My conjecture that 31 might be the first rotationally symmetric set of radii inexpressible in radicals is on the ropes. Corey and I worked out the equations for David's 27 (http://www2.stetson.edu/~efriedma/cirRcir/), but Reduce can't solve them algebraically anytime soon. Good old FindIntegerNullVector and FindRoot (999 digits) quickly give degree 20 (again?!) polynomials for the radii of all but the light orange disks, where PSLQ complains: FindIntegerNullVector::lowpr: FindIntegerNullVector cannot find an integer null vector because of insufficient precision of the input. >> Even with 9999 digits. LatticeReduce comes to the rescue, and says it's just a 10th degree over sqrt3, like the other 5. Here is the mysterious radius, in case you want to experiment: Root[4782969 - 1111774572 #1 + 44221126878 #1^2 - 307992419388 #1^3 - 8070585578919 #1^4 + 90167866460064 #1^5 + 150017012821404 #1^6 - 1381565181544056 #1^7 - 2541769632528930 #1^8 - 12793102409225160 #1^9 - 16223229604493784 #1^10 + 99440200315651104 #1^11 + 127666170124763526 #1^12 + 245161940234004576 #1^13 + 448033243426240932 #1^14 + 23246278469793432 #1^15 - 14112704124290727 #1^16 + 448087696965684 #1^17 + 19969942229946 #1^18 + 37004992764 #1^19 + 11771761 #1^20 &, 13] ~0.1760155767437583601645544130545779438560210712086956361545336164666108949916301221 Note that this is the only size which is surrounded by six different sizes. Finally, for David's Sum(radii) metric, Out[23]= s == Root[772690429687389147028310024964029054460615661037074186965170657489148640157424 - 1252150034158679892073592213738221544763982273185101716937850019400615469267216 #1 + 894101646306967418990714166469728854086811534040961919825626454389349568482607 #1^2 - 370535002186753148816065781791793841373361064007009093917305796349950654837894 #1^3 + 98585499109277440100892024783571503674461841813340963516397348934746393551831 #1^4 - 17537282099175471585324919382047103164067402676914160496887027103365237985296 #1^5 + 2095329496883146641591880827676789201936700111609182408312499347314971125404 #1^6 - 162581406984533745130162686637360629662639564698220123332118975994424295496 #1^7 + 7315174101074478959948591907336518346543812578462103252159725113209934156 #1^8 - 110179967256584058265325808692517286196365022221976714892569942238719952 #1^9 - 5210786844166514462554894258016492610911648226017109091287319543880958 #1^10 + 235504976723519352232932797704723384517491437953853656479644035104764 #1^11 - 822038136083999158156413079062172274783684008146700851353930742942 #1^12 - 78272319479518900556761239891321970600088764432072085939114898288 #1^13 + 560802701967280190031324433718254607649684741002636396646811692 #1^14 + 8012926781889727306623190565874256627453257915557081010606840 #1^15 - 22160201351067819572752983769650479439077975537076544188276 #1^16 - 77873705468844592325047782942025877646289806118814532768 #1^17 + 173017798526655483804206798601616162580746057680701703 #1^18 \ + 10267478177298250805153325223822351093391050035930 #1^19 + 118755914270972677050504275876817937004163071 #1^20 &, 8] In[24]:= N[%[[2]], 22] Out[24]= 4.762847452755912443159 This factors over sqrt3, but Root won't accept noninteger polynomials. rwg>May[be] he was seeing double after attempting a cholecystectomy? According to www.rhinoresourcecenter.com/ref_files/1178936542.pdf (p5?): black rhinos don't have gallbladders. (I wonder if the prosectorium was secure against Chinese herbalists.) The rhinos probably turn black when all that leaking bile collects in their skin. Somebody should cross a monotreme with a marsupial and name it Monomial monomial. --rwg