Answers 249first mention in the December 30, 1881, issue of Knowledge, a magazine
of popular science edited by the astronomer Richard Proctor. For a recent
discussion of the problem, see my Scientific American column for January 1964,
and the answer section of the column for the following month. A table giving
the integers from 1 to 100, each expressed with four fours, will be found
in L. Harwood Clarke, Fun With Figures (William Heinemann Ltd., 1954),
pp. 51-53, and Angela Dunn, Mathematical Bafflers (McGraw-Hill, 1964),
pp.5-8.
The number 64 is easily expressed with four fours: (4 + 4)(4 + 4), and
with three fours: 4 X 4 X 4. In Recreational Mathematics Magazine, No. 14
(1964), M. Bicknell and V. E. Hoggatt show 64 ways of expressing 64 with
four fours, and add a good list of references on the Four Fours problem.
D. E. Knuth, in his article "Representing Numbers Using Only One 4,"
Mathematics Magazine, Vol. 37, November-December, 1964, pp. 308-310,
shows how to represent 64 by using only one four and three kinds of symbols:
the square root sign, the factorial sign, and brackets. To express 64 by this
method requires 57 square root signs, nine factorials, and 18 brackets. A com-
puter program, Knuth reports, found that all positive integers less than 208
could be represented in ~ similar manner, and he conjectures that the method
applies to all positive integers.
Dudeney is partially correct in his assertion about the number 113. So far
as I know, no one has found a way to represent it without adopting highly
unorthodox symbols or complicated procedures such as Knuth's.-M. G.]
- THE TWO DIGITS
Of course it is entirely a matter of individual taste what arithmetical forms
and signs are admissible, but I should personally draw the line at expressions
introducing "log" and "antilog."
A few solutions are as follows:25 = 52
36 = 6 X 3!
64 = y'46 or (V4)6
25 = 5 + .2