When I was studying in the 12th grade, I was probably jobless enough to waste a considerable amount of (so called precious) time trying to solve an essentially useless problem (or I should optimistically say that no use for it has been found till now but the future is bright). What that problem is, you shall soon come to know, and you shall (hopefully) waste as much time on it as I did. Although my desultory efforts led nowhere, I enjoyed doing what I did and am doing now - asking others to crack their brains on this one.

It all started when I got myself one of those Casio calculators - fx-82 was the model which I was using and was quite popular. It saved a lot of my time which would have otherwise been spent in winnowing out numbers from log tables. Unfortunately enough it took away a lot of my time in something else. Some smart guy asked me to do something and I did it.

The fx-82 model has two sections of keys. The upper one has smaller less commonly used black keys. The lower one had larger grey keys, in which we are not interested here (my apologies!). The upper section has keys like sin(x), cos(x), tan(x), 1/x, x^2, log(x), x! and inverses of some and a couple of other equally useless keys. My good friend asked me to breathe life into them and get the number 7 on the screen using ONLY these keys and with 0 initially on the screen. Why the number 7, I don't know and you should not ask, since this question would soon become irrelevant.

I spent an evening trying to figure out how to do the job. When I finally managed to do it, I felt great but there was an empty feeling lingering. I asked myself: why only 7, and soon I got obsessed with the riddle of obtaining other more famous numbers. I made good progress and within a couple of hours I had all two digit numbers under my control. I continued making progress and in about a month's time I had the entire spectrum of terminating decimal numbers taken care of. Soon, I built the interesting theory of Calculatrics (refer Appendix).

Since then I have been teasing people with this riddle and quite a few people do very well. Some do whole numbers till 10, some till 100 and some even more with some prompting. Some smart ones have been able to quickly figure out solutions to numbers till 200 without any prompting at all, much quicker than I could have ever got. Nonetheless, rarely has anyone come up with solutions to the entire number line. Do try your hands at this one, and lets see how far you get! Check out the appendix for a detailed problem statement.

Appendix A

As some theoreticians would like to do, I will state the problem in a formal manner. Assume that you have quite a few functions at your disposal. They are:

sin(x), cos(x), tan(x), 1/x, x^2, log(x), ln(x), -x and their inverses

You also have: x! and cube_root(x) but not their inverses. And you DO NOT have any other function to use. These 16 functions are all that you will be allowed to use. Each of them corresponds to some key in that area of the calculator I was talking about. There were other keys there too, but they are all the more worthless.

Now, with an initial value of 0, apply functions from this list one after another. Or, in other words, compose one or more of these functions (with repeating of functions allowed) to come up with some function which when applied on the number 0 would yield the number 7.

It turns out that 7 is not a particularly easy one to crack. Beginners can try simple ones like 2,3,4,5,6, etc. Don't give up too soon convincing yourself that it is not possible. It can actually be proved that it is invariably possible.

Definition: Representative Function Let f1(x),f2(x),...,fn(x) stand for the functions stated above. A representative function for a number n, if it exists, is a particular composition g(x) of these functions such that g(0) = n. Moreover, if g(x) exists, then the list f1(x),f2(x),...,fn(x) is said to be able to represent n.

Fundamental Theorem of Calculatrics: Every number, r, with a finite decimal representation has a representative function.

Contact the author for a proof.

Example 1.1
Obtain the representative function for 2. In simple terms, by starting with 0 on the screen and simply punching keys on the top half of the keypad any number of times, get the number 2 on the screen.

Solution

This is how you would get 2: (log(x) o x^2 o 10^x o cos(x))(0) = 2. In layman terms, punch the cos key, then the antilog one followed by the x-squared key and finally the log key. Presto! you have 2 on the screen.


Exercise 1
Find representative functions for as many numbers as you can. Theoritically, you should be able to find one for each number with a terminating decimal representation. Nevertheless, there are engineering issues as well. When you implement your solution, the calculator inadvertantly introduces an error in each calculation. Ultimately, it is the implementation that matters and therefore you should keep the composition such that the number of functions composed is nearly the minimum possible, failing which you might end up getting 6.999999999 instead of 7 which is obviously not acceptable (we adhere to high quality standards!). It has been also proved and demonstrated that on the fx-82 calculator it is possible to actually solve for any number that could potentially appear on the screen. There is no reason why other similar calculators should be any different.

Future Work

For one thing, we could always fiddle with the list of functions allowed. Some other list of functions might probably not be so expressive and say, might be able to represent only whole numbers or maybe only integers. On the other hand there could be a list of functions that could be more expressive than the one above and which could probably be able to represent all real numbers. I would be eager to know of such a list, if it exists. There is still no satisfactory method, to my knowledge, to determine the exact space of numbers that a set of functions can represent. A lot of work has to be done in this area and I hope to give it a thought as soon as possible.

Contact me to obtain solutions for particular numbers you found really difficult
Visit my homepage for other interesting links.

For some reason, the following number of people have watched this page since 22nd February 2006