818 Mathematical entertainments
32.1 Beauty in mathematics: David Wells’ survey^1
Give each of the following theorems a score for beauty between 0 (the
least) and 10 (the most beautiful).
A B C D E
F G H I J
K L M N O
P Q R S T
U V W X
AEuler’s formula for a polyhedron:V −E+F=2.
BAny square matrix satisfies its own characteristic equation.
CIfp(n)is the number of partitions ofn, then
5((1−x^2 )(1−x^10 )(1−x^15 )···)^5
((1−x)(1−x^2 )(1−x^3 )(1−x^4 )···)^6
=p(4) +p(9)x+p(14)x^2 +···
.DThe number of primes is infinite.
EThere is no rational number whose square is 2.
FEvery prime of the form 4 n+1is the sum of two integral squares in
exactly one way.
G1+ 212 + 312 +···+n^12 +···=π
2
6.
H 2 × 31 × 4 − 4 × 51 × 6 + 6 ×^17 × 8 −···=π− 43.
Iπis transcendental.
JEvery number greater than 77 is the sum of integers, the sum of whose
reciprocal is 1.
KThe maximum area of a quadrilateral with sidesa,b,c,dis
√
(s−a)(s−b)(s−c)(s−d),
wheresis half the perimeter.(^1) Math. Intelligencer, 10:4 (1988) 31.