318 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1959 
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Ficure 4.—How to make the difficult simple. Problem: Given®as many of the small rec- 
tangles as you want, can you arrange them to cover completely the large figure? None 
of the small rectangles must overlap or jut beyond the margins of the large figure. As it 
happens, the feat is impossible, but the difficult thing is to prove conclusively that it is 
impossible. (For a mathematical solution, see fig. 5.) 
problems that used to be handled by various rules of thumb, and less 
accurately. 
MATHEMATICS OF LOGIC 
Computers have also had some effects on pure mathematics. Faced 
with the problems of instructing computers what to do and how to 
do it, mathematicians have reopened an old and partly dormant field: 
Boolean algebra. This branch of mathematics reduces the rules of 
formal logic to algebraic form. Two of its axioms are startlingly dif- 
ferent from the axioms of ordinary high-school algebra. In Boolean 
algebra at+a=a, and aXa=a. The reason becomes clear when a is 
interpreted as a statement, the plus sign as “or,” and the multiplica- 
tion sign as “and.” Thus, for example, the addition axiom can be 
illustrated by: “(this dress is red) or (this dress is red) means (this 
dress is red) .” 
Numerical analysis, a main part of the study of approximations, is 
another field that mathematicians have revived to program problems 
for computers. There is still a great deal of pure and fundamental 
mathematical research to be done on numerical errors that may arise 
