Express the number 2015 using only the digit 2 twice

Question

Can you use only two instances of the digit $2$, along with the mathematical operations below, to create an expression that evaluates to $2015$?

Allowed operations:

arithmetic operations: addition ($+$), subtraction ($-$), multiplication ($\times$), division ($\div$, or $\frac{x}{y}$), exponentiation (like $2^2$);
factorial ($!$), absolute value ($|...|$);
extraction of the root of any degree in a form $\sqrt[a]{b}$ or the square root in a form $\sqrt{b}$;
trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant
inverse trigonometric functions: arcsine, arccosine, arctangent, arccotangent, arcsecant, arccosecant
natural logarithm ($\ln b$), or logarithm with any base ($\log_a b$)

Parentheses $()$ are also allowed.

What is not allowed:

digits other than $2$, or more than two instances of the digit $2$
named constants such as $\pi, e$, etc...
defining and using your own functions
other variables

Example with three $2$s:

A couple of years ago, I managed to solve the same kind of problem, with three $2$s:

We can express any natural number $A$ using three $2$s and the above operations, like this: $$-\log_2\log_2\underbrace{\sqrt{\sqrt{...\sqrt{2}}}}_{A\,\mathrm{square\,roots}}$$

But I couldn't crack the problem using only two $2$s...

You could use the incrementing method described in this answer to get from 2 to sqrt(2015) with a lot of trigonometric operations and then use the other 2 to square it. — f'', Aug 07 '15 at 11:55
sorry but i can't understand "2 digits 2"? did you mean we can use only digit 2 for max 2times? — Saurabh Prajapati, Aug 07 '15 at 12:41
@SaurabhPrajapati, yes, you got it right. We can use only digit 2 but no more than 2 times. — Glinka, Aug 07 '15 at 12:45
@f'', I think this is what I was looking for! Could you post it as an answer? — Glinka, Aug 07 '15 at 12:46
I think $-log_2^2\underbrace{\sqrt{\sqrt{...\sqrt2}}}_{A,\mathrm{square,roots}}$ is simpler and shorter. — EKons, Jun 24 '16 at 15:38
Can either the floor function or the ceiling function be used? — Olive Stemforn, Jan 30 '20 at 17:54
@NumberBasher, yes, I think it can be included to the list of allowed operations, I wonder though whether it would meaningfully contribute to the range of solutions — Glinka, May 31 '22 at 09:10

score 27 · Accepted Answer · edited Apr 13 '17 at 12:50

27

This answer describes a method using trigonometric operations to obtain the square root of any rational number from 0. In this answer, Daniil Agashiyev notes that $\tan\arcsin\cos\arctan\cos\arctan\sqrt{n}=\sqrt{n+1}$.

Using this, we can write 2015 as:

$$(\underbrace{\tan\arcsin\cos\arctan\cos\arctan}_{2011\text{ times}}\,2)^2$$

or even with only one 2:

$$\underbrace{\tan\arcsin\cos\arctan\cos\arctan}_{2015^2-4\text{ times}}\,2$$

Unlike the linked problem, we're allowed to use the secant and cosecant here, so we can use $\sec\arctan$ or $\newcommand{\arccot}{\operatorname{arccot}}\csc\arccot$ for the same effect.

edited Apr 13 '17 at 12:50

Community

1

answered Aug 07 '15 at 13:19

f''

33,665
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In the first case why 2011 ans not 2013? And I'm not sure the second case is right... – Glinka Aug 07 '15 at 13:36
@Dennis 2011 because 2 is sqrt(4), so applying the sequence 2011 times gives sqrt(2015). – f'' Aug 07 '15 at 13:52
"using the digit 2 only twice" I can see three 2s in both your answers, counting the '2' in '2015' – Colonel Panic Mar 03 '16 at 15:45
@ColonelPanic The number under the brace is not actually written. It represents how many times the thing above it is to be written. It is being used as a shorthand because the full form would be too long to fit in this answer. – f'' Mar 03 '16 at 16:51
Shorter form:$$\mathrm{tan}^2\underbrace{\text{arcsincosarctancosarctan2tan}}_{2010\text{,\mathrm{times}}}\text{arcsincosarctancosarctan2}$$ – EKons Jun 24 '16 at 15:47
@ f' - I side with Colonel Panic. If the number under the brace is "not actually written" how would anyone know!? It's not a valid solution. And, accepting it as a valid solution is wrong. – Olive Stemforn Aug 03 '16 at 20:50
2

@OliveStemforn The formulas written in this answer are not the full form of the solution, but rather, an abbreviated form. The full solution has the trigonometric functions written out 2011 times, and was not included because it would be too long to fit in the post. – f'' Aug 04 '16 at 00:22

score 3 · Answer 2 · answered Aug 07 '15 at 14:55

3

It looks a bit like cheating but we are not defining any new function, we are just naming one that is allowed to use by the rules:

Name: m: x -> m(x) to be: x -> log2(x) (log in base 2)

Then we apply the square root trick:

- m(m( sqrt(sqrt( .... sqrt(2) ... )) )) = 2015
   (taking square roots 2015 times)

answered Aug 07 '15 at 14:55

ypercubeᵀᴹ

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2
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That is exactly why defining new functions are forbidden :) It would generate the whole series of easy to find solutions. But still thumbs up :) I would have voted, but I don't have enough points – Glinka Aug 07 '15 at 15:04
I will delete it if you think it doesn't match your specs. But your requirements is a bit arbitrary. You allow a whole range of trigonometric functions but only ln from the logarithmic/exponential function and for loga(b), one has to use 2 parameters. No exp() or any of the other related functions. For square root one parameter is required but for other roots two. – ypercubeᵀᴹ Aug 07 '15 at 15:10
I realize my answer looks a bit silly when one thinks that we could define m = 2/2 and then do m+m+...+m (2015 times). – ypercubeᵀᴹ Aug 07 '15 at 16:24
Your answer has the right to be, but the best one yet is given by f''. As I said, I would have voted up, but i haven't got this privilege. – Glinka Aug 07 '15 at 16:36

apm · Answer 3 · 2015-08-07T12:34:26.843

1

We can express 2015 using 0 digit like this:

$$\dfrac {\underbrace{e+e+e+...+e}_{2015 ~e~s} }{e}$$

Next Answer

15

edited Aug 07 '15 at 12:34

answered Aug 07 '15 at 12:11

apm

389
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Sadly, we cannot use named constants... I'm sorry I did't mention it in the question, now it's there. – Glinka Aug 07 '15 at 12:27
ok bro ;). We will think another solution – apm Aug 07 '15 at 12:31
is it correct ? – apm Aug 07 '15 at 12:35
What do you mean by that? It's just a 15 :) – Glinka Aug 07 '15 at 12:39
We wrote 7th aug 2015 as 7th aug 15. Like that we can wrote 2015 as 15 :p – apm Aug 07 '15 at 12:48
But you used other digits than "2". Anyway, that is not what question is about. I think @f'' just found the right solution, but left it in comments. – Glinka Aug 07 '15 at 12:50

Express the number 2015 using only the digit 2 twice

3 Answers3

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