees are narrow enough, which is the most confusing part.
Really this is just a geometry problem. Given the width of any given tree and its distance, it is possible to determine the angle from the shooter to the left side and the right side of the tree. Take the difference and that is the direction range that the shooter must shoot at to hit that tree. If the difference is 1 degree then there is a 1 in 360 chance of hitting that tree.
The probability of the shooter hitting the exact center of a tree approaches zero as the decimal places in the random number generated increase. If the degrees are integers then 0 90 180 and 270 degrees would be direct hits if the trees are placed exactly at the middle. The 45 degree shots would be hits as well if the trees are placed exactly in the middle of squares. I think everything else would be a guaranteed miss wouldn’t it.?
This is a variation on “Olbers’ Paradox”, which asks why the night sky isn’t solid white with stars. If space was infinite and stars were distributed throughout, there should be a star anywhere you look. Why Isn’t the Night Sky White? | RealClearScience
The probability for me would be zero. Some years ago I pulled my bow out to give it a try. Used to be fairly good. So I pulled back the string and let the arrow fly to my intended target. Glanced off the tree and went directly under the winter cover of my neighbor’s pool. I quietly put the bow away never to be used again. Neighbor moved. Maybe thought he was under attack.
At least it didn’t poke a hole in the neighbor’s pool cover … lol … .good story !!!
The relationship to Obler’s Paradox is an interesting idea. I don’t know enough about that subject to have an opinion one way or the other. I need to study up more on that. This bow and arrow puzzler is related to the work of a 19th century mathematician named Georg Cantor. I wonder if Obler’s Paradox is related to that work too?
Like I said before some thing I just don’t understand so just follow directions. You’ll go nuts trying to think about the infinite sky and what is it contained in, and if contained in something, is the something infinite? Impossible to imagine.
Instead I have been studying ancient building techniques with cranes. And water wheels. Hard to understand how they did it but easier than infinite universe.
Olbers’s paradox, also known as the “dark night sky paradox,” is the argument that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. Since this assumption by someone (Olber…) who died before the US Civil War in 1840. Heck, many folk still thought that if your sailing ship got too far from land, you’d fall off the end of the Earth…
Double heck, the concept of the Doppler effect was not even conceived until two years later and that was based on train whistles, not light. The effect on light was not discovered until much later, as many scientists still believed the speed of light was infinite…
I believe the night sky is filled with stars… Our eyes just are not sensitive enough to see the light. As the stars recede from us (expanding space…) the number pf photons of light that reach us are fewer and fewer.
Remember, before the Hubble Telescope took the Ultra Deep Field photograph, astronomers thought that area of the sky was basically empty.
For Hubble to make that photograph, it took almost 2-weeks of exposures (that’s a lot of looking) for a total of almost 400 photos that were combined to come with the final photo that showed over 3,000 distant galaxies (with each galaxy containing billions or even trillions of stars).
Now, getting back to the issue of expanding space, as space expands, the light is stretched (Doppler effect) and the light is Red Shifted so far, the oldest, most furthest light, is stretched into infrared and even further into the microwave and even into the x-ray spectrum, none of which our eyes can perceive.
That is why the James Webb Space Telescope shoots in the red shifted light spectrum, to catch light that not even the Hubble Telescope can perceive.
And in the most extreme example of the night sky being filled with stars (at least with light…); it is, have you ever seen the photograph of the Microwave Background Radiation Map? That is a photo of the whole sky, in every direction, and there is light, it’s just so red shifted, only special instruments can perceive it…
You have no concept of how long an infinite amount of time is. A blindfolded monkey firing only once in his lifetime and replace by an infinite series of monkeys that do the same would eventually hit the tree.
The most extrem and famous illustration of this was the statement that a bunch of monkeys sitting at a typewriter would eventually produce the complete works of Shakespeare.