What is the minimum square partition of an almost-square rectangle?

Question

This question is motivated by an older question about tiling an orthogonal polygon with squares. It is a generalisation of my former question about how to prove that the minimum square partition of a 3X2 rectangle has 3 squares).

Let:

An almost-square-rectangle be a rectangle that has a width $w$ and height $h=w-1$.
A square partitioning be a covering by non-overlapping squares; the entire rectangle must be covered, all the squares must be disjoint.
A minimum-square-partitioning be a square partitioning, for which is no square partitioning that is made of a lesser number of squares.

Illustration:

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^{Top row: The almost-square-rectangles of widths $3$, $4$ and $5$. Bottom row: Are these miminum-square-partitions of their corresponding rectangles?}

My question is now:

What is the minimum-square-partitioning of an almost-square-rectangle?

Can we prove ${\rm M{\small IN}S{\small QUARES}}(R_{w,h=w-1})=w$?

_{Note a follow-up question, Minimum square partitions for 4x3 and 5x4 rectangles.}

Is it just me, or is ${\rm M{\small IN}S{\small QUARES}}(R_{w,h})=\frac {\max (w, h)}{\gcd (w, h)}$? — Karolis Juodelė, Nov 08 '13 at 18:29
@KarolisJuodelė with $R_{5,3}$ that gives me $5$, but I think you can cover a $R_{5,3}$ with a $1\cdot R_{3,3} + 1\cdot R_{2,2} + 2\cdot R_{1,1} = 4\text{ squares}$. — Realz Slaw, Nov 08 '13 at 20:21
Right, I was silly. Still, the covering follows euclidean algorithm. The number of squares is the number of steps. I think... — Karolis Juodelė, Nov 08 '13 at 21:07
@KarolisJuodelė yeah I agree it is related to what I intuitively know is the minimum-square-partition; with some thought one can come up with a formula for general $R_{w,h}$. I don't just want a general formula; I want a proof (a proof along with a general formula would be nice though). — Realz Slaw, Nov 08 '13 at 21:08
Please format your questions with a lighter hand. Formatting should be functional in supporting the reader; you used a sledgehammer. — Raphael, Nov 10 '13 at 10:30

score 6 · Accepted Answer · edited Apr 13 '17 at 12:57

6

A similar question was asked on Mathoverflow. The commenters mentioned a paper of Kenyon, which shows that the minimum number of squares required to tile a $w \times (w-1)$ rectangle is $\Theta(\log w)$. See also a related paper of Walters.

You can tile a $(4t+7) \times (4t+6)$ rectangle using only $t+5$ squares (for $t \geq 0$).

Tiling a 6 by 7 rectangle Tiling a 10 by 11 rectangle Tiling a 14 by 15 rectangle Tiling a 18 by 19 rectangle

edited Apr 13 '17 at 12:57

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1

answered Nov 08 '13 at 22:19

Yuval Filmus

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Nice! Do I see room for recursion here? In the bottom right corner there is a $(2t+4)\times (2t+2)$ rectangle tiled with $t+2$ squares, in a way similar to the original solution, but scaled to a $2 \times 2$ grid. (sorry hard to express what I mean.) – Hendrik Jan Nov 09 '13 at 12:13
Have a look at Kenyon's paper - it gives an algorithm which tiles a $w \times (w-1)$ rectangle with $O(\log w)$ squares. – Yuval Filmus Nov 09 '13 at 18:45
@YuvalFilmus what about those specific rectangles I drew above? Are those minimally tiled? – Realz Slaw Nov 09 '13 at 22:50
That's probable, and should be provable using some case analysis. For more sophisticated methods, see Kenyon's paper, which uses the electrical resistance method. – Yuval Filmus Nov 09 '13 at 23:57
@YuvalFilmus I asked this as a follow-up on math.SE. – Realz Slaw Nov 10 '13 at 07:03
Another followup: What is the first w such that a rectangle, Rw×w−1 is minimally-square-partitioned by less than w squares.. – Realz Slaw Nov 17 '13 at 13:47

What is the minimum square partition of an almost-square rectangle?

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