5 ms·

I love “the tyranny of the Wagon Equation,” what a great way to put it. I’d never thought through this before, but it provides a lot of context on the way wars

I love “the tyranny of the Wagon Equation,” what a great way to put it. I’d never thought through this before, but it provides a lot of context on the way wars unfolded throughout history.

amluto 4y ago
It would have been fun to see the exponential behavior unfold. Want to supply one soldier for one week? No big deal. Want to supply one soldier for n weeks without resupply? Better have a huge amount of resources dedicated to just supply. Except I suspect the wagon equation is actually worse than the rocket equation. Rockets can have large fuel tanks, and rockets can throw away their stages en route. But if you like your army to continue to exist, you can’t throw away your wagon drivers, etc. So you have worse than exponential behavior — eventually any food-carrying resource runs out of the ability to carry its own food, and it can’t go farther even with exponential blowup. I suppose that, if setting up supply caches is in the cards, then the rocket equation might emerge again. If one person can carry enough for a 10-day round trip, then they could walk 4 days, deposit 2 days of food, and walk home. Repeat five times and there’s a full resupply at the cache. Now the person can deplete the cache by visiting it, filling their pack, walking four more days, caching two days of food, and walking all the way home. So they can go on a 4n-day one-way trip by doing ~5^n advance supply trips. At least once you boost a rocket to escape velocity, it keeps going!
- Someone 4y ago
  > eventually any food-carrying resource runs out of the ability to carry its own food, and it can’t go farther even with exponential blowup. That problem is called the “travelers across the desert problem” (https://en.wikipedia.org/wiki/Jeep_problem https://en.wikipedia.org/wiki/Jeep_problem) I would think it can reach arbitrarily far, but can’t think of an easy proof, or find one.
  - naniwaduni 4y ago
    The keys that makes the desert problem finite bounded are "can't leave supplies in the desert to pick up later" and "without starving any traveler on the way". Relax either restriction and you're back to exponential behavior, either by taking extra trips or leaving a lot of dead travelers.
    - Someone 4y ago
      I don’t see how that matters if you can have arbitrarily many carriers. Let’s say a carrier carries food for 3 days. To get one carrier 4 units out, send out 2 carriers, let them walk for one day, have one of them load up the other. The first has food for 1 day left and can return to base, the second has 3 days supplies and can walk to the finish. To get one carrier 5 days out, do the first part twice to get 2 carriers at distance 1 with a day of supplies and 2 with 3 days of supplies. The first two can return to base, the other two can walk to distance 2, reload, and let one carrier reach distance 5. The other has food to return to distance 1. There, it can be met by a carrier who left base just in time with 2 days rations left, and both can return to base. I think this scheme extends infinitely. To get one carrier N+1 days out, get 2 fully loaded N-3 days out, have one dash for the finish, and have one return to N days out, where they can be supplied with one day’ rations to return to N-1 days, etc. (Given the description of https://puzzling.stackexchange.com/questions/233/travellers-across-a-desert https://puzzling.stackexchange.com/questions/233/travellers-... and https://www.mathsisfun.com/puzzles/cars-across-the-desert-solution.html https://www.mathsisfun.com/puzzles/cars-across-the-desert-so..., I think “get at least one carrier across” is the correct interpretation of the somewhat imprecise text on Wikipedia. It certainly can’t be “get all across”, as that’s trivial if you don’t allow intermediate dumps)

I love “the tyranny of the Wagon Equation,” what a great way to put it. I’d never thought through this before, but it provides a lot of context on the way wars unfolded throughout history.

amluto 4y ago

It would have been fun to see the exponential behavior unfold. Want to supply one soldier for one week? No big deal. Want to supply one soldier for n weeks without resupply? Better have a huge amount of resources dedicated to just supply. Except I suspect the wagon equation is actually worse than the rocket equation. Rockets can have large fuel tanks, and rockets can throw away their stages en route. But if you like your army to continue to exist, you can’t throw away your wagon drivers, etc. So you have worse than exponential behavior — eventually any food-carrying resource runs out of the ability to carry its own food, and it can’t go farther even with exponential blowup. I suppose that, if setting up supply caches is in the cards, then the rocket equation might emerge again. If one person can carry enough for a 10-day round trip, then they could walk 4 days, deposit 2 days of food, and walk home. Repeat five times and there’s a full resupply at the cache. Now the person can deplete the cache by visiting it, filling their pack, walking four more days, caching two days of food, and walking all the way home. So they can go on a 4n-day one-way trip by doing ~5^n advance supply trips. At least once you boost a rocket to escape velocity, it keeps going!

Someone 4y ago
> eventually any food-carrying resource runs out of the ability to carry its own food, and it can’t go farther even with exponential blowup. That problem is called the “travelers across the desert problem” (https://en.wikipedia.org/wiki/Jeep_problem https://en.wikipedia.org/wiki/Jeep_problem) I would think it can reach arbitrarily far, but can’t think of an easy proof, or find one.
- naniwaduni 4y ago
  The keys that makes the desert problem finite bounded are "can't leave supplies in the desert to pick up later" and "without starving any traveler on the way". Relax either restriction and you're back to exponential behavior, either by taking extra trips or leaving a lot of dead travelers.
  - Someone 4y ago
    I don’t see how that matters if you can have arbitrarily many carriers. Let’s say a carrier carries food for 3 days. To get one carrier 4 units out, send out 2 carriers, let them walk for one day, have one of them load up the other. The first has food for 1 day left and can return to base, the second has 3 days supplies and can walk to the finish. To get one carrier 5 days out, do the first part twice to get 2 carriers at distance 1 with a day of supplies and 2 with 3 days of supplies. The first two can return to base, the other two can walk to distance 2, reload, and let one carrier reach distance 5. The other has food to return to distance 1. There, it can be met by a carrier who left base just in time with 2 days rations left, and both can return to base. I think this scheme extends infinitely. To get one carrier N+1 days out, get 2 fully loaded N-3 days out, have one dash for the finish, and have one return to N days out, where they can be supplied with one day’ rations to return to N-1 days, etc. (Given the description of https://puzzling.stackexchange.com/questions/233/travellers-across-a-desert https://puzzling.stackexchange.com/questions/233/travellers-... and https://www.mathsisfun.com/puzzles/cars-across-the-desert-solution.html https://www.mathsisfun.com/puzzles/cars-across-the-desert-so..., I think “get at least one carrier across” is the correct interpretation of the somewhat imprecise text on Wikipedia. It certainly can’t be “get all across”, as that’s trivial if you don’t allow intermediate dumps)