Display a number using a scientific calculator with most keys are stuck

Question

Your have a scientific calculator such that most of the keys are unable to be pressed. The only keys that work are those for the functions $$ x^2 \;\; \sqrt{x} \;\; x!\;\; \exp\;\; \ln\;\; \sin\;\; \cos\;\; \tan\;\; \cot\;\; \sec\;\; \csc\;\; \arcsin\;\; \arccos\;\; \arctan $$

When a key is pressed it is applied to whatever appears on the screen and the screen changes to the result. The calculator currently displays at $0$. How can we turn the display to read $2015$?

My try was that press $\cos$, we will get 1. Then press $\arctan$, we will get $\pi/4$. Press $\sin$, we will get $\sqrt{2}/2$. Press $x^2$, we will get $1/2$. Press $\arcsin$ then $\csc$, we will get 2. We can keep using the $x^2,x!$ and $\exp$ to get a larger number, but I don't know it is possible to get $2015$? I know $2015 = 5 * 13 * 31$. Can someone help me?

Answer 1

Using these provided operations, let's try to define primitives that can be used to build up the numbers we want. This is a constructive answer: the end result is a sequence of keys that should work.

1. To get 1 from 0, press $x!$, $e^x$ or $\cos$.
2. To get 1/2 from 1, we can use e.g. your solution ($\arctan$, $\sin$, $x^2$).
3. To get $1/x$, press $\arctan$ then $\cot$, since $\cot(x) = \frac{1}{\tan{x}}$ and $\tan(\arctan(x)) = x$ for all real $x$.
4. To get $x-1$ for $x \ge 1$, use the following sequence: $$ 1/x, \sqrt x, \arcsin, \tan, x^2, 1/x $$ This takes advantage of the identity $\tan(\arcsin(x)) = \frac{x}{\sqrt{1-x^2}}$ for $|x| \le 1$. In this case, it resolves into $\tan(\arcsin(\sqrt{1/x})) = \frac{\sqrt{\frac{1}{x}}}{\sqrt{1-\frac{1}{x}}} = \sqrt{\frac{\frac{1}{x}}{\frac{x-1}{x}}} = \sqrt{\frac{1}{x-1}} $, which we then square and invert to get $x-1$.

Now, the way is clear: we derive any number greater than 2015 (e.g. starting from 2, use $x^2, x^2, x^2, x^2$ to get 65536), then repeatedly subtract one to get to 2015. (There are lower easily-obtainable integers, such as 8! = 40320, but this answer is aimed just at finding a possible solution).

So here's one possible key sequence: $$ 0, \cos, \arctan, \sin, x^2 ( = 0.5), \\ \arctan, \cot, x^2, x^2, x^2, x^2 (=65536), \\ \arctan, \cot, \sqrt{x} (=\sqrt{1/65536}), \\ \{ \arcsin, \tan \}^{63521} (=\sqrt{1/2015}), \\ \arctan, \cot, x^2 (=2015). $$

(Note that due to numerical inaccuracy inherent to all numerical calculators, the result is probably not going to be exactly 2015. But it's a math puzzle, not a calculator puzzle, so this doesn't really matter :) )

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If we want to be a bit more clever with this, we can use an approach like the one Daniil Agashiyev used in his answer to reduce the number of uses of our primitives.

We can also observe that the inverse of our $x-1$ primitive enables us to get $x+1$. Thus, we can reduce the steps to simply $$ 0, \cos, \arctan, \sin (=1/\sqrt{2}),\\ x^2, \{\arctan, \sin\}^{3} (=1/\sqrt{7}),\\ x^2, \{\arcsin, \tan\}^{4} (=1/\sqrt{45}),\\ x^2, \{\arcsin, \tan\}^{10} (=1/\sqrt{2015}),\\ x^2, \arctan, \cot (=2015). $$ This is a total of just 43 keypresses - not bad! Here we're still not using $\exp$ or $\ln$.

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Finally, if our goal is to minimize the amount of button-mashing necessary to enter the number, we can bring in $\exp$ and $\ln$. Combining these with $x^2$ and $\sqrt{x}$ enables us to multiply and divide by 2, as fibonatic observed in his answer. With $x+1$, $x-1$, $2x$ and $x/2$ primitives at our disposal, we can get 2015 with just 20 keypresses:

$$ 0, \cos, \exp, \{\sqrt{x}\}^3, \ln \,(=1/8),\\ \arcsin, \tan, x^2\, (=1/63),\\ \exp, \{\sqrt{x}\}^5, \ln, \,(=1/2016),\\ \sqrt{x}, \arcsin, \cot, x^2 \, (=2015). $$

This has very good numerical stability to boot: it doesn't require any values beyond the range of a standard 32-bit float. Using 64-bit floats for computation, the result is 2014.9999999997042, within $3\cdot 10^{-10}$ of 2015.

Answer 2

This answer is from here.

The problem is a weaker version of USAMO 1995 Problem 2.

So that it has an answer here without having to leave the website.

We can break all the keys except for the trigonometric functions and still we should be able to get any square root of a rational number. In particular $2015$.

To prove the statement : If $m$ and $n$ are relatively prime nonnegative integers such that $n>0$, then the some finite sequence of buttons will yield $\sqrt{m/n}$.

To prove this statement, we apply induction on $m+n$. For our base case, $m+n=1$, we have $n=1$ and $m=0$, and $\sqrt{m/n} = 0$, which is initially shown on the screen. For the inductive step, we consider separately the cases $m=0$, $0<m\le n$, and $n<m$.

