Restating a problem into one you have solved

Some problems are a solved problem behind a change of variables. Finding the change is the work.

Why it works

Two groups, and simple algebra

Every assignment splits the values into a positive group and a negative group. Call their sums P and N. Then P minus N is the target, and P plus N is the overall total, which is fixed.

Adding those two equations gives twice P equals the target plus the overall total, so P is half of that. The question becomes: how many subsets sum to that value?

That is a counting subset-sum, which you can already write. No offsets and no negative indexes.

The algebra is the whole lesson, so here it is on a concrete input, stopping at the point where the problem has been replaced.

1values = [1, 1, 1, 1, 1]2target = 33 4total = sum(values)5wanted = (total + target) / 2          # P, the positive group's sum6 7print("overall total:", total, " target:", target)8print("the positive group must sum to:", wanted)9print("whole number?", wanted == int(wanted), " within reach?", abs(target) <= total)10print("question is now: how many subsets sum to", int(wanted), "?")

A sign choice has become a subset choice, and that is a problem already solved.

The habit worth taking

When a problem involves choices with opposite effects, or a difference that must hit a target, try writing the two groups as unknowns and solving. It frequently collapses into a standard problem.

Five minutes of algebra beat an hour of building a table around an awkward state. Do the algebra first.

Predict the output· 5 min

Predict the output: when the restated problem has no answer

The algebra turns a sign assignment into a subset sum. Three requests are checked against it before any table is built. Type the three lines it prints.

Python

1values = [1, 1, 1, 1, 1]2total = sum(values)3 4for target in (3, 2, 9):5    reachable = abs(target) <= total6    whole = (total + target) % 2 == 07    wanted = (total + target) // 2 if reachable and whole else None8    print(f"target {target}: within reach {reachable}, whole number {whole}, subset sum {wanted}")

Printed output

Problem· 28 min

Target Sum

Read the constraints first and let them tell you what complexity is expected. Derive the approach, implement it, run the tests, and submit when it passes.

1values = [1, 1, 1, 1, 1]2target = 33 4total = sum(values)5wanted = (total + target) / 2 # P, the positive group's sum6 7print("overall total:", total, " target:", target)8print("the positive group must sum to:", wanted)9print("whole number?", wanted == int(wanted), " within reach?", abs(target) <= total)10print("question is now: how many subsets sum to", int(wanted), "?")

1values = [1, 1, 1, 1, 1]2total = sum(values)3 4for target in (3, 2, 9):5 reachable = abs(target) <= total6 whole = (total + target) % 2 == 07 wanted = (total + target) // 2 if reachable and whole else None8 print(f"target {target}: within reach {reachable}, whole number {whole}, subset sum {wanted}")