Abstract: The problem 2+2=p where p is some non-trivial integer has long been the Holy Grail of integer sum mathematics. Well, we report the Grail seized. Working on a number of new analytical, experimental, numerical and quantum mechanical techniques we fix the value of p to 4 with higher accuracy than ever before. In the process, we set earlier conjectures that p=24,886 to rest.

I. Introduction

Integer number sums have been with us for a long time now. Two, three and even higher term sums have been observed and studied in the context of a plethora of every day phenomena with a great deal of approximate and exact results having been published [1]. Still, despite ongoing efforts from Mathematicians and Theoretical Physicists the vital question remains: What is the result of the 2+2 problem?

Many techniques have surfaced in the past for tackling this highly non-trivial sum, going back to arguments published on prehistoric caves [2] on what the actual outcome of adding a 2 to an existing 2 may be. In the 16th Century, a number of theories were put forward, but none was successful at providing a conclusive solution to the problem, followed by further approximate and hand-waving (as well as hand-counting) arguments in the 17th and 18th Centuries. Early in the 19th Century, modest progress was achieved by showing that the result of the sum must itself be an integer [3], but this is pretty much how things stand to this very day.

In this paper, we attempt to settle the question once and for all with a strategic attack on all aspects of the problem. We shall be presenting theoretical arguments, numerical simulations and experimental results by the end of which it shall become apparent that 2+2 equals 4 as previously suspected with a high degree of confidence.

II. The Hoicin transformation

It has been noted by many researchers on the field, that the theoretical methods employed in the very similar 1+1 sum can not be applied to the 2+2 sum [4] hence it must be through the development of new tools that a successful theory of the 2+2 sum may emerge. One such tool is the Hoicin transformation [5], a little known technique which, although limited in its power, can at least constraint the number of possible answers.

The clever trick is to appreciate the symmetry of the sum. One can convince oneself that both terms in the sum – namely the left 2 and the right 2 – are the same number. This is extremely important for the Hoicin transformation to work since, after some tedious algebraic manipulations and topological arguments that are beyond the scope of this paper, one can show that the sum can be transformed to a product thus:

2+2 = 2*2 (1)

Now, it is widely acknowledged that products are, in general, much more complicated than sums but in this particular case the reformulation of the problem as a product is highly advantageous for the following reason. Since the product is one of 2 with another integer (and, in fact, this is a doubly robust statement due to the presence of not one but two 2s), it naturally follows that the answer must not only be an integer but an even integer. Consequently, an answer such as 2+2=5 can be discarded as ludicrous, although another like 2+2=24,886 is, in principle, entirely consistent with our conclusions thus far. After the examination of further evidence in the chapters to follow, it shall also be shown that 24,886 is, however appealing at this stage, entirely ludicrous as well.

III. Quantum mechanical analysis

We also attempted the construction of the wavefunction of the result in two ways. Initially, we operated with the summation operator on two distinct 2s and, then, operated twice on a third 2 with the ladder operator.

In this latter instance, we got a result 4 when we transformed back to ordinary space but we are not entirely convinced that our operation (essentially, 2+1+1) is formally equivalent in quantum arithmetic to the original problem. This result, although tempting, will require more work which we have already undertaken.

In the original summation operation, we got the result 6+f(2)*i where f(2) is a hideously complicated function of 2 and i is as usual (-1)1/2. Although there is no reason to discard this result, we believe that the outcome of the summation must be a pure, real number. Since the calculation, numerous colleagues have pointed out to me in private communications that the discrepancy may arise for one of two reasons: either the identical nature of the two 2s leads to some yet undiscovered quantum mechanical effect or one or both of the 2s were not prepared in the state 2 at all.

IV. Numerical simulations

We have performed extensive simulations of the 2+2 sum in a variety of Operation Systems and with a number of programming languages, since, by its very nature, this problem probes the very fundamental computational operations of any given programming environment. Our simulations consisted of 105 samples where for each sample the 2+2 sum was calculated over 1017 realisations and are, to the best of our knowledge, the most thorough simulations of the sum in the scientific literature.

In the first case, we used a C++ code running on Linux to get the result of the sum to 3.998+/-0.005, which for all practical purposes must be taken as a very accurate measurement. The most unreliable results came from running the same algorithm on Windows XP were a range of results was obtained. 71% of the simulations produced a value very similar to the one quoted above, 16% of the samples observed came up with a different numeric result � namely 2+2=779 � whilst the rest 13% of the data seem to suggest that 2+2=Crash. Unfortunately, the two latter answers can not be trusted since, by the Hoicin argument of the previous chapter, 779 is not an even number and Crash is inconsistent with the Van Graat [6] principle which states that the sum of two integers can never be a letter, phrase, small vegetable or miniature fire engine.

V. Experimental results

Experimentation on sums being as complicated as it usually is, we chose to set up several distinct experiments by which to test the 2+2 sum.

Firstly, 19 postgraduate students from our group were instructed to raise two fingers from the five in their right hand. Then, a further two fingers were raised from the left hand (although the same hand can be used without compromising the outcome of the experiment) and the total number of raised fingers was recorded. The procedure was repeated again and again over a 72 hour period with very interesting results.

In the first couple of hours of readings, the overwhelming majority of data seemed to converge to 2+2=4. However, as the experiment went on some discrepancies were recorded. Periodic fluctuations were observed by all experimenters at or around pub closing times and some alphanumeric results (most, of a rather explicit nature) were also recorded as physical exhaustion begun to set in. When the experimenters were approached towards the end of the 72 hour period, most experiments had collapsed with the postgraduates raising only one of the two sets of fingers and waving them menacingly against members of staff whilst repeating a two-word mantra.

In subsequent spectroscopic, chromatographic and telefluromatic experiments, a value of 4 was reached in good agreement with numerical results.

VI. Conclusion

We have used several alternative approaches to resolve once and for all a fundamental mathematical problem. Our data seem to suggest that 2+2=4 and it would be tempting to assume at this stage that this may indeed be the case.

It would be interesting to see how the techniques we developed for the 2+2 case can be used in tackling higher order sums. Currently, we are already working on the 3+3 sum as well as asymmetric sums, for which the Hoicin principle does not apply, but it is difficult to see how more complex sums could be calculated. Some topological techniques are being developed by various research groups, however it is the authors view that an exact solution to higher-than-4 sums is still beyond the grasp of current mathematical understanding. It could be the case that the new supercomputer under construction in Cambridge may just about have enough power to also brake into three-term sums of the lowest order.