the direction of a reduction

Direction is the make-or-break detail of every hardness proof, and it is where careful students still slip. The slogan is: to prove your problem B is hard, reduce a known-hard problem A to B, not B to A. Picture it as borrowing strength. If a strong friend (the known-hard problem) can lift any weight only by handing the bar to you, then you must be at least as strong as they are. You did the lifting; their power flowed through you.

Here is the logic spelled out. Suppose A is already known to be undecidable, and you build a reduction A <= B. That reduction means: a solver for B would yield a solver for A. But A has no solver. Therefore B can have no solver either, so B is undecidable. The known-hard problem A is the source, your target problem B is the destination, and difficulty flows from A into B. Now reverse it and see why the wrong direction proves nothing: if you instead show B <= A, you have only shown that A could solve B, which (since A is unsolvable) tells you absolutely nothing about B. You learned that B is no harder than something impossible, a vacuous statement.

A reliable mantra: reduce FROM the problem you already know is hard, TO the new problem you want to indict. The known-hard problem is always the one being translated away; the new suspect is always the destination. Phrased with the <= symbol, you want A <= B with A the established-hard problem. The exact same discipline governs NP-completeness later, where you reduce a known NP-complete problem (like 3-SAT) to a new candidate to prove the candidate NP-hard; only the resource being tracked changes.