Form expressions equalling each integer from 1 to 112 from four 4's, which may be concatenated to form a...
arithmetic/digits/equations/four.4 Added material
Problem
Form expressions equalling each integer from 1 to 112 from four 4's, which may be concatenated to form a single number (e.g., 44 or 444), the decimal point, and the operators +, -, *, /, sqrt() and ! (factorial).
Show solution
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.
Additional material
[ Editor: the table above leaves 103 and 107 blank. Both can be done
with a repeating decimal: ]
103 = (44 / .|4) + 4, where .|4 = .44444444...
107 = (4! + 4! - .|4) / .|4
[Haven't double checked these solutions to make sure they're right. --b]
[ Editor: they check: .|4 = 4/9, so 44/(4/9) + 4 = 99 + 4 = 103 and
(48 - 4/9)/(4/9) = 108 - 1 = 107. ]
Actually, that solutions were there for positive n up to 112 was known
no later than Henry Dudeney. Two books of his puzzles were packaged
together a couple of decades ago (see citation below [Editor: the Dudeney reference in the entry
above]); this
is puzzle number 109. The puzzle really concerns making 64 from two
fours, but he introduces it by mentioning that he published the four
fours problem in 1899, but that he had since found it in the first
volume of _Knowledge_ (1881). He goes on to say that all numbers up to
112 can be done, but not 113. He doesn't claim to have been the first
to find all these solutions.
In the solution to this problem, Martin Gardner adds to the _Knowledge_
reference. He mentions that Don Knuth found a way to represent numbers
with only one four. He uses square root (legitimate), factorial
(legitimate) (but no double factorial!), and brackets (boo hiss). (I
assume that by brackets, he means floor.) To express 64 with one four
requires 57 sqrt, 9 !, and 18 []. M G leaves the case of 113 open
without the floor function; no one has proven it can't be done.
-- Brian Tung <brian@darkstar.isi.edu>
In article <62lnbl$ktj@lace.colorado.edu>, arnow@newton.Colorado.EDU
(Jim Arnow) wrote:
>Using exactly four 4's and any standard Algebraic manipulations, find
>expressions which equal the first 100 natural numbers, for as many as
>possible. Allowed operations are: addition, subtraction,
>multiplication, division, exponentiation, square roots (since they do
>not require additional numerals) and factorial. Also, numerals such
>as 44 and 444 are allowed, so long as the numeral 4 appears only four
>times in the expression.
>
>My students and I have been unable to come up with a number of values.
>Among these, the first two we are stuck on are 31 and 33.
31 isn't really very difficult, even with your rather restrictive
rules.
33 is harder; it helps to know how to express 64 with only two 4s.
Without decimals and repeating decimals, I can't find a way to get 39
or 41, or more than five of the odd numbers between 50 and 100.
31 = 4! + (4! + 4)/4.
64 = sqrt(sqrt(sqrt(4^(4!)))).
33 = (64 + sqrt(4))/sqrt(4).
-- Fred W. Helenius <fredh@ix.netcom.com>
Graham Underwood writes:
|> Make up sums using exactly four 4's and any number of mathematical
|> symbols (+ - * / sqrt % etc) to total up to each of the numbers
|> from 0 to 100.
If I'm allowed logarithms (Graham does say, 'etc') then I can make any
positive integer with four 4s. For positive n:
n = log_(sqrt(4)/4)(log_4(sqrt(sqrt(...sqrt(4)...))))
\----------------/
n square roots
where log_(y)(x) is log to the base y of x.
-- Gareth Rees <gdr11@cl.cam.ac.uk>
[ Editor: the following is Andy Pepperdine's note "The Four Equal
Digits Problem" (2007), which was in the TODO collection as
four-equal-digits.pdf. It is transcribed in full into plain text; the
typeset formulas are rewritten in one line with this notation: "." is
the decimal point (the source uses a midpoint); a recurring decimal is
written with the repeating digits in parentheses, so .(1) = .1111... =
1/9, .(12) = .121212... and .1(3) = .13333...; * is multiplication, /
is division (the source writes every division as a built-up fraction),
^ is a power, sqrt(x) is the square root, and root[a](b) is the a-th
root of b, i.e. b^(1/a), as in the source's fractional and negative
roots. ]
THE FOUR EQUAL DIGITS PROBLEM
The Four Equal Digits Problem
by Andy Pepperdine
2007-01-12
based on notes made 50 years ago.
