In [1]:

    

from math import *

from Goulib.notebook import *
from Goulib.table import Table
from friedman import seq


    

In [2]:

    

max=100
t=Table(range(max+1),titles=['n'])
def column(n, monadic='-s', dyadic='+-*/^_', permut=False):
    t.addcol(n)
    col = len(t.titles)-1
    for e in seq(n, monadic, dyadic, permut):
        n = int(e[0])
        # h(e[0],'=',e[1])
        if n <= max:
            e=e[1]
            old=t.get(n,col)
            if old is None or old.complexity()>e.complexity():
                t.set(n, col, e)


    

In [3]:

    

column(2018,permut=True) #yeargame


    

    




WARNING:root:19 has no solution

In [4]:

    

column(2019,permut=True) #yeargame


    

    




WARNING:root:13 has no solution

In [5]:

    

column(4444) # four-four


    

    




WARNING:root:17 has no solution

In [6]:

    

column(123456789)


    

    




WARNING:root:58 has no solution

In [7]:

    

column(999) # 999 clock


    

    




WARNING:root:5 has no solution

In [8]:

    

t


    

    
Out[8]:





n
2018
2019
4444
123456789
999

0
$${218 \cdot 0}$$
$${219 \cdot 0}$$
$${44-44}$$
$${1+23+45+6-78+\sqrt{9}}$$
$${\left(9-9\right)9}$$
1
$${28 \cdot 0+1}$$
$${20-19}$$
$${\frac{4+4-4}{4}}$$
$${-\frac{1+23+45+6-78}{\sqrt{9}}}$$
$${\frac{\frac{9}{\sqrt{9}}}{\sqrt{9}}}$$
2
$${20-18}$$
$${2+19 \cdot 0}$$
$${\left(4+4-4\right)-\sqrt{4}}$$
$${1+23+4+56+7-89}$$
$${\frac{9+9}{9}}$$
3
$${-\left(\frac{10}{2}-8\right)}$$
$${2+10-9}$$
$${\frac{4+4+4}{4}}$$
$${\left(1+23+4+56-78\right)-\sqrt{9}}$$
$${\left(9+\sqrt{9}\right)-9}$$
4
$${2+10-8}$$
$${\frac{2+10}{\sqrt{9}}}$$
$${\left(4+4-\sqrt{4}\right)-\sqrt{4}}$$
$${-\frac{12+3+45+6-78}{\sqrt{9}}}$$
$${\frac{9+\sqrt{9}}{\sqrt{9}}}$$
5
$${\frac{10\sqrt{2}}{\sqrt{8}}}$$
$${10-2-\sqrt{9}}$$
$${\frac{4+4+\sqrt{4}}{\sqrt{4}}}$$
$${\frac{1+2+34+56-78}{\sqrt{9}}}$$
 
6
$${\left(0+\sqrt{18}\right)\sqrt{2}}$$
$${2+0+1+\sqrt{9}}$$
$${\frac{4+4+4}{\sqrt{4}}}$$
$${\left(1+2+34+56-78\right)-9}$$
$${\frac{9+9}{\sqrt{9}}}$$
7
$${\left(0+\sqrt{81}\right)-2}$$
$${\frac{20+1}{\sqrt{9}}}$$
$${\frac{44}{4}-4}$$
$${-\left(1+23+45+6+7-89\right)}$$
 
