Editing Busy beaver (section)

=== Uncomputability of space(n) and num(n) ===
Both <math>\text{space}(n)</math> and <math>\text{num}(n)</math> functions are uncomputable.<ref name=":6" /> This can be shown for <math>\text{space}(n)</math> by noting that every tape square a Turing machine writes a one to, it must also visit: in other words, <math>\Sigma(n) \leq \text{space}(n)</math>.<ref name=":6" />  The <math>\text{num}(n)</math> function can be shown to be incomputable by proving, for example, that <math>\text{space}(n) < \text{num}(3n + 3)</math>: this can be done by designing an ''(3n+3)''-state Turing machine which simulates the ''n''-state space champion, and then uses it to write at least <math>\text{space}(n)</math> contiguous ones to the tape.<ref name=":6" />