This work is released under the Creative Commons Attribution -
Share-alike 2.5 license [1].
Make all integers from 0 upwards using the simplest mathematical
operators, as defined in some way, and four equal digits.. W. W. Rouse
Ball gives this in his book Mathematical Recreations and Essays[2]
under the heading of the "Four Fours" problem, but goes on to give
limits he has found for 4 of each of the digits (except 0).
This note confirms his limits, and gives examples of each number for
each of the digits, and I tried to stick to the types of operations
that he describes in the order in which he introduces them.
Allowing indices, and fractional roots, enables more numbers to be
obtained, which are listed as far I have gone.
As problems before you go to the tables, try to get 38 with four twos,
49 with four threes, 75 with four fours, 61 with four fives, 29 with
four sixes, 22 with four sevens, and 67 with four nines.
Some alternative methods are noted, too.
Notation
The midpoint (.) is used for the decimal point. [ Editor: written
here as an ordinary decimal point. ]
Recurring decimals are written with dots over the first and last digit
of the recurring sequence. So .(1) is the same as .1111... (i.e. one
ninth), .(12) is the same as .121212... (i.e. twelve ninety-ninths, or
four thirty-thirds), and .1(3) is the same as .133333... (i.e. twelve
ninetieths, or two fifteenths). Some care is required to see these
dots and some printers may not be able to print them well. [ Editor:
the dotted digits are written here in parentheses, as shown. ]
Addition, subtraction and multiplication are as normal, but division
is always given as a fraction.
The factorial function is represented as usual by a ! giving 3!=6,
4!=24 and by definition 0!=1.
Square roots are also as usual, with sqrt(4)=2 and sqrt(.(1))=1/3.
To get higher numbers than the limits Rouse Ball gives when restricting
the operators not to include any powers, I have assumed the notation
of an n-th root where n is not integral, as seemed necessary to reach
his limits for the four digits 1, 2, 3 and 4. So root[.1](2) = 2^10 =
1024, and negative roots are the equivalent of reciprocal powers, so
root[-.2](.(3)) = (1/3)^-5 = 3^5 = 243.
References
[1] Creative Commons License, http://creativecommons.org/licenses/by/2.5/
[2] W.W. Rouse Ball, revised by H.S.M. Coxeter, Mathematical
Recreations and Essays, 11th edition, 1939, reprinted in 1967 by
Macmillan, p15ff.
Four zeros
All numbers up to and including 7. [2] gives no limit for the case
of 0.
0 0+0+0+0
1 0!+0+0+0
2 0!+0!+0+0
3 0!+0!+0!+0
4 0!+0!+0!+0!
5 (0!+0!+0!)!-0!
6 (0!+0!+0!)!+0
7 (0!+0!+0!)!+0!
Four ones
All numbers up to and including 34, which is the limit in [2].
0 11-11
1 1*1*1*1
2 1+1+1-1
3 1*(1+1+1)
4 1+1+1+1
5 1/(.1*(1+1)) or (1+1+1)!-1
6 1+1/(.1+.1)
7 1/.(1)-1-1
8 1/.1-1-1
9 11-1-1
10 11*1-1
11 11+1-1
12 11*1+1
13 11+1+1
14 11+1/sqrt(.(1))
15 1/(.1*(1-sqrt(.(1))))
16 1/.1+(1/sqrt(.(1)))!
17 (1+1)/.(1)-1
18 1*(1+1)/.(1)
19 (1+1)/.1-1
20 1*(1+1)/.1
21 11+1/.1
22 11+11
23 (1+1/sqrt(.(1)))!-1
24 (1/.(1)-1)/sqrt(.(1))
25 (1+1/sqrt(.(1)))!+1
26 1/(.(1)*sqrt(.(1)))-1
27 (1+1+1)/.(1)
28 1/(.(1)*sqrt(.(1)))+1
29 1/(.1*sqrt(.(1)))-1
30 (1+1+1)/.1
31 1/(.1*sqrt(.(1)))+1
32 11/sqrt(.(1))-1
33 1*11/sqrt(.(1))
34 11/sqrt(.(1))+1
Four twos
All numbers up to and including 36, which is the limit given in [2].
