1 = (4+4-4)/4 = 44/44

2 = (4*4)/(4+4) = (4+4+4)/4

3 = (4*4-4)/4 = 4-4^(4-4)

4 = (4-4)*4+4

5 = (4*4+4)/4

6 = ((4+4)/4)+4

7 = (4+4)-(4/4) = 44/4-4

8 = (4*4)-(4+4) = 4+4+4-4

9 = (4/4)+4+4

10 = (44-4)/4

11 = 44/(sqrt 4+sqrt 4)

12 = (44+4)/4

13 = 44/4+sqrt 4 = 4+4+4+sqrt 4

14 = 4!/4+4+4 = 4!-(4+4+sqrt 4) = 4*4(4/sqrt 4)

15 = 44/4+4

16 = 4+4+4+4

17 = 4*4+4/4

18 = (4!+4!+4!)/4

19 = 4!-4-4/4

20 = (4/4+4)*4

21 = 4!-4+(4/4) = 4*4+4+sqrt 4

22 = 4/4(4!)-sqrt 4 = 4!-((4+4)/4) = 44/4*sqrt 4

23 = 4!-sqrt 4+4/4 = 4!-4^(4-4)

24 = 4*4+4+4

25 = 4!+sqrt 4-4/4 = 4!+4^(4-4)

26 = 4/4(4!)+sqrt 4 = 4!+sqrt (4+4-4)

27 = 4!+4-4/4

28 = (4+4)*4-4 = 4sqrt 4*4-4

29 = 4!+4+4/4

30 = 4!+4+4-sqrt 4

31 = ((4+sqrt 4)!+4!)/4!

32 = (4*4)+(4*4)

33 = 4!+4+sqrt 4/.4 = (4*4*sqrt 4)+sqrt 4

34 = 4!+(4!/4)+4 = sqrt(4^4)*sqrt 4+sqrt 4

35 = 4!+44/4

36 = (4+4)*4+4 = 44-4-4

37 = 4!+(4!+sqrt 4)/sqrt 4

38 = 44-(4!/4)

39 = 4!+4!/(4*.4)

40 = (4!-4)+(4!-4)

41 = (4!+sqrt 4)/.4-4!

42 = 44-4+sqrt 4 = (4!+4!)-(4!/4)

43 = 44-(4/4)

44 = 44+4-4

45 = 44+4/4

46 = 44+4-sqrt 4 = (4!+4!)-(4/sqrt 4)

47 = (4!+4!)-4/4

48 = (4*4-4)*4

49 = (4!+4!)+4/4

50 = 44+(4!/4)

51 = (4!-4+.4)/.4

52 = (44+4)+4

53 = 4!+4!+sqrt 4/.4

54 = 4!+4!+4+sqrt 4

55 = (4!-4+sqrt 4)/.4

56 = 4!+4!+4+4

57 = (4!-sqrt 4)/.4+sqrt 4

58 = ((4!+4)*sqrt 4)+sqrt 4

59 = (4!-sqrt 4)/.4+4

60 = 4*4*4-4 = 4^4/4-4

61 = (4!+sqrt 4)/.4-4

62 = 4*4*4-sqrt 4

63 = (4^4-4)/4

64 = 4sqrt 4*4sqrt 4

65 = (4^4+4)/4

66 = 4*4*4+sqrt 4

67 = (4!+sqrt 4)/.4+ sqrt 4

68 = 4*4*4+4 = 4^4/4+4

69 = (4!+sqrt 4)/.4+4

70 = (4+4)!/(4!*4!) = 44+4!+sqrt 4

71 = (4!+4.4)/.4

72 = 44+4!+4

73

74 = 4!+4!+4!+sqrt 4

75 = (4!+4+sqrt 4)/.4

76 = (4!+4!+4!)+4

77

78 = 4*(4!-4)-sqrt 4

79 = 4!+(4!-sqrt 4)/.4

80 = (4*4+4)*4 = (4-(4/4))^4

81 = (4!/(4*sqrt 4))^4

82 = 4*(4!-4)+sqrt 4

83 = (4!-.4)/.4+4!

84 = 44*sqrt 4-4

85 = (4!+4/.4)/.4

86 = 44*sqrt 4-sqrt 4

87

88 = 4*4!-4-4 = 44+44

89 = 4!+(4!+sqrt 4)/.4

90 = 4*4!-4-sqrt 4

91 = 4*4!-sqrt 4/.4

92 = 4*4!-sqrt 4-sqrt 4

93

94 = 4*4!+sqrt 4-4

95 = 4*4!-4/4

96 = 4*4!+4-4 = 4!+4!+4!+4!

97 = 4*4!+4/4

98 = 4*4!+4-sqrt 4

99

100 = 4*4!+sqrt 4+sqrt 4

101 = 4*4!+sqrt 4/.4

102 = 4*4!+4+sqrt 4

103

104 = 4*4!+4+4

105 = (44-sqrt 4)/.4

106 = 4*4!+4/.4

107

108 = 4*(4!+4)-4

109 = (44-.4)/.4

110 = (4!+4!-.4)/.4

111 = 444/4

112 = (44/.4)+sqrt 4

References

M Bicknell and V E Hoggatt, Recreational Mathematics Magazine, No 14 (1964). Shows 64 ways to make 64 with four fours. Good list of references.

L Harwood Clarke, Fun With Figures, William Heinemann Ltd, 1954, pp 51-53. Contains a table for the four fours problem from 1 to 100.

Henry Ernest Dudeney, 536 Puzzles and Curious Problems, Charles Scribner's Sons, 1967. SBN 684-71755-7, Library of Congress Card Number 67-15488. Puzzle 109. Cites Knowledge article below.

Angela Dunn, Mathematical Bafflers, McGraw?-Hill, 1964, pp 51-53. Also contains a table for the four fours problem from 1 to 100.

Martin Gardner, "Mathematical Games." In Scientific American, January 1964. Contains a more recent discussion of the problem. The following month's issue contains the solutions to posed questions.

Don Knuth, "Representing Numbers Using Only One 4." In Mathematics Magazine, Vol 37, November/December 1964, pp. 308-10. Shows how numbers up to 208 may be represented using only one four, square roots, factorials and brackets floor?. Conjectures that all integers can be represented in this way.

Knowledge, ed Richard Proctor, 30 December 1881. A magazine of popular science edited by astronomer Proctor.

PHP Warning

lib/DbaDatabase.php:154: Warning: dba_optimize() [<a href='function.dba-optimize'>function.dba-optimize</a>]: You cannot perform a modification to a database without proper access

lib/DbaDatabase.php:134: Warning: dba_replace() [<a href='function.dba-replace'>function.dba-replace</a>]: You cannot perform a modification to a database without proper access