Let's assume you have mastered the game of Nim, because Truning Turtles is really Nim in disguise. The way it's presented at the site it's a Nim with eleven rows (or heaps) that contain 1, 2, 3, ..., 10, 11 checkmarks (or any other kind of items - pebbles, for example.)O in the square #N means that the row has exactly N items. X in the square #N means that the row is empty.
To remove N items from row #N, click on the square #N.
To remove M < N items from row #N, click on squares #N and #(N-M). Why? There are two possibilites.
1. Square #(N-M) contains O. This means that row #(N-M) is empty.
2. Square #(N-M) shows X. This means that row #(N-M) is full, i.e. contains (N-M) items.
Forget for a moment about other rows. Your goal is to remove M items from row #N that at this point in the game contains N items. In other words you plan to leave a row with (N-M) items instead of N.
In the first case, you have row #N with N items and row #(N-M) empty. This means that square #N contains a O, while square #(N-M) contains X. After you click on these two squares, square #N will contain X, and square #(N-M) will contain O. This corresponds to row #N being empty and row #(N-M) being full with (N-M) items. Which means that from a configuration of two rows
empty/N you moved to a configuration (N-M)/empty, which is in fact what you wanted (except that along the way you swapped two rows, but who cares?)
In the second case, when you only look at rows #(N-M) and #N, the first one contains (N-M) elements, the second contains N elements. You wish to remove N elements from row #N leaving it with (N-M) elements. This would give you two rows of (N-M) elements each. But XOR on two equal numbers is 0. So leaving (N-M) items in row #N and having (N-M) items in row #(N-M) is the same as removing both rows. This is exactly equivalent to clicking on both boxes: #(N-M) and #N.
In class, you do not have to use 11 boxes. Take a smaller number. 3, 4, 5 will do very well.
For 3 boxes,
OOO is the same as Nim with 3 rows with number of elements being 1, 2, 3. OXO is Nim with 3 rows of 1, 0, 3 items, respectively.