@christianp@two_star What about going from a to b, where a and b are rational numbers (not sure it would work for real numbers)?
For example, from 0 to 1/2
Expand: 0 = 1 + 1-2
Swap: 1 + 1 - 2 >>> 1 - 1/2
Evaluate: 1 - 1/2 = 1/2
For real numbers, if more operations are allowed then, from example, from 0 to sqrt(2):
Convert 0 to 1/2 like previously shown
Expand: 1/2 = 1+1 - 2+(1/2)
Swap: 1+1 - 2 + (1/2) = 1-1 + 2^(1/2)
Evaluate: 1-1 + 2^(1/2) = sqrt(2)
What about going from 0 to pi, e and other numbers? Maybe allowing certain operations it would fork.
Also, if it is possible to go from a to b, then it may be done in only one iteration of expand - swap - evaluate. For example, from 0 to sqrt(2) in one step
Notice that I have just composed 0 to 1/2 and 1/2 to sqrt(2) in one step. So if we can demostrate that there is a way to go from a real to b real, it means that every real number has a equivalent formula in which we can replace the operation of it to obtain another real number we want. And that is a pretty nice theorem :)
I'm happy that you can get from any integer to any other integer, without thinking too hard about how to do that in detail.
I'm also happy that any game can be arbitrarily long, both in number of moves and length of the expression at any point in time.
Is there a constraint I could add that would make the previous statements sometimes false?
If I looked at shortest sequences of moves between two integers, would I find anything surprising?
Is it true that shortest(a,b) = shortest(b,a) for all a,b? I think so: given a sequence from a to b, reverse the order of moves and swap "expand" and "evaluate", and you get a valid path from b to a.
Would that be fun to formalise? Could I do it without saying what "a valid mathematical expression" means?
@christianp You should define "valid mathematical expression" better. Is division allowed? Is integer division allowed? Are negative integers allowed? Can one uses factorial, or other non-alphabetic operator?
A short sequence from n from/to n + 1, for every integer n, is:
@jcastroarnaud for example, I think that under a certain reasonable definition, the following is the shortest route from a to b for any a and b:
Square brackets mean "the number equivalent to the contained expression"
a
expand: b + ([a - b] + 0)
swap: b + ([a - b] × 0)
evaluate: b
Only positive integers, addition, and multiplication are permitted
Multiplication by 1 (which is trivial) is disallowed
I believe that in this case 1,2,3, and 4 are in unique single element equivalence classes. 5 and higher integers are in an equivalence class via the relation (2+3)->(2*3), iterated as needed. The shortest move between higher numbers becomes a bit tricky to calculate?
@christianp@Colinvparker@jcastroarnaud No, in my restriction I’ve forbidden operation swap when any of the arguments is 1. But that may not even be enough to make finding a path hard, as I said in the next toot.
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