Squared Squares - Numberphile
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange askers cite it as the starting point when posing new questions about tiling squares with distinct squares, such as prime side lengths, real-valued sides, or irrational packings.
Recommended by 0 people across 0 different months, 2017 to 2019.
What it is
A Numberphile interview with Dr James Grime about squared squares, squares tiled entirely by smaller integer-sided squares of all different sizes. It covers the smallest known example, the 21-square solution, and the history of the problem including the link to electrical networks.
What people said
Cites volcanrb · 2017-06 · Mathematics Stack Exchange · on “Can the sum of the first n squares of primes be a perfect square?”
So far, all I have found is for $n=1$, for which the square is 4. (Rather trivial) To give some context, I recently watched the Numberphile video Squared Squares [this video] . Basically, a squared square is a square that is entirely made up of other integral squares (preferably all of different sizes). I wanted to see whether there exists a squared square made of $n$ squares that is entirely composed of squares with the side-lengths of all primes up to the $n$th prime. By finding an example that satisfies the question, it will be shown that there is at least the possibility for one of these squares existing. If it can be proven that there are no examples other than $n=1$, then it follows that no
Cites orlp · 2017-06 · Mathematics Stack Exchange · on “Square made from other real squares”
A recent Numberphile video [this video] discusses integer sided squares made out of unique smaller integer sided squares. The smallest such square has been proven to use 21 rectangles. It's clear that for rational sided squares that result is also optimal, because if there was an arrangement that used less than 21 rectangles every side could be multiplied by the least common multiple of all denominators to form a solution where every side length is integer. What about the real number case? Without loss of generality, what is the smallest amount of unique squares with real sides such that they can be arranged to exactly cover the unit square?
Cites Rócherz · 2018-03 · Mathematics Stack Exchange · on “Mathematical inspiration from my bathroom floor”
1. Sums over the grid. The sum of the first $n$ odd numbers is actually $n^2$: $$1+3+5+\ldots+(2n-3)+(2n-1)=\sum^n_{k=1} (2k-1) = n^2.$$ [LINK] Moreover, you can express any odd number as a difference of squares: $$2n-1=n^2-(n-1)^2.$$ Or any sum of consecutive odd numbers for that matter: $$\sum^n_{k=m} (2k-1) = \left(\sum^n_{k=1} -\sum^{m-1}_{k=1}\right) (2k-1) = n^2 -(m-1)^2.$$ Or even better: the sum of the first $n$ cubes equals the squared sum of the first $n$ integers: $$1^3+2^3+3^3+\ldots+(n-1)^3+n^3 =\sum_{k=1}^n k^3 =\left(\sum_{k=1}^n k\right)^2 =\frac{n^2(n+1)^2}{4}.$$ This can also be seen on a grid: [LINK] from: [LINK] Here's more. If you have a sum over a square grid (as in ass
Cites Shaun · 2019-01 · Mathematics Stack Exchange · on “Irrational packing of Euclidean spaces (with no gaps).”
This appears to be a new question on MSE. The only post [LINK] on here after a search using the string irrational "packing" does not mention (explicitly) what I have in mind. A Google search [LINK] doesn't return anything specific worth noting to an amateur like me. Finally, MO is empty in this regard too [LINK] (but that's without looking into the minutiae of the posts that show up). The Question: Is it possible to pack $\Bbb R^n$ for some/each $n$ , with copies of a regular shape and no gaps, where each copy is of a distinct, positive irrational defining length? Thoughts: I'm guessing the answer is "no". Why? I don't know: pure speculation. It just seems typical of the behaviour of irratio