## FANDOM

10,835 Pages

There are two different operations you’re allowed to do:

• Remove a coin from box i and add two coins to box i + 1.
• Remove a coin from box i and swap the contents of boxes i + 1 and i + 2.

In the beginning, each box will contain 1 coin.

## Values

$$f(1) = 1$$, of course

$$f(2) = 3$$

$$f(3) = 7$$

$$f(4) = 28$$

$$f(5) \geq 2^{2^{14}+1}$$

## $$f(n)$$

$$u_1 = 2^{14}$$

$$u_n = 2 \uparrow^{n}(u_{n-1})$$

$$f(n) > 2 \uparrow^{n-4}(u_{n-4}+1)$$

## $$x(n)$$

At step n, there are two different operations you’re allowed to do:

• Remove a coin from box i and add n+1 coins to box i + 1.
• Remove a coin from box i and swap the contents of boxes i + 1 and i + 2.

In the beginning, each box will contain 1 coin.

### Values

$$x(1) = 1$$

$$x(2) = 3$$

$$x(3) = 18$$

$$x(4) > 3,41 * 10^{64}$$

$$x(5) > x(4)[4]$$

$$x(6) > x(5)[5]$$

$$x(n+1) > x(n)[n]$$

x(4) ≥ 34134236497030018171294712492526709945875247912229530146490668315

## $$y(n)$$

At step n, there are three different operations you’re allowed to do:

• Remove a level 1 coin from box i and add n+1 coins to box i + 1.
• Remove a level 1 coin from box i and swap the contents of boxes i + 1 and i + 2.
• Remove a level k coin from box i and add n+1 level (k-1) coins to box i.

In the beginning, each box will contain 1 level n coin.

### Values

$$y(1) = 1$$

$$y(2) = 17$$

$$y(3) > x(6)$$

$$y(4) > x(x(x(6)))$$

$$y(5) > x_3(6)$$

$$y(n+2) > x_n(6)$$

Grow rate: $$f_{\omega2}(n)$$

## $$z(n)$$

At step n, there are three different operations you’re allowed to do:

• Remove a level 1 coin from box i and add 1 level n+1 coin to box i + 1.
• Remove a level 1 coin from box i and swap the contents of boxes i + 1 and i + 2.
• Remove a level k coin from box i and add n+1 level (k-1) coins to box i.

Grow rate: $$f_{\omega^2}(n)$$