We give a possible explanation for the mystery of a missing number in the
statement of a problem that asks for the non-negative integers to be
partitioned into three subsets. We interpret the missing number as one of the
clues that can lead to a more standard solution to the problem, using only
congruence modulo five, and we give the details to the new solution, which is
based on an algorithm inspired by noticing alternating differences between sums
of elements of the same rank in the three sets. Our new solution is equivalent
to the partition consisting of numbers with remainders one or three modulo
five, two or four modulo five, and multiples of five, which we call the
standard partition. We then find all other similar statements with the same
pattern of sums, we apply the algorithm to them, and we describe all the
partitions obtained, up to a certain equivalence. There are 279936 different
such statements, they produce twenty different partitions (other than the
standard one) whose sets of the first five columns are not permutations of each
other, and only one of them (the one produced by the original statement of the
problem we study) is equivalent to the standard partition. Finally, we
construct infinitely many partitions equivalent to the standard one, and we
give a possible generalization and a sample partition problem asking for the
non-negative integers to be partitioned into four sets.Comment: 18 pages, 6 figure
