Based on Pollock's conjectures regarding sums of Platonic numbers, the functions test whether every positive integer in a given range can be written as at most:

• 5 tetrahedral numbers
• 7 octahedral numbers
• 9 cube numbers
• 13 icosahedral numbers
• 21 dodecahedral numbers

All were conjectured to output the empty set, however we observe non-empty outputs for the icosahedral and dodecahedral cases. The smallest counterexample to the icosahedral case is 47, and the smallest counterexample to the dodecahedral case is 79.