0 22-22
1 22/22
2 2+2*(2-2)
3 (2+2+2)/2
4 2+2+2-2
5 2+2+2/2
6 2*(2+2)-2
7 2+2/(2*.2)
8 2+2+2+2
9 22/2-2
10 (22-2)/2
11 2/2+2/.2
12 (22+2)/2
13 22/2+2
14 2/.2+2+2
15 (2+2/2)/.2
16 2*2*2*2
17 (2+2-.(2))/.(2)
18 22-2-2
19 (2+2-.2)/.2
20 2/.2+2/.2
21 22-2/2
22 22+2-2
23 22+2/2
24 2*(2+2/.2)
25 2/(2*.2*.2)
26 22+2+2
27 (2+2+2)/.(2)
28 (2+2)!+2+2
29 ((sqrt(2/.(2)))!-.2)/.2 or .2^-2+2+2
30 (2+2+2)/.2
31 22+2/.(2)
32 22+2/.2
33 22/sqrt(2*.(2))
34 (2+2)!+2/.2
35 (2/.(2)-2)/.2
36 2*(2+2)/.(2)
37 root[.2](2)+sqrt((.2)^-2)
38 sqrt(2*(((sqrt(2/.(2)))!)!+2))
Four threes
All numbers up to and including 64. Without powers, the limit is 46,
as given in [2].
0 33-33
1 33/33
2 (3*3-3)/3
3 (3+3+3)/3
4 (3*3+3)/3
5 3+3-3/3
6 3+3+3-3
7 3+3+3/3
8 33/3-3
9 (33-3)*.3
10 (33-3)/3
11 3/3+3/.3
12 (33+3)/3
13 (3*.3+3)/.3
14 33/3+3
15 3*(3+3)-3
16 3+3+3/.3
17 (3+3)/.3-3
18 3*(3*3-3)
19 3*3+3/.3
20 3/.3+3/.3
21 3*(3+3)+3
22 33*(.(3)+.(3))
23 33-3/.3
24 33-3*3
25 (3!-.3)/.3+3!
26 3*(3*3-.(3))
27 33-3-3
28 3*(3*3+.(3))
29 (3*3-.3)/.3
30 3*3*3+3
31 (3*3+.3)/.3
32 33-3/3
33 33+3-3
34 33+3/3
35 33+3!/3
36 3*(3*3+3)
37 3!*3!+3/3
38 3!*(3+3+.(3))
39 33+3+3
40 (3*3+3)/.3
41 (3!+3!+.3)/.3
42 33+3*3
43 33+3/.3
44 3!*(3!+.(3))+3!
45 3*(3*3+3!)
46 3!*3!+3/.3
47 3^3+3!/.3
48 3*(3!+3/.3)
49 sqrt(((3+3)!+.3)/.3)
50 (3!+3*3)/.3
51 33+3!*3
52 3!*(3*3-.(3))
53 33+3!/.3
54 3*3*(3+3)
55 33/(.3+.3)
56 3!*(3*3+.(3))
57 3!*3*3+3
58 3!*(3/.3-.(3))
59 (3!*3-.3)/.3
60 3!*(3+3)/.3 [ Editor: as printed this gives 36/.3 = 120, not 60;
3!*3/.3 = 60 uses only three 3s. ]
61 (3!*3+.3)/.3
62 3!*(3/.3+.(3))
63 3*(3!*3+3)
64 (3+3/3)^3
Four fours
All numbers up to and including 112, as in [2].
