Project: Model-based geometry reconstruction from TEM images (MATH+ Emerging Field Project EF3-1)
Challenge:
Last Christmas a truly terrible thing happened. Like every year, Santa brought presents to children all over the world. However, some of them were stolen by the Grinch. The mean Grinch unwrapped the presents, stole the gifts, and left the empty packaging behind. Consequently, many children did not get their presents—leaving not only them, but also Santa, quite miserable. In fact, Santa was shocked by what had happened and vowed that this would never happen again. Thus, Santa asked his elves to find a solution.
After a lot of pondering, the elves came up with an idea: they want to create boxes that can be opened only by precisely the child the present is addressed to. Of course, this scheme will require magic, but luckily they can consult their fairy friend. Indeed, the fairy agrees to help them and provides them with some magic glue. The elves fill the gift boxes with toys and seal them with the fairy's magic glue. However, after some boxing and gluing, the elves realize that the magic glue will be used up before they are finished with all the presents. Therefore, they ask their fairy friend for help once again. The fairy is eager to help, but only has exactly 2020 g of the magic glue left. Instead of panicking, the elves decide to use their mathematical superpowers and calculate how many boxes they will manage to seal with this amount of magic glue.
The boxes are cubes of different sizes. The elves need to apply the glue only at the boundary of the top face (a square). Let us denote these squares by A_{1}, A_{2}, A_{3}, ... The elves know that the area of the square A_{k} of the k-th box is given by
For efficiency reasons, the boxes are stacked into one another, and the elves have to process the boxes in decreasing order of their size. From earlier packaging experiences, the elves know that they need exactly 0.15 g of glue for a line of 1 cm.
How many boxes will the elves be able to seal with the 2020 g of magic glue?
Artwork: Friederike Hofmann
PDF download
Possible answers:
Project relevance:
The project MATH+ EF3-1 Model-based geometry reconstruction from TEM images analyzes images from transmission electron microscopy (TEM) obtained by experimental physicists at TU Berlin. The mathematical task is to reduce the electronic SchrÃ¶dinger equation by a finite-dimensional approximation. To evaluate the approximation error one uses similar estimates as in this puzzle.