Four Fours/Historical Note

Historical Note on Four Fours

As Henry Ernest Dudeney put it in his $58$. - The Two Fours in his $1926$ Modern Puzzles:

Dudeney reports:

I am perpetually receiving inquiries about the old "Four Fours" puzzle.

I published it in $1899$, but have since found that it first appeared in the first volume of Knowledge ($1881$).

It has since been dealt with at some length by various writers.

Martin Gardner locates that original article in Knowledge as being the December $30$th issue.

He then goes on to cite a number of more recent discussions on the subject, including his exposition in his own column in Scientific American for January $1964$.

He finishes with a reference to an article by Donald Ervin Knuth in which it is proved that all positive integers up to $208$ can be expressed with nothing but one $4$, instances of the square root sign, the factorial sign, and the floor function.

Because it is possible to express $4$ using four $4$s, it is hence possible to represent $113$ using four $4$s, although this representation may be somewhat complicated.

Ian Stewart's admittedly whimsical Professor Stewart's Casebook of Mathematical Mysteries from $2014$ delivers a deep analysis of the problem, delivering the final solution as the punchline to a particularly pointless shaggy-dog story.