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The Five Fives Problem: Recreational Mathematics

By Edited Sep 11, 2016 1 1

The Five Fives problem is a variant of the classical math puzzle Four Fours, in which you must use the number 4 exactly four times to produce every integer from 0 to 100 (or 200) without using any other digits.  You are allowed to use the basic mathematical operations of × (written here as *), ÷ (written here as /), +, -, along with exponentiation ^, square root, and factorial n!.  

You can also concatenate numbers (e.g., 44 and 444), use decimal points (.4), and use the overline to represent a repeating decimal (.4). Some sets of rules also allow you to use more obscure functions, such as the subfactorial !n, gamma function Γ(n), Euler's totient function φ(n), or the percent sign % (division by 100).  

Logarithms, the ceiling function, and the floor function are prohibited since they lead to trivial solutions which undercut the challenge and fun of the Four Fours problem.

Some examples of solutions to the Four Fours problem are

The Number 5
  • 0 = 4 + 4 - 4 - 4
  • 1 = (4/4)*(4/4)
  • 2 = 4 + 4 - 4 - sqrt(4)
  • 3 = (4 + 4 + 4)/4
  • 4 = sqrt(4*4)*(4/4)
  • 5 = 4!/4 - 4/4
  • 6 = 4 + 4 - 4 + sqrt(4)

The Five Fives puzzle has the same rules, except instead of using the digit 4 four times, you must use the digit 5 five times.  For any given integer, there are often many solutions; two people playing competitively can try to devise either the simplest or most complex expression. The answers to the Five Fives problem for the numbers 0 through 55 are given here.  You can submit more solutions (or alternate solutions) in the comments below.

0 = sqrt(5*5) + 5 - 5 - 5

1 = (5 + sqrt(5*5))/(5 + 5)

2 = [(5+5)/5]*(5/5)

3 = (5+5)/5 + 5/5

4 = (5+5+5+5)/5

5 = (5/5)*(5/5)*5

6 = (5*5)/5 + 5/5

7 = (5*5 + 5 + 5)/5

8 = (5+5+5)/5 + 5

9 = (5*5 - 5)/5 + 5

10 = (5*5 + 5*5)/5

11 = (55/5)*(5/5)

12 = [5!/(5+5)]*(5/5)

13 = 5!/(5+5) + 5/5

14 = (5! - 5*5)/5 - 5

15 = [5 + sqrt(5)]*[5 - sqrt(5)] - 5

16 = sqrt(5! + 5/5) + sqrt(5*5)

17 = (55 + 5)/5 + 5

18 = (5/.5)*[(5+5)/5]

19 = 5!/5 - 5 + 5 - 5

20 = (5! - 5*5 + 5)/5

21 = sqrt(5! + 5/5) + 5 + 5

22 = 5.5*(5 - 5/5)

23 = 5*5 - (5+5)/5

24 = (5! + 5)/5 - 5/5

25 = [(5! + 5)/5]*(5/5)

26 = (5! + 5)/5 + 5/5

27 = [(55/5)/.5] + 5

28 = .5*(55 + 5/5)

29 = 5!/5 + 5 + 5 - 5

30 = 5*5 + 5 + 5 - 5

31 = 5*5 + 5 + 5/5

32 = (5+5+5)/.5 + 5

33 = 5.5*(5 + 5/5)

34 = (5! + 5*5)/5 + 5

35 = 5!/(5 + 5/5) + 5

36 = (5/.5)*(5 - 5/5)

37 = 5!/(5 - .5) + 5 + 5

38 = 5!/5 + 5 + 5/.5

39 = 5!/5 + 5 + 5 + 5

40 = .5*(55 + 5*5)

41 = 5*(5+5) - 5/.5

42 = (5! - 55/.5)/.5

43 = 55 - 5!/(5+5)

44 = 5!/5 + 5*5 - 5

45 = [(5*5)/.5]*(5/5)

46 = (5*5)/.5 + 5/5

47 = (5!/5)/.5 - 5/5

48 = (5!/5)*(5+5)/5

49 = 5*(5+5) - 5/5

50 = 55 - 5 + 5 - 5

51 = 55 - 5 + 5/5

52 = 5/(5%) - (5!/5)/.5

53 = 55 - (5+5)/5

54 = 55 - sqrt(5*5)/5

55 = 55*sqrt(5*5)/5

Some useful hints and tips for applying functions: 

3! = 6

4! = 24

5! = 120

.5 = 5/9

5/.5 = 9

!4 = 9

!5 = 44

φ(5) = 4

φ(55) = 40

Γ(5) = 4! = 24

 

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Comments

Mar 22, 2013 1:15am
adragast
Nice one. Thumbs again
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