x^x^x^2017=2017 – A fun problem with a unique and sad history

I was sent this problem a few weeks ago by a physicist who I work with privately on weekly basis (on the side of my school studies). I found it to be a lot of fun to think about, and was eager to write about the solution.

Before getting into the solution, from what I understand this problem is part of a collection of difficult problems known as “coffins“. The history of the collection emerges from anti-Semitic discrimination, in which the elite mathematics school in Russia, the Mathematics Department of Moscow State University, had devised the problems as a way of “actively trying to keep Jewish students (and other “undesirables”) from enrolling”.

In short, Jewish students as well as others considered “undesirable” were given a different set of problems on the entry oral exams, and these problems were designed to be very difficult.

According this summary of the history, “These problems were carefully designed to have elementary solutions (so that the Department could avoid scandals) that were nearly impossible to find. Any student who failed to answer could be easily rejected, so this system was an effective method of controlling admissions. These kinds of math problems were informally referred to as “coffins”. “Coffins” is the literal translation from Russian; in English these problems are sometimes called “killer” problems.

I recall reading about a similar practice by schools elsewhere Europe. In terms of the “coffins” in particular, a more detailed account can be read here.

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With the unique history noted, what about the solution?

One might instinctively think of taking logs until an identity pops out, similar, perhaps, to when setting up for composite exponential function differentiation. This was my most immediate thought. But, as far as I am aware of, it doesn’t work. And things quickly become very convoluted.

In some respects, we can also think of tetration and “power towers”, which are cool in themselves

    \[ a^{a^{a^{a^{a}}}} \]

But the emphasis, I think, is to simplify one’s approach a little bit; because the solution itself is actually deceivingly simple.

For example, what if we just think of x^{2017} and equate that to 2017? We would get x^{2017}=2017.

Building from this, a pattern starts to develop with a very cool and fun order of substitution beginning with the higher powers and then, in a manner of speaking, working our way down cancelling as we go.

The complete solution I’ve prepared in latex.

I think it is beautiful how the equation simplifies to a cascade of higher powers that cancel!