Count Up with the Four Fours Puzzle

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username5243
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Count Up with the Four Fours Puzzle

Postby username5243 » Sun Dec 06, 2015 9:08 pm UTC

In this thread we count up using the classic four fours puzzle: how high can we get with only four fours. You are allowed to use standard operations, square root, and any other operation you can think of (so long as it doesn't in't involve other numbers). We'll see how high we can go.

I'll start with 0:

0 = 44-44
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Re: Count Up with the Four Fours Puzzle

Postby emlightened » Sun Dec 06, 2015 9:16 pm UTC

Sure.

1 = !(!4-4)/44

!k is the subfactorial, and counts derangements. The first few values (k = 0, 1, 2...) are 1, 0, 1, 2, 9, 44, 265...
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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Sun Dec 06, 2015 9:54 pm UTC

2 = 4/4+4/4

I prefer just using single digit 4s with addition, subtraction, multiplication, division, and exponentiation.

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Re: Count Up with the Four Fours Puzzle

Postby username5243 » Sun Dec 06, 2015 10:20 pm UTC

3 = (4+4+4)/4
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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Sun Dec 06, 2015 10:25 pm UTC

4 = 4 + 4*(4-4)

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Re: Count Up with the Four Fours Puzzle

Postby username5243 » Sun Dec 06, 2015 10:33 pm UTC

5 = (4*4+4)/4
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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Sun Dec 06, 2015 10:33 pm UTC

6 = 4+(4+4)/4

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Re: Count Up with the Four Fours Puzzle

Postby patzer » Sun Dec 06, 2015 10:59 pm UTC

7 = 4-4/4+4
If it looks like a duck, and quacks like a duck, we have at least to consider the possibility that we have a small aquatic bird of the family Anatidae on our hands. –Douglas Adams

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Re: Count Up with the Four Fours Puzzle

Postby username5243 » Sun Dec 06, 2015 10:59 pm UTC

8 = 4*4-4-4
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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Sun Dec 06, 2015 11:01 pm UTC

9 = 4+4+4/4

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Re: Count Up with the Four Fours Puzzle

Postby username5243 » Sun Dec 06, 2015 11:06 pm UTC

(44-4)/4 = 10
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Re: Count Up with the Four Fours Puzzle

Postby emlightened » Sun Dec 06, 2015 11:25 pm UTC

I believe the only positive integers <100 we can make with the basic 5 operations and 4 4's (no concentration) are 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 16, 20, 28, 32, 36, 48, 60, 63, 64, 65, 68 and 81. A few more with sqrt, but not very much further.

11 = 44/√(4*4)
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Re: Count Up with the Four Fours Puzzle

Postby username5243 » Sun Dec 06, 2015 11:38 pm UTC

12 = 4*(4-4/4)
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Re: Count Up with the Four Fours Puzzle

Postby emlightened » Mon Dec 07, 2015 5:23 pm UTC

13 = ((√4+√4)!+√4)/√4
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Re: Count Up with the Four Fours Puzzle

Postby patzer » Mon Dec 07, 2015 6:27 pm UTC

14 = ⌊((4!)/(∜(4+4)))⌋
If it looks like a duck, and quacks like a duck, we have at least to consider the possibility that we have a small aquatic bird of the family Anatidae on our hands. –Douglas Adams

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Re: Count Up with the Four Fours Puzzle

Postby username5243 » Mon Dec 07, 2015 8:36 pm UTC

That's a fourth root and floor function, by the way.

15 = (44/4)+4
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Re: Count Up with the Four Fours Puzzle

Postby emlightened » Mon Dec 07, 2015 9:24 pm UTC

16 = (4!-4‼)+4-4 = 4+4+4+4

That's factorial and double factorial.
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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Tue Dec 08, 2015 1:01 am UTC

17 = 4*4+4/4

Yay, another one that can be done without concatenation or unary operators :)

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Re: Count Up with the Four Fours Puzzle

Postby SirGabriel » Tue Dec 08, 2015 2:13 am UTC

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

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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Tue Dec 08, 2015 4:13 am UTC

