How to let the creative mind play its part in solving problems
Many years ago I was much influenced by one of those little talks Radio 3 (the Third Programme in those days) used to sandwich into the intervals in live broadcast concerts. It was about the divination method of the Dogon tribe of Mali in West Africa. (My memory of details may be a little inaccurate after all this time, but the principle holds good.)
Apparently they have two gods, Nommo and the Fox. Nommo represents order and the beginnings of things, and is fixed and rather crystalline. The Fox represents disorder and change. A person with a question will draw a diagram in the dust near the road, the diagram being a box divided horizontally into upper and lower sections, representing respectively the world at large and one's immediate neighbourhood or household. The box is also divided into three vertically, representing respectively Nommo, the questioner's own place in the world, and the Fox. One might think of this division as representing past, present and future, but that perhaps is more our own way of thinking in linear time. Thus we have six cells each representing a part of our world.
The person asking the question will place various sticks and pebbles in the various cells, representing aspects of the problem or people involved in the problem. The interpretation of the diagram to anyone else will not be obvious because the enquirer doesn't want everyone to know his or her business. He or she will then also place some pieces of meat around the diagram, and go home and stop thinking about it, confident in the knowledge that the question will now be answered.
In the night the foxes will come, eat the meat and in the process scatter the sticks and pebbles around. The enquirer returns in the morning and interprets the diagram. A crude example might be, should I make the journey to the distant village or not? If the pebble representing the enquirer has moved from the lower middle cell to the upper right cell, the answer is obviously 'yes.' However the interpretation will usually be subtler than that.
The point of this process is that the enquirer has gone through an intellectual process of wrestling with and defining the problem in clear enough terms that the question can be expressed using the diagram. Then the intellectual mind rests, because at this stage reason would otherwise produce an unending stream of 'yes, but' contradictions.
Meanwhile the creative part of the mind works quietly on the problem, hence our modern advice to 'sleep on it.' The solution is allowed to emerge during the interpretation of the oracle.
Similarly, Archimedes wrestled with the problem of how to determine the composition of the king's crown without breaking it. He had done all the intellectual groundwork. The creative part of his mind presented him with the solution while he was relaxing in the bath.