This blog post is kindly written by Ehud Friedgut.
My daughter, Shiri, who’s in seventh grade, had the following question in a math exam:
How many cubes of 2×2×2 fit into a box of size 8×4×3?
Shiri divided the volumes, getting the answer 12, but it was, of course, a trick question, and the intended answer was 8.
But I thought to myself – is eight really the best? (I’m pretty sure the answer is yes, but I’m only 99% sure.)
Test your intuition: Is the answer 8 ?
This gave birth to the following question (for which I know the answer):
Is there an integer n such that you can fit n+1 squares of size 1 (with disjoint interiors) into a rectangle of size (n+0.999) × 1.9999 ?How about a rectangle of size (n+0.999) × 2.9999 ?
Knowing me, (and/or my type) you can probably guess, using pure logic, what the answers are to these two last questions (otherwise, why ask both?). Hint: for the first question use 
, for the second one, try n=2.
, for the second one, try n=2.I’m happy that I could use my iPad to drag and rotate little squares and try to squish them into the rectangle, it’s helpful to be hands-on.
Illustration by Alef.
Combinatorics and more 
Haven’t though about it too much, but is the answer: no such integer exists for the first question, and n=2 for the second?
Reminds me of this:
https://mathenchant.wordpress.com/2019/09/16/sphere-packing/
Related is this paper of Erdős and Graham on packing squares into squares: https://www.sciencedirect.com/science/article/pii/0097316575900990
Not the same problem, but a great related web site:
https://mathformillennials.wordpress.com/2019/04/09/square-in-squares/
Dear Greg, this is a great site and it is very relevant. Of the same author Erich Friedman (do you know him?) , I also found this site https://erich-friedman.github.io/packing/index.html and this paper https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS7/html_1
Dear Gil,
No – no connection that I know of. A colleague showed me this web site recently.