0 44-44
1 44/44
2 4/4+4/4
3 (4+4+4)/4
4 4+4*(4-4)
5 (4*4+4)/4
6 4+(4+4)/4
7 44/4-4
8 4+4+4-4
9 4+4+4/4
10 (44-4)/4
11 4/4+4/.4
12 (44+4)/4
13 (4-.4)/.4+4
14 4*(4-.4)-.4
15 44/4+4
16 4+4+4+4
17 4*4+4/4
18 44*.4+.4
19 (4+4-.4)/.4
20 4*(4+4/4)
21 (4+4+.4)/.4
22 (4+4)/.(4)+4
23 4/(.4*.4)-sqrt(4)
24 4+4+4*4
25 4*4+4/.(4)
26 4*4+4/.4
27 (4*4-4)/.(4)
28 44-4*4
29 4/(.4*.4)+4
30 (4*4-4)/.4
31 4!/(.(4)+.(4))+4
32 4*4+4*4
33 44/sqrt(4*.(4))
34 44-4/.4
35 (4+4/.4)/.4 or 44-4/.(4)
36 44-4-4
37 (4*4+.(4))/.(4)
38 44-4-sqrt(4)
39 (4*4-.4)/.4
40 4*4/.(4)+4
41 (4*4+.4)/.4
42 44-4+sqrt(4)
43 44-4/4
44 44+4-4
45 44+4/4
46 44+4-sqrt(4)
47 44+sqrt(4/.(4))
48 4*(4*4-4)
49 44+sqrt(4)/.4
50 (4*4+4)/.4
51 (4!-4+.4)/.4
52 44+4+4
53 44+4/.(4)
54 44+4/.4
55 44/(.4+.4)
56 4*(4+4/.4)
57 (4!+.4)/.4-4
58 (4!-.4-.4)/.4
59 4!/.4-4/4
60 44+4*4
61 4!/.4+4/4
62 4*4*4-sqrt(4)
63 (44-sqrt(4))/sqrt(.(4)) or 4!/.4+sqrt(4/.(4))
64 (4+4)*(4+4)
65 (44-sqrt(.(4)))/sqrt(.(4))
66 4*4*4+sqrt(4)
67 (44+sqrt(.(4)))/sqrt(.(4)) or (4!+4)/.(4)+4
68 4*4*4+4
69 (44+sqrt(4))/sqrt(.(4)) or 4!/.4+4/.(4)
70 44/sqrt(.(4))+4
71 (4!+4+.4)/.4
72 4*(4+4)/.(4)
73 (4!*sqrt(4)+sqrt(.(4)))/sqrt(.(4))
74 (4!+4)/.4+4
75 (4-sqrt(.(4)))/(.(4)-.4) or 44/.(4)-4!
76 4*(4!-4)-4
77 4!/(.(4)*sqrt(.(4)))-4 or (4!-.(4))/.(4)+4!
78 4*(4!-4)-sqrt(4)
79 4!/(.(4)*sqrt(.(4)))-sqrt(4) or (4!+.(4))/.(4)+4!
80 4*(4*4+4)
81 (4-.4)/(.(4)-.4)
82 4*(4!-4)+sqrt(4)
83 4!/(.(4)*sqrt(.(4)))+sqrt(4)
84 44*sqrt(4)-4
85 (4!+4/.4)/.4
86 44*sqrt(4)-sqrt(4) or 4/(.(4)-.4)-4
87 4!*4-4/.(4)
88 44+44
89 (4!/sqrt(.(4))-.4)/.4 or (4!+sqrt(4))/.4+4!
90 (44-4)/.(4)
91 4!*4-sqrt(4)/.4
92 44*sqrt(4)+4
93 4!*4-sqrt(4/.(4))
94 4/(.(4)-.4)+4
95 44/.(4)-4
96 (44+4)*sqrt(4) or 4!*4+4-4
97 44/.(4)-sqrt(4)
98 (44-.(4))/.(4)
99 4!*4+sqrt(4/.(4)) or (4!*sqrt(4)-4)/.(4)
100 44/.44
101 44/.(4)+sqrt(4)
102 4!*4+4+sqrt(4)
103 44/.(4)+4
104 44+4!/.4
105 (44-sqrt(4))/.4
106 44/.4-4
107 (4!*sqrt(4)-.(4))/.(4)
108 (44+4)/.(4)
109 (44-.4)/.4
110 44/sqrt(.4*.4)
111 444/4
112 44/.4+sqrt(4)
Four fives
All numbers up to and including 67. Without powers, 36 can be
reached, as in [2].
