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Can’t Decide? Undecide!
Chaim GoodmanStrauss
In my mathematical youth, when I frst learned oF G¨odel’s Theorem, and computational undecidability,
I was at once Fascinated and strangely reassured oF our limited place in the grand universe: incredibly
mathematics itselF establishes limits on mathematical knowledge. At the same time, as one digs into the
Formalisms, this area can seem remote From most areas oF mathematics and irrelevant to the e±orts oF most
workaday mathematicians. But that’s just not so! Undecidable problems surround us, everywhere, even
in recreational mathematics!
1 Three Mysterious Examples
Somehow these simple questions seem diﬃcult to resolve:
1.1 Mysterious Example #1
Tilings are a rich source oF combinatorial puzzles. We can ask, For a given tile, whether or not it
admits a
tiling
: that is, does there exist a tiling oF the plane by copies oF this tile?
²or many examples, this is utterly trivial: clearly the tile at leFt in the fgure below does admit a tiling, and
the tile at middle leFt does not. One might discover a simple prooF that the tile at middle right does not
admit a tiling either
1
, though it is more diﬃcult to work out just how large a region you can cover beFore
getting stuck. But it’s a reasonable bet that you will not be able to discover whether or not the tile at right,
discovered by J. Myers in 2003, admits a tiling, at least not without resorting to some sort oF bruteForce
calculation on a computer! Try this For yourselF! A downloadable fle with tiles to cut out and play with has
been placed at
http://mathfactor.uark.edu/downloads/myers
tile.pdf
In general, then, we have
Input:
At
i
le
.
and a
Decision Problem:
Does the given tile admit a tiling of the plane?
1
Hint: the tile, discovered by C. Mann, can be viewed as a cluster of hexagons, with some edges bulging inwards and some
bulging out — but there are more bulging inwards than outwards. Etc.
. . .
1
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View Full DocumentWith enough bruteforce eFort, in some circumstances, we can answer this problem:
We might simply enumerate all possible con±gurations admitted by the tile, covering larger and larger disks.
If the tile
does not
admit a tiling, eventually there will be some sized disk we can no longer cover, we run
out of con±gurations to enumerate, and we then know the answer to our problem:
No
, the tile fails to admit
a tiling. If a tile does not admit a tiling, the “Heesch number” is a measure of the complexity of such a tile,
as the largest possible combinatorial radius of disks it can cover (in other words, the maximum number of
concentric rings copies of the tile can form); C. Mann discovered the current world record examples, with
Heesch number 5 [22, 23]. But how can we determine whether an arbitrary given tile
does
admit a tiling?
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 Spring '09
 Prof.Ceasar

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