On July 16, 2012, at 0950 mountain daylight time, a Cessna 172N, N5204K, experienced a partial loss of engine power during level cruise flight. The pilot subsequently initiated a forced landing in an open field, and came to rest inverted near Rockford, Idaho. The certified flight instructor and student pilot were not injured. The airplane sustained substantial damage to the wings and the tail section. The airplane was registered to Teton Leasing, LLC, and operated by the pilot under the provisions of 14 Code of Federal Regulations (CFR) Part 91 for the instructional cross-country flight. Visual meteorological conditions prevailed, and a visual flight rules (VFR) flight plan was filed. The flight originated from Idaho Falls Regional Airport (IDA), Idaho Falls, Idaho, at 0835. The flight instructor reported he had intended to conduct a cross-country flight from IDA to Arco-Butte County Airport (AOC), Arco, Idaho, into McCarley Field Airport (U02), Blackfoot, Idaho, and back to IDA. The flight from IDA to AOC was uneventful. About 12 miles northwest of U02, the engine began to vibrate, and the engine revolutions dropped to approximately 1,900 rpm. Carburetor heat was applied, the mixture was set to rich, and the fuel selector was in the BOTH position. The airplane could not maintain altitude, and the flight instructor took control, and decided to divert for a landing at Rockford Municipal Airport (2U4), Rockford. Upon arrival at 2U4, the flight instructor determined the winds were calm, and elected to land on runway 16. During the landing sequence, the altitude and airspeed were too high to safely land on the runway. He subsequently aborted the landing, and initiated a forced landing to a wheat field. During the touchdown, the airplane collided with the wheat; the airplane's nose pitched down, and came to rest inverted. A postaccident examination by a Federal Aviation Administration (FAA) inspector was conducted on the engine. The visual inspection of the engine revealed no catastrophic failures; however, during the compression check, the numbers two and three cylinders had no compression. A closer inspection of the number three cylinder revealed damage to the piston head, and the exhaust valve was not present. The intake tubes for the numbers two and three cylinders, as well as, the exhaust system for the engine were removed. With the exhaust section removed, the FAA inspector was able to see the head of the missing valve embedded into the cylinder head. The valve stem was missing and not located. Inspection of the number two cylinder revealed a large amount of metal dust and debris in the intake and exhaust ports.