8
$${21 \cdot 0+8}$$
$${\frac{10}{2}+\sqrt{9}}$$
$${4+4+4-4}$$
$${\frac{12+34+56-78}{\sqrt{9}}}$$
$${9+\frac{9}{-9}}$$
9
$${0+\frac{18}{2}}$$
$${2+10-\sqrt{9}}$$
$${\frac{44}{4}-\sqrt{4}}$$
$${1+23+4+56-78+\sqrt{9}}$$
$${9+9-9}$$
10
$${2+0+1 \cdot 8}$$
$${20-1-9}$$
$${\frac{44-4}{4}}$$
$${\frac{1+23+45+6+7+8}{9}}$$
$${9+\frac{9}{9}}$$
11
$${20-1-8}$$
$${20 \cdot 1-9}$$
$${\frac{\frac{44}{\sqrt{4}}}{\sqrt{4}}}$$
$${1+2+34+56+7-89}$$
$${\frac{99}{9}}$$
12
$${20 \cdot 1-8}$$
$${20+1-9}$$
$${\frac{4+44}{4}}$$
$${\left(1+2+34+56-78\right)-\sqrt{9}}$$
$${9+\frac{9}{\sqrt{9}}}$$
13
$${20+1-8}$$
 
$${\frac{44}{4}+\sqrt{4}}$$
$${\frac{12+34+56+7+8}{9}}$$
 
14
$${10+\frac{8}{2}}$$
$${\frac{10}{2}+9}$$
$${4+4+4+\sqrt{4}}$$
$${\frac{12+34+5+67+8}{9}}$$
 
15
$${\frac{120}{8}}$$
$${2+10+\sqrt{9}}$$
$${4+\frac{44}{4}}$$
$${\left(12+34+56-78\right)-9}$$
$${9+9-\sqrt{9}}$$
16
$${0+18-2}$$
$${20-1-\sqrt{9}}$$
$${4+4+4+4}$$
$${\frac{1+23+45+67+8}{9}}$$
 
17
$${0+2 \cdot 8+1}$$
$${20 \cdot 1-\sqrt{9}}$$
 
$${\frac{1+23+45+6+78}{9}}$$
 
18
$${28-10}$$
$${20+1-\sqrt{9}}$$
$${\frac{44}{\sqrt{4}}-4}$$
$${\frac{1+23+4+56+78}{9}}$$
$${\left(9-\sqrt{9}\right)\sqrt{9}}$$
19
 
$${29-10}$$
 
$${\frac{1+2+34+56+78}{9}}$$
 
20
$${2+0+18}$$
$${\left(0+1+9\right)2}$$
$${\frac{44-4}{\sqrt{4}}}$$
$${\frac{12+34+56+78}{9}}$$
 
21
$${21+0 \cdot 8}$$
$${2+0+19}$$
$${\frac{44-\sqrt{4}}{\sqrt{4}}}$$
$${\left(12+34+56-78\right)-\sqrt{9}}$$
$${9+9+\sqrt{9}}$$
22
 
$${20-1+\sqrt{9}}$$
$${\frac{44\sqrt{4}}{4}}$$
$${\frac{123+4+56+7+8}{9}}$$
 
23
 
$${20+1\sqrt{9}}$$
$${\frac{44+\sqrt{4}}{\sqrt{4}}}$$
$${\frac{123+4+5+67+8}{9}}$$
 
24
$${\left(2+0+1\right)8}$$
$${20+1+\sqrt{9}}$$
$${\frac{4+44}{\sqrt{4}}}$$
$${\frac{123+4+5+6+78}{9}}$$
$${9\sqrt{9}-\sqrt{9}}$$
25
 
 
 
$${-\left(12+34+5+6+7-89\right)}$$
 
26
$${10+2 \cdot 8}$$
$${\left(10+\sqrt{9}\right)2}$$
$${4+\frac{44}{\sqrt{4}}}$$
$${\left(12+3+4+5+6+7-8\right)-\sqrt{9}}$$
 
27
$${20-1+8}$$
$${\left(2+0+1\right)9}$$
 
$${\frac{123+45+67+8}{9}}$$
$${9+9+9}$$
28
$${20+1 \cdot 8}$$
$${20-1+9}$$
$${44+4\left(-4\right)}$$
$${\frac{123+45+6+78}{9}}$$
 