19 = 4! - 4 - 4/4

I wrote some code to find all the answers to this. All the numbers it's possible to generate using addition, subtraction, multiplication, division, exponentiation (with the maximum exponent limited to 16384), and concatenation are in the spoiler below:
Spoiler:
0: 4+4-4-4
1: (4+4-4)/4
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: 4+4-4/4
8: 4+4+4-4
9: 4+4+4/4
10: (44-4)/4
12: (44+4)/4
15: 4*4-4/4
16: 4+4+4+4
17: 4*4+4/4
20: (4/4+4)*4
24: 4+4+4*4
28: (4+4)*4-4
32: 4*4+4*4
36: (4+4)*4+4
43: 44-4/4
44: 4-4+44
45: 4/4+44
48: (4+4+4)*4
52: 4+4+44
60: 4*4*4-4
63: (4^4-4)/4
64: (4+4)*(4+4)
65: (4^4+4)/4
68: 4*4*4+4
80: (4*4+4)*4
81: (4/4-4)^4
88: 44+44
111: 444/4
128: (4+4)*4*4
160: (44-4)*4
172: 44*4-4
180: 44*4+4
192: (44+4)*4
212: 4^4-44
240: 4^4-4*4
248: 4^4-4-4
255: 4^4-4/4
256: (4+4-4)^4
257: 4/4+4^4
264: 4+4+4^4
272: 4*4+4^4
300: 4^4+44
352: (4+4)*44
440: 444-4
448: 444+4
512: 4^4+4^4
625: (4/4+4)^4
704: 4*4*44
1008: (4^4-4)*4
1020: 4^4*4-4
1024: (4+4)^4/4
1028: 4^4*4+4
1040: (4^4+4)*4
1776: 444*4
1936: 44*44
2048: (4+4)*4^4
4092: (4+4)^4-4
4096: 4*4*4^4
4100: (4+4)^4+4
4444: 4444
11264: 4^4*44
14641: (44/4)^4
16384: (4+4)^4*4
20736: (4+4+4)^4
65532: (4*4)^4-4
65536: 4^4*4^4
65540: (4*4)^4+4
160000: (4*4+4)^4
262144: (4*4)^4*4
937024: 44^4/4
1048576: ((4+4)*4)^4
2560000: (44-4)^4
3748092: 44^4-4
3748100: 44^4+4
4194304: 4^(44/4)
5308416: (44+4)^4
14992384: 44^4*4
16777216: (4+4)^(4+4)
959512576: (44*4)^4
1073741824: (4^4)^4/4
4032758016: (4^4-4)^4
4294967292: (4^4)^4-4
4294967296: (4*4)^(4+4)
4294967300: (4^4)^4+4
4569760000: (4^4+4)^4
17179869184: (4^4)^4*4
38862602496: 444^4
1099511627776: (4^4*4)^4
14048223625216: 44^(4+4)
281474976710656: ((4+4)^4)^4
18446744073709551616: ((4*4)^4)^4
1208925819614629174706176: 4^(44-4)
77371252455336267181195264: 4^44/4
197352587024076973231046656: (44^4)^4
309485009821345068724781052: 4^44-4
309485009821345068724781060: 4^44+4
1237940039285380274899124224: 4^44*4
79228162514264337593543950336: 4^(44+4)
340282366920938463463374607431768211456: ((4^4)^4)^4
5444517870735015415413993718908291383296: (4+4)^44
95780971304118053647396689196894323976171195136475136: (4*4)^44
2050773823560610053645205609172376035486179836520607547294916966189367296: 44^44
9173994463960286046443283581208347763186259956673124494950355357547691504353939232280074212440502746218496: (4^4)^44
52374249726338269920211035149241586435466272736689036631732661889538140742474792878132321477214466514414186946040961136147476104734166288853256441430016: 4^(4^4-4)
3351951982485649274893506249551461531869841455148098344430890360930441007518386744200468574541725856922507964546621512713438470702986642486608412251521024: 4^4^4/4
13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084092: 4^4^4-4
13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084100: 4^4^4+4
53631231719770388398296099992823384509917463282369573510894245774887056120294187907207497192667613710760127432745944203415015531247786279785734596024336384: 4^4^4*4
3432398830065304857490950399540696608634717650071652704697231729592771591698828026061279820330727277488648155695740429018560993999858321906287014145557528576: 4^(4^4+4)
1552518092300708935148979488462502555256886017116696611139052038026050952686376886330878408828646477950487730697131073206171580044114814391444287275041181139204454976020849905550265285631598444825262999193716468750892846853816057856: (4+4)^4^4
2063650512248692368563827284830142994214247367328599695812346519635444931862206482321942405811160890213571855442410658901884170154307365379884917884620857722298385484371113610034107490923540785363375909797699954703703235518560788042337487885808736236287260081631789056: 4^444
