text: # Move matches and make largest possible number
  
  a) Move three matches to make the largest possible number.
  
  b) Do so moving four matches.
  
  • Does a 1 require 2 matches to form, or is 1 sufficient? Commented Aug 3, 2020 at 13:10
  • puzzling.stackexchange.com/questions/100767/… Commented Aug 3, 2020 at 15:29
  • @rtaft 2 matches Commented Aug 3, 2020 at 16:42
  • Math Nerd points out there is no "largest possible number" Pick your infinity and then generate $2^{yourInfinity}$ . :-) //yes I know that's not the point here Commented Aug 5, 2020 at 12:00
  
  I can't beat @Excited Raichu's a) but for b):
  
  • If g is allowed, then 3 matches gets one $g1173$, the 1173rd value in the sequence that includes Graham's Number at $g64$. It's slightly bigger than that result. (-: Commented Aug 4, 2020 at 0:57
  • @JdeBP I don't know, anything that needs half a pargaraph and a wikipedia link for explanation doesn't really work IMO, :-P Commented Aug 4, 2020 at 8:05
  
  a) Not certain this is the biggest, but I can't find anything bigger than
  
  b) Moving 4 matches allows
  
  which is pretty big. Not sure if it's the absolute biggest, but it's up there.
  
  This is, of course, assuming that the digit 1 must be two sticks high. There's definitely a higher ceiling if it can only be one.
  
  • For b) you could do $7^{43^9}$. Commented Aug 2, 2020 at 23:55
  • @PaulPanzer how is that? Making a 7 and a 9, both on other lines besides the main one, would require 9 movements. Commented Aug 3, 2020 at 0:02
  • You only offset half a line which is more within typographic conventions anyway. So for the $7$ you keep two sticks of the first $4$ exactly where they are and for the $9$ three sticks of the second $3$. Commented Aug 3, 2020 at 0:04
  • Ok, ok. If you absolutely don't want it I will make my own answer ;-) Commented Aug 3, 2020 at 0:36
  • @Chronocidal You can look at the first revision of my answer for that. Note that one of the new sticks in the $9$ does not come from the $3$ but is left over from the $4$ that becomes a $7$. Commented Aug 3, 2020 at 8:21
  
  a)
  Is it:
  
  4431111
  
  b)
  Is it:
  
  44771111
  I added 4 extra digits which multiply the digit by 10 and add 1 each time it does so. I added them on the back since 4 is a bigger number than 1. (Obviously, but for explanation's sake it's here). This logic goes for a) as well.
  
  • For b) you could do $7^{43^9}$ - sorry, I meant to comment on the other answer. Commented Aug 2, 2020 at 23:51
  • yes i could, but why do that when you could post an answer yourself? I don't really want to leach off of other people. Im sure you understand ;) Commented Aug 2, 2020 at 23:55
  • As I said I meant to comment on the other answer and since my contribution is just a minor optimization on their major idea I thought it wouldn't warrant an answer of my own. Commented Aug 2, 2020 at 23:58
  
  For a)
  
  For b)
  
  Turn the first 4 into two 1s to make 1,111,731
  
  Moving 4 matches to form
  
           _   _
  | | | | |_  |_
  | | | |  _|  _|
  
  
  and viewing it upside down gives S S 1111 where S 1111 is the maximum shifts function of the busy beaver game with 1111 states http://en.wikipedia.org/wiki/Busy_beaver#Maximum_shifts_function_S . S 6 is 3.515E+18267 and S 7 is already 10^10^10^10^18705353.
  
  S S 1111 is the maximum shifts function of the busy beaver game with S 1111 states, a number uncomputably large, but still finite.
  
  (b) Move 4 matches to form
  
              _  _
  |_| /|\ /|\  | _|
  |          | _|
  
  
  or 4 ↑↑ 73, where ↑ is Knuth's up-arrow notation http://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
  
  4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4 (73 4's) is too large to compute in Wolframalpha.
  
              _  _
  |_| /|\ |||  | _|
  |          | _|
  
  
  A convention allows multiple ↑'s to be specified with a superscript, so 4 ↑¹¹¹ 73 = 4 ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑ 73 (111 ↑'s), a number too large to represent even with power towers.
  
  Hard to go wrong with
  
  4477
  
  As obviously
  
  there is no such thing as a "largest possible number", but infinity is a common misconception ;)
  
  111153
  
  Moving 2 Match sticks, the 4s can be made into 11s. And the third match stick would be moved to get 5.
  
  Moving 3 matches, I'd
  
  take them all from the last 3 to make a 1
  
  giving
  
  74431
  
  My best guess for 4 moves is to
  
  borrow 2 from each 3, and use them to make 1s
  
  to get
  
  447711
  
  I think that's the highest so far, if each one needs two matchsticks.
  
  • Hi Cristobol. These are certainly not higher than those given by Paul Panzer and Excited Raichu above.
  – user69943
  Commented Aug 3, 2020 at 19:03
  • I keyed off the OP asking for a number, not an equation. Commented Aug 3, 2020 at 19:05
  • Those are expressions, not equations. Commented Aug 3, 2020 at 19:58
  • Okay, got me there...still not numbers, even if the operators aren't visible on their own. Commented Aug 3, 2020 at 20:03
  • Cristobol, what about mine? Commented Aug 3, 2020 at 21:17
  
  (b) Move 4 matches to form
  
        _      _
  | |_| _|  / |_
  |   | _| /  |_
  
  
  The slash has to be squeezed in between the 3's to make 143/ε. The last character represents epsilon, an arbritarily small positive quantity used in the definition of limit in calculus, which is greater than zero to avoid division by zero.
  
  (a) Move 3 matches to form
  
       _  _     _
  |_| | | _| / |_
  |   | _|    _
  
  
  or 473/ε. The vertical match of ε is centered over the middle bar.
  
  (b) Move 4 matches to form
  
     _   _
  | |_| |_|  |  |
  |   |   | _| _|
  
  
  which viewed upside down is Γ Γ 661, where Γ is the gamma function http://en.wikipedia.org/wiki/Gamma_function
  
  Γ 661 ~ 10^1576 and Γ Γ 661 ~ 10^10^1579
  
  which is slightly larger than 7^7^473 ~ 10^10^400.
  
  Interpreting it as Γ Γ bb1 where bb1 in hexadecimal is 2993 in decimal,
  
  Γ 2993 ~ 10^9103 and Γ Γ 2993 is too large to compute in Wolframalpha but is larger than Γ Γ 661.
  
  Even better are
  
      _   _
  |_| |_|  |  |
  ||   |   | _| _|
  
  
  Γ Γ 66¹¹ and Γ Γ bb¹¹, respectively.