Write as many numbers as possible, from 0 upward, using only the digits 1, 9, 9 and 4, each exactly once...

In article <22DEC199312401316@csa5.lbl.gov>, sichase@csa5.lbl.gov
(SCOTT I CHASE) writes:

> Then on January 1 you can try to do the "Annual puzzle," which is to
> construct all the numbers from 0 to 100 that you can, from the
> numbers 1,9,9, and 4, preferably in order, using the same symbols
>
> 0 = (1*9-9)/4
> 1 = (1*9/9)^4
> 2 = sqrt(1*(9/9)*4)

Using + - * / () sqrt ! and concatenation only, ie no floor, ceiling,
repeated decimal or other trickery:

3 = -1 + 9 - 9 + 4
4 = 1*9/9*4
5 = 1 + 9 - 9 + 4
6 = 1*9/sqrt(9)*sqrt(4)
7 = 1*9/sqrt(9) + 4
8 = 1*9 + sqrt(9) - 4
9 = 1 + 9 + sqrt(9) - 4
10 = 1*9 - sqrt(9) + 4
11 = 1 + 9 - sqrt(9) + 4
12 = 1*9/sqrt(9)*4
13 = 1 + 9/sqrt(9)*4
14 = 1 + sqrt(9)*sqrt(9) + 4
15 = 1*9 + sqrt(9)*sqrt(4)
16 = 1 + 9 + sqrt(9)*sqrt(4)
17 = 1 + 9 + sqrt(9) + 4
18 = 1*9 + sqrt(9)^sqrt(4)
19 = -1 + 9 + 9 + sqrt(4)
20 = 1*9 + 9 + sqrt(4)
21 = -1 + 9 + 9 + 4
22 = 1*9 + 9 + 4
23 = 1 + 9 + 9 + 4
24 = 19 + 9 - 4
25 = (1 + 99)/4
26 = (1 + sqrt(9) + 9)*sqrt(4)
27 = 1/sqrt(9)*sqrt(9)^4
28 = 19 + sqrt(9)^sqrt(4)
29 = -1 + sqrt(9)*(sqrt(9)! + 4)
30 = 1*sqrt(9)*(sqrt(9)! + 4)
31 = 1 + sqrt(9)*(sqrt(9)! + 4)
32 = (-1 + sqrt(9))^(9 - 4)
33 = 1*sqrt(9)*(9 + sqrt(4))
34 = (-1 + 9 + 9)*sqrt(4)
35 = -1 + (9 + 9)*sqrt(4)
36 = 1*(9 + 9)*sqrt(4)
37 = 1 + (9 + 9)*sqrt(4)
38 = (1 + 9 + 9)*sqrt(4)
39 = 1*sqrt(9) + 9*4
40 = (-1 + 9)*(9 - 4)
41 = -1 + sqrt(9)!*(sqrt(9) + 4)
42 = 1*sqrt(9)!*(sqrt(9) + 4)
43 = 1 + sqrt(9)!*(sqrt(9) + 4)
44 = -1 + 9*(9 - 4)
45 = 1*9*(9 - 4)
46 = 1 + 9*(9 - 4)
47 = -1 + (sqrt(9)!/sqrt(9)) * 4!
48 = (1 + 9/9)*4!
49 = 19 + sqrt(9)! + 4!
50 = (1 + 9)*(9 - 4)
51 = 1*9*sqrt(9) + 4!
52 = (1 + 9 + sqrt(9))*4
53 = 19*sqrt(9) - 4
54 = 1*9*sqrt(9)*sqrt(4)
55 = 1 + 9*sqrt(9)*sqrt(4)
56 = (-1 + 9)*(sqrt(9) + 4)
57 = 1*9*9 - 4!
58 = 1*9*sqrt(9)! + 4
59 = 1 + 9*sqrt(9)! + 4
60 = (1 + 9)*sqrt(9)*sqrt(4)
61 = 19*sqrt(9) + 4
62 = -1 + 9*(sqrt(9) + 4)
63 = 1*9*(sqrt(9) + 4)
64 = 1 + 9*(sqrt(9) + 4)
65 = -1 + sqrt(9)!*(9 + sqrt(4))
66 = 1*sqrt(9)!*(9 + sqrt(4))
67 = 1 + sqrt(9)!*(9 + sqrt(4))
68 = (-1 + 9 + 9)*4
69 = -(1*sqrt(9)) + sqrt(9) * 4!
70 = (-1 + 9)*9 - sqrt(4)
71 = -1 + 9/sqrt(9)*4!
72 = 1*9/sqrt(9)*4!
73 = 1 + 9/sqrt(9)*4!
74 = (-1 + 9)*9 + sqrt(4)
75 = 1*99 - 4!
76 = (-1 + 9)*9 + 4
77 = 1*9*9 - 4
78 = 1 + 9*9 - 4
79 = 1*9*9 - sqrt(4)
80 = 1 + 9*9 - sqrt(4)
81 = 1*9*sqrt(9)^sqrt(4)
82 = 1 + 9*sqrt(9)^sqrt(4)
83 = 1*9*9 + sqrt(4)
84 = 1 + 9*9 + sqrt(4)
85 = 1*9*9 + 4
86 = 1 + 9*9 + 4
87 = 1*sqrt(9)! + sqrt(9)^4
88 = 1 + sqrt(9)! + sqrt(9)^4
89 = -1 + 9*(sqrt(9)! + 4)
90 = 1*9*(sqrt(9)! + 4)
91 = 1 + 9*(sqrt(9)! + 4)
92 = 1 - sqrt(9) + 94
93 = -1^9 + 94
94 = -1 + 99 - 4
95 = 19*(9 - 4)
96 = 1 + 99 - 4
97 = 1*sqrt(9) + 94
98 = 1 + sqrt(9) + 94
99 = -1 + sqrt(9)! + 94
100 = 1*sqrt(9)! + 94