29
$${20+\sqrt{81}}$$
$${20+1 \cdot 9}$$
 
$${\frac{123+4+56+78}{9}}$$
 
30
 
$${20+1+9}$$
 
$${\frac{1+23+45+6+7+8}{\sqrt{9}}}$$
$${9\sqrt{9}+\sqrt{9}}$$
31
 
 
 
$${1+23+4+56+7\left(-8\right)+\sqrt{9}}$$
 
32
 
$${2+10\sqrt{9}}$$
 
$${\frac{123+45+6-78}{\sqrt{9}}}$$
 
33
 
 
 
$${12+34+56-78+9}$$
$${\frac{99}{\sqrt{9}}}$$
34
 
$${\frac{102}{\sqrt{9}}}$$
 
$${\frac{1+234+56+7+8}{9}}$$
 
35
 
 
 
$${\frac{123+4+56-78}{\sqrt{9}}}$$
 
36
$${0+18 \cdot 2}$$
$${\left(2+10\right)\sqrt{9}}$$
$${44-4-4}$$
$${\frac{1+234+5+6+78}{9}}$$
$${9+9\sqrt{9}}$$
37
 
 
 
$${12+34+56+7+8\left(-9\right)}$$
 
38
$${20+18}$$
$${0+19 \cdot 2}$$
$${44-4-\sqrt{4}}$$
$${12+3+45+67-89}$$
 
39
$${\frac{80}{2}-1}$$
$${20+19}$$
 
$${\frac{12+34+56+7+8}{\sqrt{9}}}$$
 
40
$${\frac{10}{2}8}$$
$${\frac{120}{\sqrt{9}}}$$
$${44-\sqrt{4}-\sqrt{4}}$$
$${1+2+34+56+7\left(-8\right)+\sqrt{9}}$$
 
41
$${1+\frac{80}{2}}$$
 
 
$${\frac{1+234+56+78}{9}}$$
 
42
 
 
$${\left(44+\sqrt{4}\right)-4}$$
$${\frac{12+34+5+67+8}{\sqrt{9}}}$$
 
43
 
 
$${44+\frac{4}{-4}}$$
$${\left(12+34+56+7\left(-8\right)\right)-\sqrt{9}}$$
 
44
 
$${\frac{90}{2}-1}$$
$${4+44-4}$$
$${\left(1+2+34+5+6+7-8\right)-\sqrt{9}}$$
 
45
 
$${\frac{10}{2}9}$$
$${44+\frac{4}{4}}$$
$${\frac{12+3+45+67+8}{\sqrt{9}}}$$
 
46
 
$${1+\frac{90}{2}}$$
$${4+44-\sqrt{4}}$$
$${12+34+5+67+8\left(-9\right)}$$
 
47
 
 
 
$${1+23+45+67-89}$$
 
48
 
 
$${44+\sqrt{4}+\sqrt{4}}$$
$${\frac{12+345+67+8}{9}}$$
 
49
 
 
 
$${\frac{12+345+6+78}{9}}$$
 
50
 
 
$${4+44+\sqrt{4}}$$
$${1+2+34+5+6+7-8+\sqrt{9}}$$
 
51
 
 
 
$${\frac{1+23+45+6+78}{\sqrt{9}}}$$
 
52
 
 
$${4+4+44}$$
$${-\left(12+3+4+5+6+7-89\right)}$$
 
53
 
 
 
$${\frac{1+2+3+456+7+8}{9}}$$
 
54
$${\frac{108}{2}}$$
 
 
$${\frac{1+23+4+56+78}{\sqrt{9}}}$$
$${\left(9+9\right)\sqrt{9}}$$
55
 
 
 
$${\frac{1+23+456+7+8}{9}}$$
 
56
 
 
 
$${123+4+5+6+7-89}$$
 
57
 
$${\left(20-1\right)\sqrt{9}}$$
 
$${\frac{1+2+34+56+78}{\sqrt{9}}}$$
 
58
 
 
 
 
 
