By "hard" problems, I meant, at least in part, those question in philosophy and its ugly stepsister, politics, that have seemingly defied conclusive answers, and have certainly defied widely accepted consensus answers. But it also means any problem one is having difficulty coming up with an answer for by the usual means, whether in science, computer programming, personal finance, relationships, wherever such problems arise.
Sideways thinking brings back into the question the most important aspect of context, the thing that anchors all other contexts. It is a question left implicit in the process of abstraction, but eventually ignored altogether. The question is "why do you want to solve this problem?" The arguments become heated because each participant has his own answer to that question, his own context, yet both sides implicitly agree that what they are seeking is a wider abstraction that transcends that question.
This is not to say that in such arguments, both sides are right. In fact both sides are wrong, but that does not imply the opposite, that either side is right because one is wrong. What it says is that they have divided the problem space along an axis that is perpendicular, or orthogonal to both positions. An answer is impossible to arrive at, because, in a very real sense, they are not arguing about the same question. Nor does thinking sideways mean bringing the context down to only the most concrete and immediate one, eliminating any abstractions from the realm of possible answers. Abstraction is a continuum from concrete to fully abstract. Bringing in the question of "why?", the context of value, does not obviate abstraction, it merely brings it down from the furthest extreme of that spectrum where the answers are "pure" abstractions.
Take the question of the existence of God, for instance. That is a very "hard" question by the criteria above, though most people think they know the answer, and that it is not hard at all. I am actually in that group, coming down on the side of "no", but that does not mean that the problem cannot be looked at sideways, and that doing so cannot produce useful results.
Think of the question as a space, an area like, say, a piece of paper. This is the "problem space". In this space, all the possible answers lay, as well as all the facts, ideas, beliefs that inform the debate. The usual approach to the problem is to metaphorically draw a vertical line down the center of the paper, label it "God's existence", and then argue about whether any given aspect of the problem, including the "right" answer is on the "yes" or "no" side of the divided space.
Just to arbitrarily label the two halves of the space, on the left is "yes", and the right is "no". God lives in the left half of the page, he is absent from the right half. So here's how to think sideways about the problem: Cut God in half. Draw a horizontal line across the page, intersecting the axis of "existence". This new axis is orthogonal to the first axis. It is sideways relative to it. On both sides of this divide, God exists, and He doesn't exist. We are, in effect, assuming the conclusion, or rather, assuming both conclusions. Since the axis no longer divides existence from non-existence, it is assumed by our choice of a perpendicular axis that both are true, for the purpose of argument.
Where do we draw this line? We have a lot of choices here. We're making up an entirely new axis, and it can be anything we want it to be. In the course of an argument over God, one side may bring up objections or positive assertions that seemingly have nothing to do with the question of existence. They are usually questions of purpose, of value, such as "God created the universe", or "God keeps us moral", or "Religious belief in God leads to oppression and violence against those with other beliefs." These are in fact not arbitrary, but are indicators of useful sideways axes.
Let's take that first statement and make it an axis, called "Created the Universe". Actually, lets change the name a little to reflect the orthogonality of this axis to the one of existence, and call it "Caused the Universe to Exist". By itself, this does not seem to help us answer the question of God's existence. Both "God exists", and "God does not exist" inhabit the newly formed half of the problem space that represents "Did cause the universe to exist", and they both occupy the half of the problem space that represents "did not cause the universe to exist". It is not meant to answer the question of His existence. We've, temporarily at least, abandoned that axis in favor of our new one.
It may, however, help us think about the problem space in a new way that may or may not lead us to answers to the question of existence. Sideways thinking is not a tool for coming to an answer to contextless "ultimate abstraction" questions, it is a way of re-framing the problem entirely to keep it in context, and it leads to multiple re-framings in multiple contexts. The ultimate answer may arise out of the aggregate of these new divisions, but it is not necessary that it do so. The out of context abstraction may in fact not be answerable, at least not in any useful way.
So what do we do with this new axis? Well, we've cut God - or at least the idea of God - in half. There's that portion of God-that-does-exist that lays on the "did cause the universe to exist" half of the page, and that portion of God-that-does-exist that lays on the "did not cause the universe to exist" half of the page. Our job then, is to discover which parts of God are which. We assume that God exists, and by our new axis, we define "God" as "that which caused the universe to exist". What parts of God fit the description? What properties are required for causing a universe to exist?
Pre-existing the universe is certainly one of those necessary properties. It is also a necessary property of any non-God thing that might cause a universe to exist. Is being a living, conscious being also a necessary property? Probably not. Though those who believe certainly believe that God is both a living, conscious being and that he created the universe, one is not strictly necessary to the other.
It is possible by this process that we will find no properties that are both necessary to causing a universe to exist and that are actually possible. In any such division along a context-based axis, the empty or null set is one possible outcome. This would not, though, disprove God's existence. It would in fact show that there is no possible thing, God or otherwise, that caused the universe to exist, leaving only the possibility that the universe is uncaused, Maybe because it always existed, or maybe because it sprang into existence literally from no thing and no cause.
It is also possible that properties that can exist, but that are not part of what anyone thinks of as properties of God will be found to be necessary to cause a universe to exist. This would still not prove that God does not exist - we are not concerned with that axis now - but it would show that, existing or not, He didn't create the universe. And of course, it is a technically possible outcome to discover that the only thing that could possibly have created a universe had to have been living, conscious, all-loving, all-powerful, and all-knowing, and that these properties are in fact possible to all exist.
I'm not going to solve the question of God's existence or the creation of the universe today. I just use them to illustrate a technique I find helpful. There are many other questions that can be approached the same way. "Does the universe even exist, or is it a figment of our imagination?" The "Matrix" question. Look at it sideways. The universe is that which I am conscious of. What properties must it have? Even if I am conscious of nothing but my own consciousness, that is something that exists, and thus my consciousness is the entire universe. (I don't believe that, but examining that is outside the scope of this work.)
"What is the universal human nature?" becomes "What properties of humans are universal?", what properties fall on the "is human nature" side of the new axis. Certainly all humans breathe, and eat, and excrete, but those aren't really the kinds of things we're after. The question of value that must be brought in is usually that of elements of human nature that support theories of morality, ethics, and rights, so a better axis might be to assume a universal human nature that supports those theories does in fact exist, and then examine what properties it must contain, and whether or not those are in fact universal properties of human beings. It could always be a null set, meaning there is no "human nature" in a philosophically relevant sense. In any case, this new axis provides a rich vein of thought to mine for useful answers.
"How big should government be?" is another popular subject of debate. Arguments like "who will build the roads?", "Who will defend us from invasion?", "Government violates rights" commonly come up. So instead, start with "protects our rights" or "builds roads" or "defends against invasion" as possible axes, either side of which contain both "giant all-controlling government" and "no government" and look at which properties are necessary to build roads or protect rights. Is "collects taxes involuntarily" a necessary property? That government that should exist, at least for the purposes above, is whatever entity has the necessary properties.
It's not my purpose to address these questions in detail, nor to conclusively settle millennia old arguments. This is merely to illustrate a way of thinking about such problems that, while it may not provide ultimate answers, can provide useful answers. Sometimes, especially in philosophy, we have to settle for useful rather than ultimate. I personally prefer it that way, since I see philosophy as a tool for living the best possible life. The ultimate questions and answers are less interesting to me than questions that help me find answers to how to do that. Of course, this question can itself be subject to sideways thinking. I leave how to do that as an exercise for the reader.