179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586298239947245938479716304835356329624224137216: (4*4)^4^4
5295234290518813958694052696765703539877636129016726673614641118920249401528833695219291788853023674320519052888070721543349779449470353315509176840272326589840282387004815049507387058834387690545681298091791630344876593393965914765953202265859793057074640778689676074897426540352087177411464898988145449760179672865153814333080118198079695296777241118464609887804811597512038167059947328040161798508872824087556816633856: 44^4^4
32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638215525166389437335543602135433229604645318478604952148193555853611059596230656: (4^4)^4^4
1090748135619415929462984244733782862448264161996232692431832786189721331849119295216264234525201987223957291796157025273109870820177184063610979765077554799078906298842192989538609825228048205159696851613591638196771886542609324560121290553901886301017900252535799917200010079600026535836800905297805880952350501630195475653911005312364560014847426035293551245843928918752768696279344088055617515694349945406677825140814900616105920256438504578013326493565836047242407382442812245131517757519164899226365743722432277368075027627883045206501792761700945699168497257879683851737049996900961120515655050115561271491492515342105748966629547032786321505730828430221664970324396138635251626409516168005427623435996308921691446181187406395310665404885739434832877428167407495370993511868756359970390117021823616749458620969857006263612082706715408157066575137281027022310927564910276759160520878304632411049364568754920967322982459184763427383790272448438018526977764941072715611580434690827459339991961414242741410599117426060556483763756314527611362658628383368621157993638020878537675545336789915694234433955666315070087213535470255670312004130725495834508357439653828936077080978550578912967907352780054935621561090795845172954115972927479877527738560008204118558930004777748727761853813510493840581861598652211605960308356405941821189714037868726219481498727603653616298856174822413033485438785324024751419417183012281078209729303537372804574372095228703622776363945290869806258422355148507571039619387449629866808188769662815778153079393179093143648340761738581819563002994422790754955061288818308430079648693232179158765918035565216157115402992120276155607873107937477466841528362987708699450152031231862594203085693838944657061346236704234026821102958954951197087076546186622796294536451620756509351018906023773821539532776208676978589731966330308893304665169436185078350641568336944530051437491311298834367265238595404904273455928723949525227184617404367854754610474377019768025576605881038077270707717942221977090385438585844095492116099852538903974655703943973086090930596963360767529964938414598185705963754561497355827813623833288906309004288017321424808663962671333528009232758350873059614118723781422101460198615747386855096896089189180441339558524822867541113212638793675567650340362970031930023397828465318547238244232028015189689660418822976000815437610652254270163595650875433851147123214227266605403581781469090806576468950587661997186505665475715792896: 4^(4+4)^4
-1: (4-4-4)/4
-2: (4+4)/4-4
-3: (4/4)^4-4
-4: (4-4)*4-4
-7: 4/4-4-4
-8: 4+4-4*4
-10: (4-44)/4
-12: (4/4-4)*4
-15: 4/4-4*4
-16: (4-4-4)*4
-28: 4*4-44
-36: 4+4-44
-43: 4/4-44
-44: 4-4-44
-48: (4-4*4)*4
-60: 4-4*4*4
-63: (4-4^4)/4
-160: (4-44)*4
-172: 4-44*4
-212: 44-4^4
-240: 4*4-4^4
-248: 4+4-4^4
-255: 4/4-4^4
-256: 4-4-4^4
-440: 4-444
-1008: (4-4^4)*4
-1020: 4-4^4*4
-4092: 4-(4+4)^4
-65532: 4-(4*4)^4
-3748092: 4-44^4
-4294967292: 4-(4^4)^4
-309485009821345068724781052: 4-4^44
-13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084092: 4-4^4^4
Last edited by faubiguy on Tue Dec 08, 2015 4:43 am UTC, edited 2 times in total.

