## Beauty in Mathematics

La rivista The Mathematical Intelligencer nel n. 4/1988 propose ai suoi lettori la seguente lista di 24 risultati al fine di effettuare una indagine sulle formule ritenute più belle.

A | Euler’s formula for a polyhedron: V + F =E + 2. | M | At any party, there is a pair of people who have the same number of friends present. |

B | Any square matrix satisfies its characteristic equation | N | The number of partitions of an integer into odd integers is equal to the number of partitions into distinct integers. |

C | 5{(1 – x5)(1 – x10)(1 – X15) . . . }5 {(1 – x)(1 – x2)(1 – x3 )(1 – x4). . . }6 = p(4) + p(9)x+ p(14)x2 +. . . where p(n) is the number of partitions of n. |
O | If the points of the plane are each coloured red, yellow, or blue, there is a pair of points of the same colour of mutual distance unity. |

D | The number of primes is infinite. | P | Every plane map can be coloured with 4 colours. |

E | There is no rational number whose square is 2 | Q | A continuous mapping of the closed unit disk into itself has a fixed point. |

F | Every prime of the form 4n + 1 is the sum of two integral squares in exactly one way. | R | Write down the multiples of root 2, ignoring fractional parts, and underneath the numbers missing from the first sequence. 1 2 4 5 7 8 9 11 12 … 3 6 10 13 17 20 23 27 30 … The difference is 2n in the nth place. |

G | 1 + 1/2^{2} + 1/3^{2} + 1/4^{2} + 1/5^{2} + … = ? ^{2}/6. |
S | A regular icosahedron inscribed in a regular octagon divides the edges in the Colden Ratio. |

H | 1 – 1 + 1 2 x 3 x 4 4 x 5 x 6 6 x 7 x 8- ….. =(?-3)/4 . |
T | The number of representations of an odd number as the sum of 4 squares is 8 times the sum of its divisors; of an even number, 24 times the sum of its odd divisors. |

I | ? is transcendental. | U | The word problem for groups is unsolvable. |

J | Every number greater than 77 is the sum of integers, the sum of whose reciprocals is l. | V | The order of a subgroup divides the order of the group. |

K | The maximum area of a quadrilateral withsides a, b, c, d is {(s-a)(s-b)(s-c)(s-d)}½, where s is half the perimeter. | W | e^{i?} = – 1. |

L | There is no equilateral triangle whose vertices are plane lattice points. | X | There are 5 regular polyhedra. |