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!*

## Comments

## (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