Solution

While this problem can be solved both geometrically and numerically, the geometric solution is definitely much more elegant (shorter, and provides more insight to the geometry of image formation). On the other hand, all the purely geometrical solutions were similar, so the award for the best solution is this time distributed evenly between all these seven purely geometrical solutions which were submitted before the appropriate hints were published (see below). Out of these seven, six were suitable for presenting on this web-page. According to the rules, the best solutions are published without additional bonus factor. Here, however, the divided bonus becomes small (), and there is a need to differentiate between the published and non-published awarded solutions. Therefore, there published best solutions received this time also the publication bonus of 1.1.

Let us start with the solution of Nadezhda Vartanian, which provides a short, yet clear enough motivation and explanation of the geometrical construction.

Exactly the same construction is provided by Jakub Šafin, who shows how to draw everything using only a ruler and a compass (as is typically done in the case of mathematical construction tasks). His solution is best viewed using his .pdf file, because changing pages yields an animation effect. His explanation is as follows.

The next solution with exactly the same construction is provided by Ilie Popanu. He found a nice program for easy drawing the required geometrical elements with high precision, geogebra. Owing to this, he was able to reconstruct the original square with a high precision; here we provide only his drawing (the method has been already explained well enough, see above).

Among the best solution award winners, there was one more solution with the same construction as above, sent by Andrew Zhao; however, his presentation style is somewhat sketchy, not suitable for publication.

It appears that the centre of the lens can be found as an intersection of circles, different from those shown above. Previously, the circles were drawn around diameters; Szabó Attila and Ion Toloaca drew these around segments with a central angle of 90 degrees. Attila's solution is provided below, Ion's solution can be downloaded in the pdf-format

Last but not least – the solution of Nikita Sopenko, which is actually the simplest one (and thus my favourite) as it substitutes one circle with a straight line.

Next, let us consider some solutions which involve calculations. All these are relatively long and therefore are provided  only as downloadable files. First, Aleksandra Vasileva performed calculations only to motivate certain geometrical constructions (so, no numerical measurements and numerical calculations are needed). Second, Adrian Nugraha Utama constructed one circle and was thus very close to a fully geometrical solution. Third, Petar Tadic measured angles and using these values calculated two angles required for determining the position of the centre of the lens. Fourth, Selver Pepič measured angles and calculated the focal distance together with the tangential offset of the centre of the lens; the highlight of his solution (in the form of a slideshow) is the error analysis.

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Jaan Kalda, Academic Committee of IPhO-2012