The fact that both 9 and 4 are square made this realatively easy, in
fact most of these have several good representations.  I think next
year will be harder, not to mention year 2000

-- Stein Kulseth <stein@hal.nta.no>

1994 puzzle

Puzzle = write as many numbers as possible using only the digits 1,9,9,
and 4. Each digit must be used exactly once. Two digit numbers may
appear in the formula, but the individual digits of a multidigit number
should not themselves be the result of computation.

This puzzle was presented to members of the public at a University open
day. Formulas for all numbers between 1 and 100 were found. Messing
around later I computed solutions up to 111 quite easily. The puzzle
can be done with the digits of any year, but this year seems
particularly fruitful because 1,4,9, and 49 are all perfect squares.
When the year 2000 rolls around things are likely to get a lot more
difficult. It seems that there must be some smallest number that cannot
be written in this way. Finding it could be a good net.puzzle well
suited to multiple contributers. If the public can get up to 100, I
wouldn't mind betting rec.puzzles readers can take the list well beyond
200.

RULES OF THE GAME:

1. Numbers must be computed in sequence. Posting solutions for numbers
beyond an uncomputed number is strictly forbidden.

2. A combined list of solutions may be posted from time to time. Names
of contributors should be appended. Shortest formulas (smallest ascii
length) for each number are sought. The square root operation will be
denoted as for example %49 to denote the square root of 49. Other
operations are fairly self explanatory. Brackets are neccessary only
where ambiguity might arise.  For example %4! is ambiguous.
Mathematical rules for order of operations are to be followed.

1994 Puzzle - combined list (5th May)

1        = 1^994
2        = 4-1-9/9
3        = 94-91
4        = 4x1x9/9
5        = 4+1x9/9
6        = 4+1+9/9
7        = %9+%9+%4-1
8        = %9+%9+%4x1
9        = %9+%9+4-1
10        = %9+%9+4x1
11        = %9+%9+4+1
12        = %9x%9+4-1
13        = 14-9/9
14        = 19-9+4
15        = 9+9-4+1
16        = %9x4+%9+1
17        = 9+9-1^4
18        = 9+9x1^4
19        = 9+9+1^4
20        = 9+9+%4x1
21        = 9+9+4-1
22        = 9+9+4x1
23        = 9+9+4+1
24        = %9x9-4+1
25        = %9x9-%4x1
26        = 9x4-9-1
27        = 9x4x1-9
28        = 9x4-9+1
29        = %9x9+%4x1
30        = 49-19
31        = %9x9+4x1
32        = %9x9+4+1
33        = 4x9-%9x1
34        = 4x9-%9+1
35        = 4x9-1^9
36        = 4x9x1^9
37        = 4x9+1^9
38        = 4x9+%(%9+1)
39        = 49-9-1
40        = 49-9x1
41        = 49-9+1
42        = (%9)!x%49x1
43        = 49-(%9)!x1
44        = 9x4+9-1
45        = 9x4x1+9
46        = 9x4+9+1
47        = 49-%9+1
48        = 49-1^9
49        = 49x1^9
50        = 49+1^9
51        = 49+%9-1
52        = 49+%9x1
53        = 49+%9+1
54        = (1+4)x9+9
55        = 4x9+19
56        = %49x(9-1)
57        = 49+9-1
58        = 49+9x1
59        = 49+9+1
60        = %9x%4x(9+1)
61        = (%9)!x((%9)!+4)+1
62        = %49x9-1
63        = %49x9x1
64        = %49x9+1
65        = (%9)!x(9+%4)-1
66        = (%9)!x(9+%4)x1
67        = 19x4-9
68        = 49+19
69        = (4x(%9)!-1)x%9
70        = %49x(9+1)
71        = 9^%4-9-1
72        = 9^%4-9x1
73        = 9^%4-9+1
74        = (9-1)x9+%4
75        = 94-19
76        = 9x9-4-1
77        = 9x9-4x1
78        = 9x9-4+1
79        = 9x9-%4x1
80        = 9x9-1^4
81        = 9x9x1^4
82        = 9x9+1^4
83        = 9x9+%4x1
84        = 94-9-1
85        = 94-9x1
86        = 94-9+1
87        = 9x9+(4-1)!
88        = (9-1)x(9+%4)
89        = (%9)^4+9-1
90        = 94-%9-1
91        = 94-%9x1
92        = 94-%9+1
93        = 94-1^9
94        = 94x1^9
95        = 94+1^9
96        = 99+1-4
97        = 99-%4x1
98        = 99-1^4
99        = 99x1^4
100        = 99+1^4

