For a number n, let f(n) be the set of numbers gotten by splitting n^2 at the 0 digits. For example

29648^2 = 879003904

so f(29648) = { 4, 39, 879 }

Let S be the smallest set of numbers containing 2 and fixed by f. What is the largest element of S?

[ meaning "if s in S , then f(s) is a subset of S " ] a simple depth-first search finds the 103-digit number 6748117221949113364929892272889264116674288119996291 959417364228984144237685142782381391415539921278251 in S , at generation 37. that's the largest found so far, checked through 40 generations. i will try to push the search somewhat further. mike