A perfect cube refers to any number multiplied by itself three times.

(i) Continuing Odd Numbers

Successive Cubes can be written as a series of continuing and consecutive odd numbers.

13 = 1 = 1

23 = 8 = 3 + 5

33 = 27 = 7 + 9 + 11

43 = 64 = 13 + 15 + 17 + 19

53 = 125 = 21 + 23 + 25 + 27 + 29

63 = 216 = 31 + 33 + 35 + 37 + 39 + 41

73 = 343 = 43 + 45 + 47 + 49 + 51 + 53 + 55

83 = 512 = 57 + 59 + 61 + 63 + 65 + 67 + 69 + 71

93 = 729 = 73 + 75 + 77 + 79 + 81 + 83 + 85 + 87 + 89

(ii) Dudeney Numbers

A Dudeney number is named after the Famous English Mathematician, Henry Dudeney.

A Dudeney number is one which when cubed contains digits, which sum to itself.

For example: 27 is a Dudeney number.

27 = 1 + 9 + 6 + 8 + 3   and 273 = 19683

There are only six Dudeney numbers. Here are the remaining 5 Dudeney numbers.

1 = 1 and 13 = 1

8 = 5 + 1 + 2 and 83 = 512

17= 4 + 9 + 1 + 3 and 173 = 4913

18 = 5 + 8 + 3 + 2 and 183 = 5832

26 = 1 + 7 + 5 + 7 + 6 and 263 = 17576

(iii) Curious Cubes

37 x (3 + 7) = 33 + 73

48 x (4 + 8) = 33 + 73

111 x (11 + 1) = 113 + 13

147 x (14 + 7) = 143 + 73

148 x (14 + 8) = 143 + 83