For any game or number G, tiny G (denoted by ⧾G in many texts) is the game {0|{0|-G}}, using the bracket notation for combinatorial games in which the left side of the vertical bar lists the game positions that the left player may move to, and the right side of the bar lists the positions that the right player can move to.
Symmetrically, miny G (analogously denoted ⧿G) is tiny G's negative, or {{G|0}|0}.
Specifically, though ⧾G is infinitesimal with respect to ↑ for all positive values of x, ⧾⧾⧾G is equal to up.
While the expression appears daunting, some careful and persistent expansion of the game tree of ⧾⧾⧾G + ↓ will show that it is a second player win, and that, consequently, ⧾⧾⧾G = ↑.
Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "↑ is the unique solution of ⧾G = G." Conway's assertion is also easily verifiable with canonical forms and game trees.