Talk:Mythical tree problem

Some comments:

The solution that Friedman is calculating is actually the number of tree branches grown on the final day, not the total number of tree branches.

Friedman calculates that the answer for k=2 is 2^39 * 40 - 2 = 21,990,232,555,518, but the correct answer is 2^39 * 41 - 2.

The number 2^^(2^2^95) is a lower bound not for k = 4, but for k = 2 where the squirrel can go from any treetop to a root in 5 steps rather than 4.

The problem as stated does not have Ackermannian growth rate; rather, it is the generalization of the problem to trees of height n that have Ackermannian growth rate.

If we set \(t_n(k)\) equal to the maximum number of tree branches grown on the final day, assuming the tree grows k branches on the first day and the squirrel can go from any treetop to the root in n steps or less, then we can define the following recursions:

\(T_0(k) = k+1\)

\(T_{n+1}(k) = T_n^k(k+1)\)

\(t_n(k) = T_n(k) - 1\)

From this we can deduce that \(t_2(k) = 2k, t_3(k) = 2^k(k+2) - 2\). In comparison with the Fast-growing Hierarchy, we have for \(n \ge 2, F_{n-1}(k) \le t_n(k) < F_{n-1}(k+1)\). Deedlit11 (talk) 05:49, October 10, 2013 (UTC)