Zeolites are porous aluminosilicate materials that have found applications in many different technologies. However, although simulations suggest that there are millions of possible zeolite topologies, only a little over 200 zeolite frameworks of all compositions are currently known, of which about 50 are pure silica materials. This is known as the zeolite conundrum--why have so few of all the possible structures been made? Several criteria have been formulated to explain why most zeolites are unfeasible synthesis targets. Here we demonstrate the synthesis of two such 'unfeasible' zeolites, IPC-9 and IPC-10, through the assembly-disassembly-organization-reassembly mechanism. These new high-silica zeolites have rare characteristics, such as windows that comprise odd-membered rings. Their synthesis opens up the possibility of preparing other zeolites that have not been accessible by traditional solvothermal synthetic methods. We envisage that these findings may lead to a step change in the number and types of zeolites available for future applications.