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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: \(f_{\omega^2}(n)\)

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