0 55-55
1 55/55
2 5/5+5/5
3 (5+5+5)/5
4 (5*5-5)/5
5 5+5*(5-5)
6 55/5-5
7 5+(5+5)/5
8 5*.5+5+.5
9 5+5-5/5
10 (55-5)/5
11 5+5+5/5
12 (55+5)/5
13 5*5*.5+.5
14 (5-.5)/.5+5
15 5*5-5-5
16 55/5+5
17 (5+5-.(5))/.(5)
18 5/.(5)+5/.(5)
19 (5+5-.5)/.5
20 5+5+5+5
21 (5+5+.5)/.5
22 55/(5*.5)
23 5*(5-.5)+.5
24 5*5-5/5
25 5*5+5-5 or .5*(55-5)
26 5*5+5/5
27 55*.5-.5
28 55*.5+.5
29 (5-5/5)!+5
30 55-5*5
31 55-5!/5
32 5!/(5-.(5))+5
33 5!/5+5/.(5) or 55/sqrt(5*.(5))
34 5*5+5/.(5)
35 5*5+5+5
36 (5*5-5)/.(5)
37 .5^-5+sqrt(5*5)
38 (5!/5-5)/.5
39 5!-root[.5](5/.(5))
40 (5*5-5)/.5
41 5+sqrt(((sqrt(5/.(5)))!)!/.(5)) or .5^-5+5/.(5)
42 5+5+.5^-5
43 5!/(5*.5)-5
44 (5*5-.(5))/.(5)
45 55-5-5
46 55-5/.(5)
47 (5!/.5-5)/5
48 5!/(5-5*.5)
49 (5*5-.5)/.5
50 5*5+5*5
51 (5*5+.5)/.5
52 55-sqrt(5/.(5))
53 5!/(5*.5)+5
54 55-5/5
55 55+5-5
56 55+5/5
57 (5!-5)*.5-.5
58 55+sqrt(5/.(5))
59 5!*.5-5/5
60 (5*5+5)/.5
61 5!*.55-5
62 (5!+5)*.5-.5
63 (5!+5)*.5+.5
64 55+5/.(5)
65 55+5+5
66 (sqrt(5*5))!*.55
67 5!/sqrt(5*.(5))-5
Four sixes
All numbers up to and including 30, as in [2].
0 66-66
1 66/66
2 6/6+6/6
3 (6+6+6)/6
4 6-(6+6)/6
5 66/6-6 or (6*6-6)/6
6 6+6*(6-6)
7 (6*6+6)/6
8 6+(6+6)/6
9 6/.(6)+6-6
10 (66-6)/6
11 6+6-6/6
12 6+6+6-6
13 6+6+6/6
14 (6+6)/.6-6 or 6+6+sqrt(6*.(6))
15 6/(6/6-.6)
16 6+6+sqrt(6/.(6)) or (6+6*.6)/.6
[ Editor: as printed the first gives 6+6+3 = 15, not 16. ]
17 66/6+6
18 .6*(6*6-6)
19 (6+6-.6)/.6
20 6/.6+6/.6
21 6*6*.6-.6
22 6/.6+6+6
23 (6*.(6))!-6/6
24 6+6+6+6
25 6/(.6-.6*.6)
26 (6+6)/.6+6
27 (6+6+6)/.(6)
28 (6*6+6)*.(6)
29 sqrt(6!+(6!+6)/6)
30 66-6*6
Four sevens
All numbers up to and including 25, as in [2].
0 77-77
1 77/77
2 7/7+7/7
3 (7+7+7)/7
4 77/7-7
5 7-(7+7)/7
6 (7*7-7)/7
7 7+7*(7-7)
8 (7*7+7)/7
9 (7+7)/7+7
10 (77-7)/7
11 7/.7+7/7
12 (77+7)/7
13 7+7-7/7
14 7+7+7-7
15 7+7+7/7
16 7+(7-.7)/.7
17 (7*.7+7)/.7
18 77/7+7
19 (7+7-.7)/.7
20 7/.7+7/.7
21 7*(7/.7-7)
22 (7*sqrt(7)+sqrt(.(7)))/sqrt(.(7))
23 7+7+7/.(7)
24 7+7+7/.7
25 (7+7)/.(7)+7
Four eights
All numbers up to and including 36, as in [2].