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patzer
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Re: Count Up with the Four Fours Puzzle

Postby patzer » Tue Dec 08, 2015 4:30 am UTC

20 = 4*(⌈(√((((√4)!)((4/(√4))!))!))⌉)
If it looks like a duck, and quacks like a duck, we have at least to consider the possibility that we have a small aquatic bird of the family Anatidae on our hands. –Douglas Adams

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faubiguy
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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Tue Dec 08, 2015 4:44 am UTC

21 = 4! - 4 + 4/4

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Re: Count Up with the Four Fours Puzzle

Postby patzer » Tue Dec 08, 2015 11:00 am UTC

22 = 44/(4-√4)
If it looks like a duck, and quacks like a duck, we have at least to consider the possibility that we have a small aquatic bird of the family Anatidae on our hands. –Douglas Adams

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faubiguy
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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Tue Dec 08, 2015 11:33 am UTC

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

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Re: Count Up with the Four Fours Puzzle

Postby Elvish Pillager » Tue Dec 08, 2015 2:41 pm UTC

24 = 4*4 + 4 + 4

Using unary operators feels a bit weird to me. Especially square root: we can't use cube root because it has a 3 in it, but we can use square root just because the symbolic notation doesn't have a number drawn in it...? And the successor operator must obviously be forbidden, but there's no obvious rule about where to draw the line.

Because I was curious, I wrote a script to see what numbers are accessible using only a SINGLE four, by repeatedly applying the unary operators that have already been used in this thread (floor, square root, factorial, subfactorial, and double factorial).

The script:
Spoiler:

Code: Select all

import math
from functools import reduce
result = {}
result [4] = "4"
frontier = [4]
next_frontier = []
sub_factorial = [0, 0, 1, 2, 9, 44, 265, 1854, 14883]
def record (number, representation):
  if number not in result:
    result [number] = representation
    next_frontier.append (number)
while len(frontier) >0:
  for input in frontier:
    root = math.floor (math.sqrt (input))
   
    record (root, ("" if root*root == input else "floor.") + "sqrt(" + result [input] + ")")
    if input >0 and input <100:
      record (math.factorial (input), "(" + result [input] + ")!")
      record (reduce(lambda x,y: y*x, range(input,0,-2)), "(" + result [input] + ")!!")
    if input <len( sub_factorial):
      record (sub_factorial [input], "!(" + result [input] + ")")
  frontier = next_frontier
  next_frontier = []
 
for number in range (0, 200):
  if number in result:
    print (str( number) + " = " + result [number])
  else:
    print ("...")

The results:
Spoiler:

