Write the number 2011 using only the digit 4 and any of the operations of addition, subtraction, multiplication, division, exponentiation, taking a square root and factorial. You can use any number composed of the digit 4, even if it's decimal, so 44 and 44.44 are both allowed. You're also allowed to use brackets.
Have fun!
This puzzle was contributed by Paulo Ferro, a maths teacher in Oporto, Portugal. For more of Paulo's puzzles, visit his website in English or Portuguese. If you have a puzzle you think might interest Plus readers, please email us!
Solution link
(44*(44+4))-4444/44
(44*(44+4))-4444/44
Solution
4444/2=2222
2222-(4x44+44)=2002
2002+4+4= 2010
2010+(4/4)=2011
44*44+(4*4*4)+4+4+4-(4/4)
44*44+(4*4*4)+4+4+4-(4/4)
one possible answer
4*4*4*4*4*((4/4)+(4/4))-4*4*((4/4)+(4/4))-4-(4/4)=2011.
4*4*4*4*4*2-4*4*2-4-1=2011.
2048-32-5=2011.
Searched using Haskell
Least 4s required maybe.
(4^4-4)*(4+4)-(4+4/4)
(4+4)!/(4!-4)-(4+4/4)
40320/20 - 5
From Bassel
2011=44^SQRT(4)+4^(4-4/4)+44/4
A refinement for less 4s
2011=44^SQRT(4)+4*4*4+44/4
Another solution
((4^(4 + (4/4)))*(4^1/2)) - 4! - (4*(4^1/2) + (4/4)) - 4
4444 - ((4*4*4*4*4) +
4444 - ((4*4*4*4*4) + (4*4*4*4*4)) - (4*4*4*4) - ((4*4*4) + (4*4*4)) -4/4
Possible solution
444*4 + 44*(4 + 4:4) + 4*4 - 4:4 =
1776 + 220 + 16 - 1 = 2011
=(4^4)*(4+4)-44+4+4-4/4 by
=(4^4)*(4+4)-44+4+4-4/4
by tanks
Donald Knuth variant
This puzzle reminds me of a conjecture made by Donald Knuth. I learned about it from the book "Artificial Intelligence" by Russel & Nordvig, and I quote it from there:
Knuth conjectured that, starting with the number 4, a sequence of factorial, square root, and floor operations will reach any desired positive integer. For example, we can reach 5 from 4 as follows:
Floor(Sqr(Sqr(Sqr(Sqr(Sqr((4!)!)))))) = 5
It would be nice to see if anyone could write 2011 using only one 4 and the mentioned functions!
4+4+4+4+4+4...
...+4+4+4 + 44/4
= 2000 + 11
Is it cheating if I have to use an ellipsis or sigma notation?
Yes!
Yes!
why are people afraid of the decimals?
((((4+4) /.4) * (4/.4)) * (4/.4)) + (44/4) = 2011
broken down:
8 / .4 = 20
4 / .4 = 10
so (20 * 10) * 10 = 2000
44/4 = 11
Other solution (10 fours)
2011 = (4^4 * 4 * sqrt 4) + (44/4) - 44 -4
By Krani Lupus
8 fours
sqrt 4 * (4^4 * 4 - 4!) + 44/4
By Krani Lupus
twelve fours
44^((4+4)/4) + 4!(4-4/4) + 4 - 4/4
2011 in fours
((4(44/4)-(4/4))*((4((4/44)-(4/4)) + ((44/4)-(4/4)))) + (44/4)
4^4*4*(spuare root of
4^4*4*(spuare root of 4)-44+4+4-4/4
10 4's
2011 in fours (eight fours!)
factorial(4)=24
4*24*24=2304
4*(factorial(4))to_the_power(squareroot(4))=2304
factorial(4)/squareroot(4)=288
(4+4/4)=5
2304-288-5=2011
Solution in eight fours: 4*(factorial(4))to_the_power(squareroot(4))-factorial(4)/squareroot(4)-(4+4/4)
Barry Daniels
Ref: plus.maths card in Xmas 2011 New Scientist
4*4*4*4*4+4*4*4*4*4-44+4+4-(4\4)
4*4*4*4*4+4*4*4*4*4-44+4+4-(4\4)=
1024+1024-44+4+4-1=
2048-44+4+4-1=2011
2011
(4^4)*4+(444+444)+(44+44)+(4+4+4)-(4/4)
(256)*(4)+(888)+(88)+(12)-1
1024+888+88+11
2000+11
2011
On all fours
((4^4) * 4 * sqrt(4)) - (4!) - (4*4) + 4 - (4/4)
[500/((4/4)/4)]+(40/4)+(4/4)
[500/((4/4)/4)]+(40/4)+(4/4)
uh oh
i do belive that 5 and 0 are not the number 4
An amazing answer
How to get to 2011 using all 4's.
4444 - (444*4) = 2668
2668 - 444 = 2224
2224 - (44*4) = 2048
2048 - 44 = 2004
2004 + 4 + 4 = 2012
2012 - 4/4 = 2011
Solution: = 4444/4 + 4444/4 +
Solution:
= 4444/4 + 4444/4 + 44/4 - 444/4 - 4444/4
= 1111 + 1111 +11 - 111 -111
= 2233 - 222
= 2011
An excellent answer Choi
An excellent answer
Choi
2011 in 4
I feel there should be a limit on number of times 4 is used. Otherwise, the simplest solution would be to add (4/4) 2011 times ( 4/4+4/4+..................=2011). :-)
Anil Sharma
4+4/4(4^4 x4) - 4+4/4(4x4) -
4+4/4(4^4 x4) - 4+4/4(4x4) - 4 - 4/4
Using only the number 4 (as opposed to digit)
2011 = 4 * 502 + 3
502 = 125 * 4 + 2
125 = 31 * 4 + 1
31 = 4 * 4 * 2 - 1
1 = 4/4
2 = sqrt(4)
3 = 4-4/4
--> 2011 = (((4 * 4 * sqrt(4) - 4/4) * 4 + 4/4) * 4 + sqrt(4)) * 4 + 4 - 4/4
2011 in 4's
2011 = (444+44+4)*4+44-(4/4)
Also:
2012 = (444+44+4)*4+44
2013 = (444+44+4)*4+44+(4/4)
another solution
(4444/4)+(4444+4)-(444/4)-(444/4)+(44/4)
i love open ended questions like this...
(256 x 16)/2 - 37 = 2011
- 256 = 4 to the power of 4
- 16 = 4 x 4
- 2 = sqrt 4
- 37 = 4 x 4 x sqrt4 + 4 + 4/4
Like I said... I love puzzles with infinite right answers :D
6 fours
8000=SQRT(SQRT(SQRT(4!-4)^4!)))
2011=(8000+44)/4