Here are my solutions to the 1994 puzzle in the range 0-200.  There are
slight changes from what I earlier promised due to new solutions and a
couple of errors.

I have solutions for all number except 157 and 166. Furthermore I have
ordered solutions (the digits 1,9,9,4 appear in order) for all the
others except 155,165, and 194.

I am violating my own rules by posting solutions for numbers beyond 157
(the first unknown). But having such a complete list with so few
exceptions was so tempting I could not resist.

1        = 1^994
2        = -1-9/9+4
3        = -1x9/9+4
4        = 1x9/9x4
5        = 1+9-9+4
6        = 19-9-4
7        = -1+%9+%9+%4
8        = 1x%9+%9+%4
9        = -1+%9+%9+4
10        = 1x%9+%9+4
11        = 1+%9+%9+4
12        = 19-%9-4
13        = 1x%9x%9+4
14        = 19-9+4
15        = 1+9+9-4
16        = 1+%9+%9x4
17        = -1^9+9x%4
18        = 1^9x9x%4
19        = 1^9+9x%4
20        = 1x9+9+%4
21        = -1+9+9+4
22        = 1x9+9+4
23        = 1+9+9+4
24        = 1+%9x9-4
25        = 1x%9x9-%4
26        = -1-9+9x4
27        = -1x9+9x4
28        = 1-9+9x4
29        = 1x%9x9+%4
30        = 1+%9x9+%4
31        = 1x%9x9+4
32        = 1+%9x9+4
33        = -1x%9+9x4
34        = 1-%9+9x4
35        = -1^9+9x4
36        = 1^9x9x4
37        = 1^9+9x4
38        = -1+%9+9x4
39        = 1x%9+9x4
40        = 1+%9+9x4
41        = -1+(%9)!+9x4
42        = -1+9+9+4!  [ Editor: -1+9+9+24 = 41, not 42. ]
42 [Editor: was 43]        = 1x9+9+4!
43        = 1+9+9+4!
44        = -1+9+9x4
45        = 1x9+9x4
46        = 1+9+9x4
47        = -1+((%9)!/%9)x4!
48        = (1+9/9)x4!
49        = 1+((%9)!/%9)x4!
50        = (1+9)x(9-4)
51        = 1+(%9)!x9-4
52        = (1+9+%9)x4
53        = -1+(%9)!x%9^%4
54        = 1x(%9)!x%9^%4
55        = 19+9x4
56        = (19+9)x%4
57        = -1+(%9)!x9+4
58        = 1x(%9)!x9+4
59        = 1+(%9)!x9+4
60        = (1+9)x%9x%4
61        = 1+(%9)!x((%9)!+4)
62        = -1+9x(9-%4)
63        = 1x9x(9-%4)
64        = 1+9x(9-%4)
65        = -1+(%9)!x(9+%4)
66        = (1+9)x9-4!
67        = 1-(%9)!+%9x4!
68        = -1-%9+%9x4!
69        = -1x%9+%9x4!
70        = (1+9)x(9-%4)
71        = -1-9+9^%4
72        = -1x9+9^%4
73        = 1-9+9^%4
74        = -1+99-4!
75        = -19+94
76        = -1+9x9-4
77        = 1x9x9-4
78        = 1+9x9-4
79        = 1x9x9-%4
80        = 1+9x9-%4
81        = 1^9x9^%4
82        = 1^9+9^%4
83        = 1x9x9+%4
84        = -1-9+94
85        = -1x9+94
86        = 1-9+94
87        = -1-(%9)!+94
88        = -1x(%9)!+94
89        = -1+9+9^%4
90        = 1x9+9^%4
91        = 1+9+9^%4
92        = 1-%9+94
93        = -1^9+94
94        = 1^9x94
95        = 1^9+94
96        = 1+99-4
97        = 1x99-%4
98        = 1+99-%4
99        = -1+(%9)!+94
100        = -1+99+%4