59
$${80-21}$$
$${20\sqrt{9}-1}$$
 
$${12+34+5+6+7-8+\sqrt{9}}$$
 
60
 
$${20 \cdot 1\sqrt{9}}$$
$${44+4 \cdot 4}$$
$${\frac{12+34+56+78}{\sqrt{9}}}$$
 
61
$${81-20}$$
$${20\sqrt{9}+1}$$
 
$${\frac{12+3+456+78}{9}}$$
 
62
 
 
 
$${\frac{1+23+456+78}{9}}$$
 
63
 
$${\left(20+1\right)\sqrt{9}}$$
 
$${\frac{123+45+6+7+8}{\sqrt{9}}}$$
 
64
$${\left(10-2\right)8}$$
 
 
$${1+23+45+67+8\left(-9\right)}$$
 
65
 
 
 
$${1+23+4+5+6+\frac{78}{\sqrt{9}}}$$
 
66
 
 
 
$${\frac{123+456+7+8}{9}}$$
 
67
 
$${\frac{201}{\sqrt{9}}}$$
 
$${\frac{1+23+4+567+8}{9}}$$
 
68
$${80-12}$$
 
 
$${\frac{1+2+34+567+8}{9}}$$
 
69
 
$${90-21}$$
 
$${\frac{12+34+567+8}{9}}$$
 
70
 
$${\frac{210}{\sqrt{9}}}$$
 
$${1+23+4+5+6+7+8\sqrt{9}}$$
 
71
 
$${91-20}$$
 
$${\frac{1+234+56-78}{\sqrt{9}}}$$
 
72
$${82-10}$$
$${\left(10-2\right)9}$$
 
$${\left(12+34+56-78\right)\sqrt{9}}$$
$${9 \cdot 9-9}$$
73
 
 
 
$${\frac{123+456+78}{9}}$$
 
74
 
 
 
$${1+2+34+5+6+\frac{78}{\sqrt{9}}}$$
 
75
 
 
 
$${12+34+5+6+7+8+\sqrt{9}}$$
 
76
 
 
 
 
 
77
$${80-2-1}$$
 
 
$${\frac{1+2+3+4+5+678}{9}}$$
 
78
$${10 \cdot 8-2}$$
$${90-12}$$
 
$${\frac{123+4+567+8}{9}}$$
$${9 \cdot 9-\sqrt{9}}$$
79
$${0+81-2}$$
 
 
$${\frac{1+23+4+5+678}{9}}$$
 
80
 
 
$${\left(44-4\right)\sqrt{4}}$$
$${\frac{1+2+34+5+678}{9}}$$
 
81
$${2+80-1}$$
 
 
$${\frac{12+34+5+678}{9}}$$
$${9\sqrt{9}\sqrt{9}}$$
82
$${2+10 \cdot 8}$$
$${92-10}$$
 
$${\frac{12+3+45+678}{9}}$$
 
83
$${2+0+81}$$
 
 
$${\frac{1+23+45+678}{9}}$$
 
84
 
 
$${\left(44-\sqrt{4}\right)\sqrt{4}}$$
$${\frac{123+45+6+78}{\sqrt{9}}}$$
$${9 \cdot 9+\sqrt{9}}$$
85
 
 
 