0 88-88
1 88/88
2 8/8+8/8
3 (8+8+8)/8 or 88/8-8
4 8*8/(8+8)
5 8/(8-8*.8)
6 8-(8+8)/8
7 (8*8-8)/8
8 8+8*(8-8)
9 (8*8+8)/8
10 (88-8)/8
11 8/.8+8/8
12 (88+8)/8
13 8/(.8+.8)+8
14 8/.8+sqrt(8+8)
15 8+8-8/8
16 8+8+8-8
17 8+8+8/8
18 (8*.8+8)/.8
19 88/8+8
20 8/.8+8/.8
21 (8+8+.8)/.8
22 88/sqrt(8+8)
23 (sqrt(8+8))!-8/8
24 88-8*8
25 (8+8)/(.8*.8)
26 8/.8+8+8
27 (8+8+8)/.(8)
28 (8+8)/.8+8
29 ((sqrt(8+8))!-.8)/.8
30 (8+8+8)/.8
31 ((sqrt(8+8))!+.8)/.8
32 8+8+8+8
33 88/sqrt(8-.(8))
34 (sqrt(8+8))!+8/.8
35 (sqrt(8+8))!/.(8)+8
36 8*8/(.(8)+.(8))
Four nines
All numbers up to and including 132, without powers, 130 is reached
as in [2].
I've tried to avoid using the fact that .(9)=1 although sometimes there
are some much simpler expressions when that identity is allowed. The
troublesome ones are 103, 106, 115, 124, 127, 128 and 130
0 99-99
1 99/99
2 99/9-9
3 (9+9+9)/9
4 9-9/(.9+.9)
5 9-sqrt(9)-9/9
6 9-sqrt(9)+9-9
7 9-(9+9)/9
8 (9*9-9)/9
9 9+9*(9-9)
10 (99-9)/9
11 9+(9+9)/9
12 (99+9)/9
13 9+sqrt(9)+9/9
14 9+9/(.9+.9)
15 9+(9+9)/sqrt(9)
16 9/.9+9-sqrt(9)
17 9+9+9/9 [ Editor: as printed this gives 19, the same as the
entry for 19; 9+9-9/9 = 17 was presumably meant. ]
18 99-9*9
19 9+9+9/9
20 99/9+9
21 (9+9+.9)/.9
22 9+sqrt(9)+9/.9
23 (9+9)/.9+sqrt(9)
24 99/sqrt(9)-9
25 9/.9+9+(sqrt(9))!
26 9*sqrt(9)-9/9
27 9*sqrt(9)+9-9
28 9/.9+9+9
29 (9+9)/.9+9
30 (9+9+9)/.9
31 (9*sqrt(9)+.9)/.9
32 (99-sqrt(9))/sqrt(9)
33 99*sqrt(9)/9
34 (99+sqrt(9))/sqrt(9)
35 (99+(sqrt(9))!)/sqrt(9)
36 9+9+9+9
37 9*sqrt(9)+9/.9
38 (sqrt(9))!*(sqrt(9)+sqrt(9)/.9)
39 9*sqrt(9)/.9+9
40 (9+9*sqrt(9))/.9
41 ((sqrt(9))!*(sqrt(9))!+.9)/.9
42 99/sqrt(9)+9
43 ((sqrt(9))!)!/(9+9)+sqrt(9)
44 (sqrt(9))!*9-9/.9
45 9*9/(.9+.9)
46 (sqrt(9))!*(sqrt(9))!+9/.9
47 9!/(9*((sqrt(9))!)!)-9
48 (sqrt(9))!*(9-9/9)
49 ((sqrt(9))!)!/(9+9)+9
50 ((sqrt(9))!*9-9)/.9
51 9*(9-sqrt(9)/.9)
52 (sqrt(9))!*(9-9/sqrt(9)) [ Editor: as printed this gives
6*(9-3) = 36, not 52; (sqrt(9))!*(9-sqrt(9)/9) = 52. ]
53 (sqrt(9))!*9-9/9
54 9*9-9*sqrt(9)
55 99/(.9+.9)
56 9!/((9-sqrt(9))!*9)
57 (9+9/.9)*sqrt(9)
