Archimedes' Cattle Problem/Historical Note

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Historical Note on Archimedes' Cattle Problem

There are various interpretations of this problem, as the initial wording can be construed as ambiguous.

In particular, it is not completely clear in the difficult version whether the white and black bulls need to be arranged in a square or a rectangle.

While the problem is attributed to Archimedes of Syracuse, it is far from certain that he was the one who originated it.

It appears, from the arbitrary nature of the numbers selected, and the difficulty of finding a solution, that the problem was contrived with no consideration of what the solution might be, or indeed whether there is one, in the same way that a lecturer in mathematics invents a problem, on the fly, as an illustration of the use of a particular solution technique, without first having checked that the solution will be immediately tractable.

The smallest solution to the Pellian Equation $t^2 - 4 \, 729 \, 494 u^2 = 1$ was discovered by A. Amthor and B. Krumbiegel in $1880$, but the corresponding total number of cattle was not actually calculated until $1965$.

This was done by computer. It took $7$ hours and $49$ minutes of computer time, and its printout filled $46$ pages.

This was checked in $1981$ on a more powerful machine, and this took $10$ minutes.

The full solution was published in the Journal of Recreational Mathematics.