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

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## 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