101        = 1x99+%4
102        = -1+99+4
103        = 1x99+4
104        = 1+99+4
105        = 1x9x9+4!
106        = 1+9x9+4!
107        = -1+9x%9x4
108        = 1x9x%9x4
109        = 1+9x%9x4
110        = -1-9+(9-4)!
111        = -1x9+(9-4)!
112        = (19+9)x4
113        = 19+94
114        = 19x%9x%4
115        = 1-(%9)!+(9-4)!
116        = 19x(%9)!+%4
117        = -1x%9+(9-4)!
118        = (1+%9)!+94
119        = -1+(9/9+4)!
120        = 1x(9/9+4)!
121        = 1+(9/9+4)!
122        = -1+99+4!
123        = 1x99+4!
124        = 1+99+4!
125        = -1+(%9)!+(9-4)!
126        = 1x(%9)!+(9-4)!
127        = 1+(%9)!+(9-4)!  [Editor: "(%9)%" in the original]
128        = -1+9+(9-4)!
129        = 1x9+(9-4)!
130        = 1+9+(9-4)!
131        = (-1+(%9)!)!+9+%4
132        = (1+%9)x(9+4!)
133        = (-1+(%9)!)!+9+4
134        = -1+9x(-9+4!)
135        = 1x9x(-9+4!)
136        = 1+9x(-9+4!)
137        = -1-(%9)!+(%9)!x4!
138        = (-1+(%9)!)!+9x%4
139        = 19+(9-4)!
140        = -1-%9+(%9)!x4!
141        = -1x%9+(%9)!x4!
142        = 1-%9+(%9)!x4!
143        = -1+(%9+9)^%4
144        = 1x(%9+9)^%4
145        = 1+(%9+9)^%4
146        = -1+%9+(%9)!x4!
147        = 19x9-4!
148        = 1+%9+(%9)!x4!
149        = (-1+(%9)!)^%9+4!
150        = (1+9)x(-9+4!)
151        = 1+(%9)!+(%9)!x4!
152        = -1+9+(%9)!x4!
153        = 1x9+(%9)!x4!
154        = 1+9+(%9)!x4!
155        = no ordered solution.                = 149+(%9)!
156        = (-1+(%9)!)!+9x4
157        = no known solution ordered or otherwise.
158        = (-1+(%9)!!/9)x%4
159        = -1+(%9)!!/9x%4
160        = (-1+9x9)x%4
161        = -1+9x9x%4
162        = 1x9x9x%4
163        = 1+9x9x%4
164        = (1+9x9)x%4
165        = no ordered solution.                = 4!x((%9)!+1)-%9
166        = no known solution ordered or otherwise.
167        = 19x9-4
168        = (1+%9+%9)x4!
169        = 19x9-%4
170        = -1-9+(%9)!!/4
171        = -1x9+(%9)!!/4
172        = 1-9+(%9)!!/4
173        = 19x9+%4
174        = -1x(%9)!+(%9)!!/4
175        = 19x9+4
176        = -1-%9+(%9)!!/4
177        = -1x%9+(%9)!!/4
178        = 1-%9+(%9)!!/4
179        = -1+(%9)!x((%9)!+4!)
180        = (1+9)x9x%4
181        = 1+(%9)!x((%9)!+4!)
182        = -1+%9+(%9)!!/4
183        = 1x%9+(%9)!!/4
184        = 1+%9+(%9)!!/4
185        = -1+(%9)!+(%9)!!/4
186        = 1x(%9)!+(%9)!!/4
187        = 1+(%9)!+(%9)!!/4
188        = (-1+%9)x94
189        = (1+(%9)!)x(%9+4!)
190        = 19x((%9)!+4)
191        = -1+((%9)!)^%9-4!
192        = 1x((%9)!)^%9-4!
193        = 1+((%9)!)^%9-4!
194        = no ordered solution.                = (91+(%9)!)x%4
195        = 199-4
196        = (-1+99)x%4
197        = 199-%4
198        = 1x99x%4
199        = 1+99x%4
200        = (1+99)x%4

-- Ian H. <hawthorn@waikato.ac.nz>