A C172 ON AN IFR TRAINING FLT ON V1 TO JFK VOR IS ADVISED OF TFC AT 12 O'CLOCK; OPPOSITE DIRECTION AT 4800 FT. THEY ARE AT 5000 FT AND INSTRUCTED TO CLB IMMEDIATELY. TFC PASSES 100 FT BELOW AND 100 FT HORIZLY.

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WHILE ENRTE FROM CGS TO ISP ON AN IFR FLT PLAN ON V1 TO JFK AT AN ALT OF 5000 FT MSL; WE WERE ADVISED OF TFC AT OUR 12 O'CLOCK POS OPPOSITE DIRECTION AT 4800 FT. IMMEDIATELY AFTER; WE WERE ISSUED A TFC ALERT AND INSTRUCTED TO INITIATE AN IMMEDIATE CLB. JUST ABOUT THAT SAME TIME I SPOTTED THE TFC AT OUR 12 O'CLOCK POS AND WHAT APPEARED TO BE THE SAME ALT. THE OPPOSING TFC WAS NO MORE THAN +/-100 FT OFF OUR ALT OF 5000 FT AND APPEARED TO PASS AT LESS THAN 100 FT FROM OUR R WING. THIS WAS CONFIRMED BY THE PLT OCCUPYING THE R SEAT; A 200 HR INST RATED PLT. I INITIATED AN IMMEDIATE TURN TO THE L; AND A CLB AS INSTRUCTED BY ATC. THE CTLR DID AN EXCELLENT JOB OF ALERTING US OF THE TFC AND ISSUING EVASIVE ACTION IN A TIMELY FASHION. HOWEVER; WHEN WE ASKED THE CTLR IF THE OPPOSING TFC WAS TALKING TO HIM; HE SAID NO. HE INFORMED US THE TFC WAS TALKING TO THE SECTOR WE WERE HANDED OFF BY 15 MINS EARLIER. THE CTLR FOLLOWED UP BY GIVING US A LCL PHONE NUMBER TO CALL TO FIND OUT EXACTLY WHAT HAPPENED. MY QUESTIONS ARE: 1) IF THE TFC WAS IFR OR VFR? 2) WHY WAS IT NOT ADHERING TO THE PROPER CRUISING ALT? IFR -- THE ACFT SHOULD HAVE BEEN AT AN EVEN ALT ON ITS SW HDG. VFR -- THE ACFT SHOULD HAVE BEEN AT AN EVEN ALT +500 FT. 3) WHY WAS THE ACFT ALLOWED TO STRAY FROM ITS PROPERLY ASSIGNED ALT WHILE IN CLASS B AIRSPACE? 4) WHY WERE 2 ACFT IN THE SAME AIRSPACE (SECTOR) TALKING TO 2 DIFFERENT CTLRS?