Bigtone53 wrote:I am not sure that this is the case. The original faulty problem has to have two correct answers, ie the lengths of both the tangents between the circles have to be a whole number of cm. You have picked a special case where they both are but is this constraint satisfied as the circles are rolled apart? In other words, is your special case a unique answer?

My special case isn't the unique answer, as I mentioned to satisfy all the constraints there are 2 sets of combination: [

r1=7, r2=17, c=26, a=24, b=10] & [

r1=7, r2=23, c=34, a=30, b=16] (highlight between the square brackets to view them). It just so happens for these 2 sets the radius of the smaller circle remains the same, so we have a unique answer to the whole brain teaser you post.

Just have a look at this

pic for the general configuration.

If you are willing to relax one of the original constraints, e.g. r1 need not to be prime, then there are other workable combinations, where the tangents aren't perpendicular to each other. For example, in this pic, the combination I used is: [

r1=4, r2=11, c=25, a=24, b=20] (check carefully to see how they work out).

Perhaps next time I will post on how I searched upon all Pythagorean triples within the range 0..49 to find the only 2 possible combinations which satisfy all the constraints. (It isn't that hard, so I don't consider it as brute force.)

Bigtone53 wrote:There are many problems out there which can be simply solved by assuming that there has to be a unique answer or the problem would not be set (shades of UR!). Therefore picking the ideal circumstances to find

an answer therefore gives you

the answer. My favourite is

A solid spherical cheese has a cylinder of cheese removed from it, leaving a hole through the centre of the cheese from one side to the other with the axis of the hole going through the centre of the former sphere. The hole is 10cm long. What volume of cheese remains?

When solving this problem I didn't assume any uniqueness or things like that. The 2 equations I worked out are independent of whether the 2 tangents are at right angle to each other or not.

As for your spherical cheese problem, I don't think you can work out the exact answer without knowing the diameter of the hole, even if you know the radius of the sphere is 5cm, and the original volume of the cheese is 500pi/3 cm^3?