## Ehud Friedgut: How many cubes of 2×2×2 fit into a box of size 8×4×3? (TYI 49)

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.)

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.
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.
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### 6 Responses to Ehud Friedgut: How many cubes of 2×2×2 fit into a box of size 8×4×3? (TYI 49)

1. - says:

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?

2. igessel says:

Related is this paper of Erdős and Graham on packing squares into squares: https://www.sciencedirect.com/science/article/pii/0097316575900990

3. Greg Friedman says:

Not the same problem, but a great related web site:
https://mathformillennials.wordpress.com/2019/04/09/square-in-squares/