Code: Select all

0 = !(floor.sqrt(sqrt(4)))
1 = floor.sqrt(sqrt(4))
2 = sqrt(4)
3 = sqrt(!(4))
4 = 4
5 = floor.sqrt(floor.sqrt((!(4))!!))
6 = (sqrt(!(4)))!
7 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!!)))
8 = (4)!!
9 = !(4)
10 = floor.sqrt((floor.sqrt(floor.sqrt((!(4))!!)))!)
11 = sqrt(floor.sqrt(!((4)!!)))
12 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(((4)!!)!!))!!)))
13 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(((floor.sqrt(floor.sqrt((!(4))!!)))!!)!))))!!))))
14 = floor.sqrt(floor.sqrt(((4)!!)!))
15 = (floor.sqrt(floor.sqrt((!(4))!!)))!!
16 = floor.sqrt(!((sqrt(!(4)))!))
17 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!)))))
18 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!!))))
19 = floor.sqrt(((4)!!)!!)
20 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt((floor.sqrt((floor.sqrt(floor.sqrt((!(4))!!)))!))!!))!!)))))
21 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!!))))!))!)))))))!!))))
22 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((floor.sqrt(floor.sqrt((!(4))!!)))!!)!!)))!)))))
23 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!)))
24 = (4)!
25 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!))))!))))
26 = floor.sqrt(((sqrt(!(4)))!)!)
27 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!))))!!)))
28 = floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!))
...
30 = floor.sqrt((!(4))!!)
31 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!))))!)))))!!))))!)))))
32 = floor.sqrt(floor.sqrt(floor.sqrt(((floor.sqrt(floor.sqrt((!(4))!!)))!!)!)))
33 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!))))!!)))))!!)))))
34 = floor.sqrt(floor.sqrt(floor.sqrt(((4)!)!!)))
35 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!)))))
36 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!!))))!))!))))))
37 = floor.sqrt(floor.sqrt(((floor.sqrt(floor.sqrt((!(4))!!)))!!)!!))
...
...
40 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!))))!)))))!!)))
...
...
43 = floor.sqrt(floor.sqrt((floor.sqrt((floor.sqrt(floor.sqrt((!(4))!!)))!))!))
44 = !(floor.sqrt(floor.sqrt((!(4))!!)))
...
46 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!)))
...
48 = ((sqrt(!(4)))!)!!
...
50 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((!(floor.sqrt(floor.sqrt((!(4))!!))))!)))))
51 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(((sqrt(!(4)))!)!))!!)))
...
53 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((!(floor.sqrt(floor.sqrt((!(4))!!))))!!))))
54 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!))))))!!)))!))))))
...
56 = floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!!))
...
58 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((((sqrt(!(4)))!)!!)!))))))!)))))))!))))))
...
60 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((((sqrt(!(4)))!)!!)!))))))!)))))))!!)))))
61 = floor.sqrt((floor.sqrt((floor.sqrt(floor.sqrt((!(4))!!)))!))!!)
62 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!!))))!))))))
63 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!))))!)))))
...
65 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!!))))!!)))))
...
67 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((sqrt(floor.sqrt(!((4)!!))))!)))!))))))
68 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!))))!!))))
69 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!))))
70 = floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!!))))!)
...
...
...
...
...
76 = floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!))))))!!))
77 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((((sqrt(!(4)))!)!!)!))))))!))))))
78 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!!)))
79 = floor.sqrt(floor.sqrt((sqrt(floor.sqrt(!((4)!!))))!))
...
81 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((((sqrt(!(4)))!)!!)!)))))
...
...
...
...
86 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((((sqrt(!(4)))!)!!)!!))))
...
...
...
...
...
...
...
94 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!!)))))!)))
...
...
...
...
...
...
101 = floor.sqrt((sqrt(floor.sqrt(!((4)!!))))!!)
...
103 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((!(floor.sqrt(floor.sqrt((!(4))!!))))!))))))!)))))
...
105 = (floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(!((sqrt(!(4)))!)))!!))))!!
106 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt((!(4))!!))!))))
...
108 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((((sqrt(!(4)))!)!!)!!)))))!))))))
...
110 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((!(floor.sqrt(floor.sqrt((!(4))!!))))!))))))!!))))
...
...
113 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((((sqrt(!(4)))!)!!)!!)))))!!)))))
...
...
116 = floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!!)))))!!))
117 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(((sqrt(!(4)))!)!))!!))))!)))))
...
119 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt((!(4))!!))!!)))
120 = (floor.sqrt(floor.sqrt((!(4))!!)))!
121 = floor.sqrt(!((4)!!))
...
123 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(((sqrt(!(4)))!)!))!!))))!!))))
...
...
...
...
...
...
...
131 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!))))!)))))!!))))!))))))!))))
...
...
...
...
136 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(((4)!!)!!))!)))
...
...
...
...
...
...
...
144 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!))))!)))))!!))))!))))))!!)))
...
...
147 = floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(((4)!!)!!))!!))))!))
...
149 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((!(floor.sqrt(floor.sqrt((!(4))!!))))!!)))))!)))))
...
...
...
...
...
...
...
...
158 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((!(floor.sqrt(floor.sqrt((!(4))!!))))!!)))))!!))))
159 = floor.sqrt(floor.sqrt((floor.sqrt(((4)!!)!!))!!))
...
...
...
163 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(((floor.sqrt(floor.sqrt((!(4))!!)))!!)!))))!))))
...
...
...
...
...
169 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!))))))!!)))!)))))))!)))))
...
...
...
...
...
...
...
...
...
...
...
...
182 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!))))))!!)))!)))))))!!))))
...
184 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(((floor.sqrt(floor.sqrt((!(4))!!)))!!)!))))!!)))
...
...
...
...
...
...
191 = floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(((4)!!)!)))!!)))!)))))!!))))))!!)))))!))))!))))))
...
...
...
...
...
...
198 = floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt(floor.sqrt((floor.sqrt((floor.sqrt((floor.sqrt(floor.sqrt((!(4))!!)))!))!!))!!))))))!)))
...


