How often have you gone round in circles?
Resisting the temptation to look at the comments where there may be an answer to the puzzle which at first you thought was proving difficult only because you had not spotted the correct approach to the solution, you slowly begin to discern that in order to solve it it may be that you need one more piece of information. What do you do? Like every good student sitting an examination or test, you read the whole question again, carefully this time, looking for that one additional piece of information that will cause the solution to appear, rather like those pieces of art which you must examine with cross-eyes to see what is really there, out of the mass (mess?) of mathematical nonsense that already litters the page, but it is not there. You have found all of the information that is to be had.
You try again this time to establish what the missing piece might be which would allow a solution to be found. You discover that if you knew any one of three things there would be a solution. You look for all three things in the information provided, only to discover that it is still not there. At that point you open up the comments on the question to find several answers, but they are all different. How could that be? There is only one correct answer, but the respondents are quite sure about their own answers. You also see that there are others who have understood correctly that insufficient information has been given: some even point out that the assumptions made by those who have found an answer are both extraneous to the question, and unjustifiable on the information given.
In the light of this discovery it behoves those who are tempted to solve questions posted in this forum to ask whether there is sufficient information to find the answer before attempting to solve the riddle.
With that in mind, I thought I would pose one myself, not in its most generalised form, but then neither in the most simplified. We have three spheres in three dimensional space, whose sizes and locations we know. A fourth sphere lies between them and touches the surface of each of the other three. We want to know the size and location of the fourth sphere.
Without any more information we must be content to describe the locus and radii of all spheres that satisfy the touching condition.
The additional piece of information you need is that the centre of the fourth sphere lies on the same plane as the centres of the other three spheres.
Aside you will find a diagram, in which the z-axis has been rotated so that it is vertical to the plane that all four spheres hold in common.
E&OE If you do think you need more information, please raise your hand.
Ans. There are two solutions, one for an inner circle as above, the other for an outer circle. If the solution is progressed correctly both solutions should fall out:
(233/90,41/90,231/90) (~3.34010206896552,~2.49456275862069,~12.56491310344830)
