Mensa Math Puzzle Solution

 

The key to solving this math puzzle is to first re-configure the puzzle diagram as shown above to more easily apply the geometry and algebra.  We see immediately, for example, the large radius R can be expressed either as:

R = 210 +  y    or:

R = 330 + z

And:  z + 330 = y + 210

So:  z = y - 120 (solving simultaneous eqn. in y, z)

Then:  z +330 =  x + z + 165  

So:  x = 165

Also:  r = x + z

Further, from Pythagoras:

x² + y ²   =  r²  =  (x + z) ²

x² + y ²   =  x ²  + 2xz  +  z ²

è

y ²   =   2xz  +  z ²

y ²   =  330 (y - 120) +  (y -  120) ²

y ²   =  330 y  - 39,600  +  y ²  - 240 y  + 14,400

è

90 y = 25,200

y =  25,200/ 90  =  280

But:  R = 210 +  y =  210 + 280 = 490

If: r = x + z    and x = 165  

Then, from the geometry:

r =  R -  x    

r  =  490 -   165 =  325