I suspect that all natural numbers are accessible from a single 4 using only these functions. Note that my script doesn't consider taking factorials of very high numbers, or taking multiple square roots without flooring in between.
Also known as Eli Dupree. Check out elidupree.com for my comics, games, and other work.

GENERATION A(g64, g64): Social experiment. Take the busy beaver function of the generation number and add it to your signature.

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patzer
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Re: Count Up with the Four Fours Puzzle

Postby patzer » Wed Dec 09, 2015 4:33 am UTC

Good point, but there isn't really an alternative - most numbers require the use of unary operators in the solution.

25 = 4!+4/(√4+√4)
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Re: Count Up with the Four Fours Puzzle

Postby generalz » Wed Dec 09, 2015 1:26 pm UTC

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

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Re: Count Up with the Four Fours Puzzle

Postby patzer » Wed Dec 09, 2015 3:29 pm UTC

27 = √!4*√!4*√!4*⌊√√4⌋
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Lawrencelot
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Re: Count Up with the Four Fours Puzzle

Postby Lawrencelot » Wed Dec 09, 2015 3:39 pm UTC

28 = 4! + 4 + (4-4)!

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Re: Count Up with the Four Fours Puzzle

Postby faubiguy » Wed Dec 09, 2015 3:52 pm UTC

29 = 4! + 4 + (4-4)!

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Re: Count Up with the Four Fours Puzzle

Postby emlightened » Wed Dec 09, 2015 6:54 pm UTC

30 = √4*√!4*(√4+√!4) = (4+4)*4-√4

I try to limit myself to the basic ops, sqrt, and factorial variants. Sometimes I use % or start with (say) .4, but I've never liked using ceiling/floor.
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Re: Count Up with the Four Fours Puzzle

Postby JackHK » Wed Dec 09, 2015 7:14 pm UTC

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

This one was surprisingly difficult to find...

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Re: Count Up with the Four Fours Puzzle

Postby patzer » Wed Dec 09, 2015 7:23 pm UTC

32 = 4(√!4+√4)/√4
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Re: Count Up with the Four Fours Puzzle

Postby PsiSquared » Wed Dec 09, 2015 7:36 pm UTC

emlightened wrote:I try to limit myself to the basic ops, sqrt, and factorial variants. Sometimes I use % or start with (say) .4, but I've never liked using ceiling/floor.


Yeah. Allowing the floor function makes the game kind of pointless (you just use factorials and squareroots for as long as it takes to reach your desired number).

Anyway:

33 = 4!+√!4+4-√4

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Re: Count Up with the Four Fours Puzzle

Postby patzer » Wed Dec 09, 2015 8:21 pm UTC

34 = 4*4*√4+√4
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Re: Count Up with the Four Fours Puzzle

Postby JackHK » Wed Dec 09, 2015 8:57 pm UTC

35 = (((4! + 4)/4)!!)/(√!4)

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Re: Count Up with the Four Fours Puzzle

Postby emlightened » Wed Dec 09, 2015 9:08 pm UTC

36 = (4+4)*4 +4 = ((4!!)!/(4!!)!!)/√!4 + !√4
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Re: Count Up with the Four Fours Puzzle

Postby username5243 » Wed Dec 09, 2015 10:24 pm UTC

37 = 4!+sqrt(4*4)+!4
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Re: Count Up with the Four Fours Puzzle

Postby SirGabriel » Wed Dec 09, 2015 10:55 pm UTC

emlightened wrote:I've never liked using ceiling/floor.

I agree. Also, what does !4 mean?

38 = 4!!+4!+4+√4

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Re: Count Up with the Four Fours Puzzle

Postby emlightened » Thu Dec 10, 2015 7:33 am UTC

Subfactorial; it counts derangements. !0 = 1, !1 = 0, !2 = 1, !3 = 2, !4 = 9, !5 = 44 and !6 = 265. It's pretty useful, as we can get 3 = √!4 and 1 = !√4 with just unary operators.

.4√4 + !4 - √4

(That's the 0.4'th root, not the 4'th root.)
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