According to Fermat (and Andrew Wiles) no two cubes can sum to another cube. However, this does not necessarily apply to the sum of three cubes. I asked some pupils to try to find sets of three of the first dozen cubes which sum to a fourth cube. Mary and Norma told me that they had found the sets with two summands in common. Oswald found a set which had no summand in common with theirs. Peter then experimented with summands beyond the first dozen cubes and found a higher set of three, two of whose summands were sums in the previous sets, which summed to a fourth cube. What four cubes did (a)Oswald and (b) Peter find?