Henry Ernest Dudeney/Modern Puzzles/58 - The Two Fours/Historical Note

Historical Note on $58$. - The Two Fours

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 parentheses.

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.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $58$. -- The Two Fours
• Nov-Dec 1964: Donald E. Knuth: Representing Numbers using Only One Four (Math. Mag. Vol. 37: pp. 308 – 310)  www.jstor.org/stable/2689238
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $109$. The Two Fours
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $109$. The Two Fours