Finding solutions to 'balancing problems'.

This repo solves questions of the following form:

If [5, 4, 19], [8 7 31] share a pattern, fill in [2, 1, ?]:

a. 7, b. 8, c. 9, d. 10

The answer is then (of course!) pattern: a + c - 4 = 5b, answer: 7.

How does it work?

It works by generating a bunch of expressions and brute-forcing it. We assume that:

• None of the numbers are duplicated or unused in the pattern.
• There may be one 'optional' number in the pattern (e.g. 5 in the example).
• We only consider binary operators.

Reverse polish notation

We use reverse Polish notation (RPN) to parameterise each expression. An example of RPN and a normal expression is provided below:

Normal expression: (3 + 5) * 2 - 4 * 2 RPN: 235+42-

The evaluation rule is:

• Go left to right.
• When you see an operand, push it to a stack
• When you see an operator, pop two operands and evaluate. Push result back to stack.

So, in the example, we first see +, compute 3+5, then see *, so we compute (3 + 5) * 2, and so forth. Any expression can be converted to RPN, and vice versa.

Properties of RPN

An RPN expression having N operands must satisfy the following:

• There are N - 1 binary operators.
• The RPN expression is 2*N-1 long.

Suppose the expression consists of substrings vi, oi, where vi is made up of operands and oi of operators. Then, we must have:

• vi and oi must alternate. RPN is of the form v1o1v2o2 ...
• length(v1) = length(o1) + 1, and length(vi) = length(oi) otherwise.

Generating RPN

Given these rules, we can construct all possible RPNs with N operands via the following.

• Find all possible n_i such that such that sum of n_i is N.
• Create a base string with vi, oi, where length of vi is n_i found before. Length of oi is determined based on the rule from before.
• Generate all possible permutations of the operand symbols and operators. For each base type constructed from before, fill in the string with these permutations (substrings marked with v are filled with operand permutation sequentially, and the same for operators).

Brute-forcing.

With the RPNs generated, it is now simple to brute-force the problem from before. Outline:

• Generate RPNs.
• For each generated RPN type, try given examples of the pattern.
• If an RPN produces the same result for each pattern, try each candidate solution.
• If a candidate solution also produces the same result, report as answer.