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

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Modern Puzzles by Henry Ernest Dudeney: $58$

The Two Fours
The point [of the Four Fours puzzle] is to express all possible whole numbers with four fours (no more and no fewer), using the various arithmetical signs.
Thus:
$17 = 4 \times 4 + \dfrac 4 4$
and:
$50 = 44 + 4 + \sqrt 4$


All numbers up to $112$ inclusive may be solved, using only the signs for addition, subtraction, multiplication, division, square root, decimal points, and the factorial sign $4!$ which means $1 \times 2 \times 3 \times 4$, or $24$, but $113$ is impossible.
It is necessary to discover which numbers can be formed with one four, with two fours, and with three fours, and to record these for combination as required.
It is the failure to find some of these that leads to so much difficulty.
For example, I think very few discover that $64$ can be expressed with only two fours.
Can the reader do it?


Click here for solution

Historical Note

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.


Sources

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