You draw a square. Then, you draw the largest possible circle, tangent to all four sides. If the lower left corner of the square is at (0,0), and the sides of the square are parallel to the coordinate axis, the point (2,1) is on the circle, within the square.
What is the radius of the circle?
This is why I enjoy this site... :-)
Various dictionary definitions of "within" are of no help because they generally do not approach things mathematically or recognize the difference of "on" vs, "in". SO:
DEF; "Inscribe" (not used by Federico in this problem, but hopefully a more exact statement of what he intended)
- Mathematics. To draw (one figure) within another figure so that every vertex of the enclosed figure touches the outer figure.
This definition is similar to many found on the web. Note the usage of "within" and "touches". (Please allow some slack - a circle has no vertices, but we all know how to apply this definition to the problem at hand). This, it would seem continues the hazy situation.
However, falling back on an old logicians trick and taking a step back, I am leaning towards the argument (in a broad sense), that IN THIS CONTEXT, "within" means "inside". The issue then becomes how to define "inside", when the boundary has no dimension. In your house, for example, you would say something is "inside" or "within" if it does not pierce the outer wall (can't see it from the outside). However I would also agree that if I had to cut into the inner walls to make something just fit, I would say "it didn't fit inside".
I think Federico chose his words carefully (as many do on this site). He did not need to use "within". The circle was already defined as an inscribed circle. He could have left out the "within the square", but did not. The math of this problem is not the issue. Perhaps Federico was really hoping to prompt this very discussion!. If so, Bravo
BTW - given all my ranting and raving, I change my mind and agree that the only solution is 5.
Edited on November 26, 2006, 7:45 am
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Posted by Kenny M
on 2006-11-26 07:43:54 |
Circular Argument