$${123+4+5+6+7\left(-8\right)+\sqrt{9}}$$
 
86
 
 
$${44\sqrt{4}-\sqrt{4}}$$
$${1+23+4+56+7-8+\sqrt{9}}$$
 
87
 
$${90-2-1}$$
 
$${\frac{123+4+56+78}{\sqrt{9}}}$$
 
88
 
$${10 \cdot 9-2}$$
$${44+44}$$
$${12+34+5+6+7+8\sqrt{9}}$$
 
89
 
$${0+91-2}$$
 
$${\left(1+2+34+56+7-8\right)-\sqrt{9}}$$
 
90
$${\frac{180}{2}}$$
 
$${44\sqrt{4}+\sqrt{4}}$$
$${\frac{1+234+567+8}{9}}$$
$${99-9}$$
91
 
$${2+90-1}$$
 
$${123+4+5+6+7\left(-8\right)+9}$$
 
92
$${12+80}$$
$${2+10 \cdot 9}$$
$${4+44\sqrt{4}}$$
$${123+45+6+7-89}$$
 
93
 
$${102-9}$$
 
$${\left(123+45+6-78\right)-\sqrt{9}}$$
 
94
$${102-8}$$
 
 
$${\frac{123+45+678}{9}}$$
 
95
 
$${\frac{190}{2}}$$
 
$${\frac{12+345+6-78}{\sqrt{9}}}$$
 
96
$${\left(2+10\right)8}$$
 
$${\left(4+44\right)\sqrt{4}}$$
$${\left(123+4+56-78\right)-9}$$
$${99-\sqrt{9}}$$
97
 
 
 
$${12+3+45+6+7+8\sqrt{9}}$$
 
98
 
 
 
$${\left(12+34+56+7-8\right)-\sqrt{9}}$$
 
99
 
$${102-\sqrt{9}}$$
 
$${123+45+6-78+\sqrt{9}}$$
 
100
 
 
 
$${1+2+3+4+5+6+7+8 \cdot 9}$$
 

In [9]:

    

from friedman import Monadic, context
functions=context.functions


    

In [10]:

    

print(functions.keys()) #list allowed functions


    

    




SortedKeysView(SortedDict({'acos': (<built-in function acos>, 9999, None, None, '\\arccos'), 'acosh': (<built-in function acosh>, 9999, None, None, '\\cosh^{-1}'), 'asin': (<built-in function asin>, 9999, None, None, '\\arcsin'), 'asinh': (<built-in function asinh>, 9999, None, None, '\\sinh^{-1}'), 'atan': (<built-in function atan>, 9999, None, None, '\\arctan'), 'atan2': (<built-in function atan2>, 9999, None, None, None), 'atanh': (<built-in function atanh>, 9999, None, None, '\\tanh^{-1}'), 'copysign': (<built-in function copysign>, 9999, None, None, None), 'cos': (<built-in function cos>, 9999, None, None, None), 'cosh': (<built-in function cosh>, 9999, None, None, None), 'exp': (<built-in function exp>, 9999, None, None, 'e^{%s}'), 'factorial': (<built-in function factorial>, 9999, '%s!', 'fact', '%s!'), 'factorial2': (<function factorial2 at 0x0000018125328510>, 9999, '%s!', 'fact', '%s!!'), 'fmod': (<built-in function fmod>, 9999, None, None, None), 'gamma': (<built-in function gamma>, 9999, None, None, '\\Gamma'), 'gcd': (<built-in function gcd>, 9999, None, None, None), 'hypot': (<built-in function hypot>, 9999, None, None, None), 'isclose': (<built-in function isclose>, 9999, None, None, None), 'ldexp': (<built-in function ldexp>, 9999, None, None, None), 'log': (<built-in function log>, 9999, None, None, '\\ln'), 'log10': (<built-in function log10>, 9999, None, None, '\\log_{10}'), 'log2': (<built-in function log2>, 9999, None, None, '\\log_2'), 'modf': (<built-in function modf>, 9999, None, None, None), 'pow': (<built-in function pow>, 9999, None, None, None), 'remainder': (<built-in function remainder>, 9999, None, None, None), 'sin': (<built-in function sin>, 9999, None, None, None), 'sinh': (<built-in function sinh>, 9999, None, None, None), 'sqrt': (<function sqrt at 0x0000018125315598>, 9999, None, None, '\\sqrt{%s}'), 'tan': (<built-in function tan>, 9999, None, None, None), 'tanh': (<built-in function tanh>, 9999, None, None, None)}))

In [11]:

    

m=Monadic(0,functions)+Monadic(1,functions)+Monadic(2,functions)+Monadic(5,functions)+Monadic(7,functions)
for x in m.items():
    h(x[0],'=',x[1])


    