If $m=0$, then $n=1$, and we have the base case.

If $0< m \le n$, then by inductive hypothesis, $\sqrt{(n-m)/m}$ can be obtained in finitely many steps; then so can $\cos \tan^{-1} \sqrt{(n-m)/m} = \sqrt{m/n}$.

If $n<m$, then by the previous case, $\sqrt{n/m}$ can be obtained in finitely many steps. Since $\cos \tan^{-1} \sqrt{n/m} = \sin \tan^{-1} \sqrt{m/n}$, it follows that $\tan \sin^{-1} \cos \tan^{-1} \sqrt{n/m} = \sqrt{m/n}$ can be obtained in finitely many steps.

Thus the induction is complete.

The proof pretty much gives us a ready-to-use method for finding the sequence.

1. Start with an empty sequence and your number $\sqrt{m/n}$.
2. If $m\leq n$, append $\cos\arctan$ to the sequence, and change your number to $\sqrt{(n-m)/m}$, i.e. set $m' = n - m, n' = m$.
3. Otherwise, if $m>n$, append $\tan\arcsin\cos\arctan$ to the sequence, and change your number to $\sqrt{m/n}$, i.e. set $m' = n, n' = m$.
4. Repeat until your number is zero.

We have initially $m,n = m,1$, with $m=N^2$. That means that we will first apply step 3 to obtain $1,m$, then step 2 to obtain $m - 1,1$. This process will repeat $m=N^2$ times until we have $0,1$. Our sequence will therefore consist of:

$$ \underbrace{\tan\arcsin\cos\arctan\cos\arctan}_{N^2\text{ times}}\,0 $$

Which is equivalent to:

$$ \tan\underbrace{\arcsin\cos\arctan\cos}_{N^2\text{ times}}\,0 $$

For $N=2015$, that's $16\,240\,901$ button presses. Have fun!

Answer 3

We can adapt the idea used in the solution to the 1995 USAMO problem to construct an incrementor. For any positive integer $n$, we can go from $\sqrt{n}$ to $\sqrt{n+1}$ by the following sequence of calculator operations:

[$tan^{-1}$] [$cos$] [$tan^{-1}$] [$cos$] [$sin^{-1}$] [$tan$] (applying left to right)

Similarly, we can construct a decrementor that takes $\sqrt{n}$ to $\sqrt{n-1}$ by inverting and reversing the operations as follows:

[$tan^{-1}$] [$sin$] [$cos^{-1}$] [$tan$] [$cos^{-1}$] [$tan$]

As a shorthand, lets call incrementor sequence $[inc]$ and decrementor $[dec]$.

You already know how to get to 2. From there you can go to $\sqrt{7}$ by:
$[inc]^3$

Then hit [$x^2$] to get $\sqrt{49}$.

Now hit $[dec]^4$ to get $\sqrt{45}$.

Hit [$x^2$] again to get $\sqrt{2025}$.

Now $[dec]^{10}$ to get $\sqrt{2015}$.

And finally, [$x^2$] to get 2015.

There should probably be a faster way to get to 2015 taking advantage of the incrementor and decrementor, but this is the first I came up with.

Answer 4

All the posted answers haven't used $\exp$ and $\ln$, which can also be quite useful, for example the sequence: $[\exp][x^2][\ln]$ multiplies by two, similarly: $[\exp][\sqrt{x}][\ln]$ divides by two. The only downfall of this is that the intermediate results may exceed the largest number that the calculator can represent.

These functions can also be used to calculate $-x$, namely: $[\exp][x^{-1}][\ln]$, which can be achieved with for example with: $[\exp][\tan^{-1}][\cot][\ln]$. Given that: $\cos\left(\sin^{-1}\left(x\right)\right)=\sqrt{1-x^2}$, then the following eight long sequence can also be used to calculate $x-1$: $$ [\sqrt{x}][\sin^{-1}][\cos][x^2][\exp][\tan^{-1}][\cot][\ln] $$ The use for this sequence, instead of the already mentioned sequences in previous answers, will become clear in a bit. In order to get to 2015 from 0 you preferably want to use functions which keep the results integers, but increase as fast as possible. So $!$ and $x^2$ would be the best options. Since the given sequences to either achieve $x+1$ or $x-1$ have length 8, it would be best to minimize the need for them.

Factorial ($!$) does not come very close to 2015, namely $6!=720$ and $7!=5040$, therefore I did not really looked into using it much further. Since 2015 does not have very useful divisors I looked at the adjacent integers, 2014 and 2016. The second can be achieved with $63\cdot32=2016$, which can be achieved relatively easy by multiplying by two and one subtraction. The resulting sequence I came up with: $$ [\exp]^2[x^2]^3[\ln][x^2][x-1][\exp][x^2]^5[\ln][x-1] $$ Now if you substitute my sequence for $x-1$ in here, then the start and end functions cancel with the adjacent functions: $[x^2][\sqrt{x}]=[\ln][\exp]=x$, thus for the final expanded sequence you need 4 less button presses: $$ [\exp]^2[x^2]^3[\ln][\sin^{-1}][\cos][x^2][\exp][\tan^{-1}][\cot][x^2]^5[\ln][\sqrt{x}][\sin^{-1}][\cos][x^2][\exp][\tan^{-1}][\cot][\ln] $$ Thus only requiring 26 button presses, however $\exp{2016}\approx3.44886\cdot10^{875}$ which is very likely to exceed the largest number that the calculator can represent.