Pneumofoils

tao , 20 days ago (edited 20 days ago) to random

The "Wolf, goat and cabbage problem" https://en.wikipedia.org/wiki/Wolf,_goat_and_cabbage_problem is a classic logic puzzle, in which your job is to transfer a wolf, goat, and cabbage safely across a river in a boat that can only carry one of these at a time, without any of them eating the other. Counterintuitively, the solution involves temporarily bringing back over the river one of the animals that one had previously transported over the river.

I recently discovered by accident that this type of puzzle can be a helpful conceptual framework for trying to solve a system of nonlinear equations through a combination of quantifier elimination and introduction of variables. Let me illustrate this with a simple (and artificial) example problem: locate real numbers 𝑥,𝑦 obeying the constraints 𝑥>0 and 1/2<𝑥+𝑦²<√𝑥. Here, the mental model to keep in mind is that 𝑥,𝑦 are currently on one side of a "river" (representing unknown quantities) and the task is to get them to the other side of the "river" (representing quantities that can be solved for, possibly
in terms of other variables).

One can solve the problem using this framework as follows. The quantity 𝑧:=𝑥+𝑦² implicitly appears in the problem but is currently on the other side of the "river", since it is expressible in terms of 𝑥,𝑦. So let us "bring it back to our side of the river" by introducing it explicitly as a new variable; now we want to find real numbers 𝑥,𝑦,𝑧. that solve the constraints 𝑥>0, 𝑧=𝑥+𝑦² and 1/2<𝑧<√𝑥.(1/2)