34 Arithmetic & Algebraic Problems109. THE TWO FOURS
I am perpetually receiving inquiries about the old "Four Fours" puzzle. I
published it in IS99, but have since found that it first appeared in the first
volume of Knowledge (lSSI). It has since been dealt with at some length by
various writers. The point is to express all possible whole numbers with four
fours (no more and no fewer), using the various arithmetical signs. Thus
4 X 4 + % equals 17, and 44 + 4 + v'4 equals 50. All numbers up to 112
inclusive may be solved, using only the signs for addition, subtraction, multi-
plication, division, square root, decimal points, and the factorial sign 4! which
means I X 2 X 3 X 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?110. THE TWO DIGITS
Write down any two-figure number (different figures and no 0) and then
express that number by writing the same figures in reverse order, with or
without arithmetical signs. For example, 45 = 5 X 9 would be correct if only
the 9 had happened to be a 4. Or SI = (I + S)2 would do, except for the fact
that it introduces a third figure-the 2.
- DIGITAL COINCIDENCES
If! multiply, and also add, 9 and 9, I get SI and IS, which contain the same
figures. If I multiply and add 2 and 47, I get 94 and 49-the same figures. If
I multiply and add 3 and 24, I get the same figures-72 and 27. Can you find
two numbers that when multiplied and added will, in this simple manner,
produce the same three figures? There are two cases.- PALINDROMIC SQUARE NUMBERS
This is a curious subject for investigation-the search for square numbers
the figures of which read backwards and forwards alike. Some of them are