    





0 = ${\arccos\left(1\right)}$






    





0.28366218546322625 = ${\cos\left(5\right)}$






    





0.3010299956639812 = ${\log_{10}\left(2\right)}$






    





0.4161468365471424 = ${-\cos\left(2\right)}$






    





0.5403023058681398 = ${\cos\left(1\right)}$






    





0.6569865987187891 = ${\sin\left(7\right)}$






    





0.6931471805599453 = ${\ln\left(2\right)}$






    





0.6989700043360189 = ${\log_{10}\left(5\right)}$






    





0.7539022543433046 = ${\cos\left(7\right)}$






    





0.7615941559557649 = ${\tanh\left(1\right)}$






    





0.7853981633974483 = ${\arctan\left(1\right)}$






    





0.8414709848078965 = ${\sin\left(1\right)}$






    





0.8450980400142568 = ${\log_{10}\left(7\right)}$






    





0.8714479827243188 = ${\tan\left(7\right)}$






    





0.8813735870195429 = ${\sinh^{-1}\left(1\right)}$






    





0.9092974268256817 = ${\sin\left(2\right)}$






    





0.9589242746631385 = ${-\sin\left(5\right)}$






    





0.9640275800758169 = ${\tanh\left(2\right)}$






    





0.9999092042625951 = ${\tanh\left(5\right)}$






    





0.9999983369439447 = ${\tanh\left(7\right)}$






    





1 = ${\Gamma\left(2\right)}$






    





1.1071487177940904 = ${\arctan\left(2\right)}$






    





1.1752011936438014 = ${\sinh\left(1\right)}$






    





1.3169578969248166 = ${\cosh^{-1}\left(2\right)}$






    





1.373400766945016 = ${\arctan\left(5\right)}$






    





1.4142135623730951 = ${\sqrt{2}}$






    





1.4288992721907328 = ${\arctan\left(7\right)}$






    





1.4436354751788103 = ${\sinh^{-1}\left(2\right)}$






    





1.5430806348152437 = ${\cosh\left(1\right)}$






    





1.5574077246549023 = ${\tan\left(1\right)}$






    





1.5707963267948966 = ${\arcsin\left(1\right)}$






    





1.6094379124341003 = ${\ln\left(5\right)}$






    





1.9459101490553132 = ${\ln\left(7\right)}$






    





2 = ${2}$






    





2.185039863261519 = ${-\tan\left(2\right)}$






    





2.23606797749979 = ${\sqrt{5}}$






    





2.2924316695611777 = ${\cosh^{-1}\left(5\right)}$






    





2.3124383412727525 = ${\sinh^{-1}\left(5\right)}$






    





2.321928094887362 = ${\log_2\left(5\right)}$






    





2.6339157938496336 = ${\cosh^{-1}\left(7\right)}$






    





2.644120761058629 = ${\sinh^{-1}\left(7\right)}$






    





2.6457513110645907 = ${\sqrt{7}}$






    





2.718281828459045 = ${e^{1}}$






    





2.807354922057604 = ${\log_2\left(7\right)}$






    





3.380515006246586 = ${-\tan\left(5\right)}$






    





3.6268604078470186 = ${\sinh\left(2\right)}$






    





3.7621956910836314 = ${\cosh\left(2\right)}$






    





5 = ${5}$






    





7 = ${7}$






    





7.38905609893065 = ${e^{2}}$






    





24 = ${\Gamma\left(5\right)}$






    





74.20321057778875 = ${\sinh\left(5\right)}$






    





74.20994852478785 = ${\cosh\left(5\right)}$






    





120 = ${5!}$






    





148.4131591025766 = ${e^{5}}$






    





548.3161232732465 = ${\sinh\left(7\right)}$






    





548.3170351552121 = ${\cosh\left(7\right)}$






    





720 = ${\Gamma\left(7\right)}$






    





1096.6331584284585 = ${e^{7}}$






    





5040 = ${7!}$