Mathematical_Recreations-Kraitchik-2e

52 Mathematical Recreations
1. 121 represents a perfect square in every scale with
base B > 2. (B cannot = 2, since 2 is not one of the digits
in the scale with base 2.)
Proof: 121 represents B2 + 2B + 1 = (B + 1)2, which is
represented by 112.
2. Similarly, 1,331 represents a perfect cube in every
scale with base B > 3; 14,641 represents a perfect 4th power
for B > 6, and so on.
3. The number 40,001 is divisible by 221 and by 221 in
every scale having B > 4.
Proof: 40,001 represents
4B4 + 1 = (2B2 + 2B + 1) (2B2 - 2B + 1),
and the factors are represented by 221 and 221, respectively.
In the binary scale, every number is represented as a sum
of powers of 2 (including 2° = 1), and the digits are ° and 1.
Thus the number six is written 110, eleven is written 1,011,
one hundred is written 1,100,100, and so on. Or we may use
negative digits, as in 1,010 for six, 1,111 for eleven, and so
on.
In the ternary scale every number is expressed as a sum
or difference of powers of 3, using the digits 0, 1, I. Thus
sixteen (Arabic decimal) may be denoted by 121 = IITI.
A very well-known medieval problem whose solution may
be found by using the ternary scale is the following:
4. Find the least number of standard weights needed to
weigh every whole number of pounds from 1 to 40. (It is
understood that the weighing machine is the simple equalarm
balance.)
Solution: Bachet de Meziriac gave the two following solutions:
Weights of 1,2,4,8,16 and 32 pounds, totaling 63 pounds.
Weights of 1, 3, 9,27 pounds, totaling 40 pounds.