58 (sqrt(9))!*(9+(sqrt(9))!/9)
59 ((sqrt(9))!*9-.9)/.9
60 (9+9)*sqrt(9)/.9
61 ((sqrt(9))!*9+.9)/.9
62 ((sqrt(9))!)!/9-9-9
63 9*9-9-9
64 (sqrt(9))!*9+9/.9
65 9!/(9*((sqrt(9))!)!)+9
66 (sqrt(9))!*99/9
67 sqrt(9!/(9*9)+9)
68 ((sqrt(9))!)!/9-9-sqrt(9)
69 9*9-9-sqrt(9)
70 ((sqrt(9))!*9+9)/.9
71 9*9-9/.9
72 9*(9-9/9)
73 (((sqrt(9))!)!*.9+9)/9
74 ((sqrt(9))!)!/9-9-sqrt(9) [ Editor: as printed this is the same
as the entry for 68 and gives 80-9-3 = 68, not 74;
80-9+3 = 74. ]
75 9*9-9+sqrt(9)
76 (((sqrt(9))!)!-9)/9-sqrt(9) or
(sqrt(9))!*((sqrt(9))!+(sqrt(9))!/.9)
77 ((sqrt(9))!)!/9-9/sqrt(9)
78 9*(9-sqrt(9)/9)
79 ((9-sqrt(9))!-9)/9
80 9*9-9/9
81 99-9-9
82 9*9+9/9
83 9*9+(sqrt(9))!/sqrt(9)
84 9*(9+sqrt(9)/9)
85 (((sqrt(9))!)!-9)/9+(sqrt(9))!
86 ((sqrt(9))!)!/9+9-sqrt(9)
87 99-9-sqrt(9)
88 (((sqrt(9))!)!-9)/9+9
89 99-9/.9
90 99*.9+.9
91 9*9+9/.9
92 ((sqrt(9))!)!/9+9+sqrt(9)
93 99-9+sqrt(9)
94 (sqrt(9))!*(9+(sqrt(9))!/.9)
95 ((sqrt(9))!)!/9+9+(sqrt(9))!
96 99-9/sqrt(9)
97 99-(sqrt(9))!/sqrt(9)
98 99-9/9
99 99+9-9
100 99/.99
101 99/.9-9
102 99+9/sqrt(9)
103 99+sqrt(9)+.(9)
104 99/.9-(sqrt(9))!
105 99+9-sqrt(9)
106 99+(sqrt(9))!+.(9)
107 99/.9-sqrt(9)
108 99+sqrt(9*9)
109 99+9/.9
110 (99/9)!/9!
111 999/9
112 (9!+9!)/(((sqrt(9))!)!*9)
113 99/.9+sqrt(9)
114 99+9+(sqrt(9))!
115 ((sqrt(9))!-.(9))!-(sqrt(9))!-.(9) [ Editor: as printed this
gives 120-6-1 = 113, not 115; 120-6+1 = 115. ]
116 99/.9+(sqrt(9))!
117 99+9+9
118 (((sqrt(9))!)!-9-sqrt(9))/(sqrt(9))!
119 99/.9+9
120 (99+9)/.9
121 (((sqrt(9))!)!+9-sqrt(9))/(sqrt(9))!
122 (((sqrt(9))!)!+9+sqrt(9))/(sqrt(9))!
123 (9/(.9+.9))!+sqrt(9)
124 (((sqrt(9))!)!+(.(9)+sqrt(9))!)/(sqrt(9))!
125 ((sqrt(9))!-.(9))!+(sqrt(9))!-.(9) or (9/(.9+.9))^sqrt(9)
126 99+9*sqrt(9)
127 ((sqrt(9))!-.(9))!+(sqrt(9))!+.(9)
128 ((sqrt(9))!-.(9))!+9-.(9)
129 (9/(.9+.9))!+9
130 ((sqrt(9))!-.(9))!+9/.9
131 ((sqrt(9))!-.(9))^sqrt(9)!+(sqrt(9))! [ Editor: as typeset a "!"
follows the exponent sqrt(9); read as ((sqrt(9))!-.(9))^sqrt(9) +
(sqrt(9))! it gives 125+6 = 131. ]
132 9*((sqrt(9))!+9)-sqrt(9)