Consider the following variant of Nim called Split-Nim, which is played with heaps of tokens as in Nim. Like in Nim, a player can remove some tokens from one of the heaps, or else split a heap into two (not necessarily equal) new heaps. For example, a heap of size 4 can be reduced to size 3, 2, 1, or 0 as in ordinary Nim, or be split into two heaps of sizes 1 and 3, respectively, or into two heaps of sizes 2 and 2. As usual, the last player to be able to move wins.

(a) Find the Nim values (size of equivalent Nim heaps) of the single Split-Nim heaps of size 1, 2, 3, and 4 , respectively.

(b) Find all winning moves, if any, when Split-Nim is played with three heaps of sizes \(1,2,3\).

(c) Find all winning moves, if any, when Split-Nim is played with three heaps of sizes \(1,2,4\).

(d) Determine if the following statement is true or false: "Two heaps in Split-Nim are equivalent if and only if they have equal size." Justify your answer.