Problem of the Month (April 1999)

We call a shape n-convex if n non-overlapping copies of the shape can be arranged into a convex shape. A shape's spectrum is the set of n for which the shape is n-convex.

The spectrum of a square or triangle is {1,2,3,4, . . . }. The spectrum of a circle is {1}. This month's problem is "Which sets of positive integers are the spectrum of some shape?"

ANSWERS

Sectors of a circle have spectra of the form {1,2,3, . . . n} if the sectors cannot form a complete circle, and the form {1,2,3, . . . n,2n} or {1,2,3, . . . n,2n+1} if they do. A 360/n degree sector of a circle dented so that the pieces fit together into a circle, has spectrum {n}. These observations were made by Joseph DeVincentis, Mike Reid, and Tom Turrittin.

Let N={1,2,3,4, . . . }. Connecting the center of an equilateral triangle or square to its vertices by curved but symmetric lines gives shapes with spectra 2N, 3N, and 4N. This was noticed by Joseph DeVincentis.

Joseph DeVincentis found an infinite family of chevrons with spectra {4,3n+4,3n+6,3n+8, . . . } and {4n,4n+2,4n+4, . . . } for all n.

Mike Reid found an infinite family of polyominoes that have all but finitely many integers in their spectra.

Shapes with Two-Element Spectra

m\n	2	3	4	5	6	7	8
1			(Ed Pegg)	?	(Yinji Wu)	?	?
2		(Joe DeVincentis)	(Mike Reid)	?	(George Sicherman)	?	(George Sicherman)
3			?	?	?	?	?
4				?	(Mike Reid)	?	(George Sicherman)

Here are some other shapes and their spectra:

Other Finite Spectra

Shape	Spectrum	Author
	{4,16}	Dave Barlow
	{8,9}	Livio Zucca
	{1,2,3}	Erich Friedman
	{1,2,4}	Erich Friedman
	{1,2,5}	Erich Friedman
	{1,3,4}	Mike Reid
	{1,3,9}	Mike Reid
	{2,3,4}	Erich Friedman
	{2,6,10}	George Sicherman
	{1,2,3,4}	George Sicherman
	{1,2,3,5}	Erich Friedman
	{1,2,3,6}	Erich Friedman
	{1,2,3,7}	Erich Friedman
	{1,2,4,6}	Erich Friedman
	{1,2,3,4,6}	George Sicherman
	{1,2,3,4,16}	Karl Scherer
	{1,2,9,14,18}	Ed Pegg
	{1,2,4,5,6,10,20}	George Sicherman

Karl Scherer found this next tile, though George Sicherman improved its spectrum to {1,2,3,4,5,6,8,10,11,12,16,18,20,24,26,48}:

Spectra with Eventual Period 1

Shape	Spectrum	Author
	N	Erich Friedman
	N–{1}	Mike Reid
	N–{1,5}	Mike Reid
	N–{1,3,5,7,9}	Mike Reid
	N–{1,3,5,7,9,11,13,17}	Mike Reid
	N–{1,2,3,4,5,6,7,9}	Mike Reid George Sicherman
	N–{1,3,5,7,9,15,17,19,35}?	Mike Reid George Sicherman
	N–{1,3,5,7,9,19,23}	George Sicherman

Spectra with Eventual Period 2

Shape	Spectrum	Author
	2N	Erich Friedman
	{1,3} ∪ 2N	Dave Barlow
	2N–{2}	Erich Friedman

Spectra with Eventual Period 3

Shape	Spectrum	Author
	3N	Erich Friedman
	{1,2} ∪ 3N	Erich Friedman

Spectra with Eventual Period 4

Shape	Spectrum	Author
	4N	Erich Friedman
	2k+4N	Erich Friedman

Spectra with Eventual Period 5

Shape	Spectrum	Author
	3+5N	George Sicherman

Spectra with Eventual Period 6

Shape	Spectrum	Author
	{n≠5 (mod 6)} ?	Mike Reid

Spectra with Eventual Period 8

Shape	Spectrum	Author
	8N	Mike Reid
	{6,10,14,18, . . . } ∪ {24,32,40, . . . }	Mike Reid

Other Spectra

Shape	Spectrum	Author
	N–{1,9,odd primes}	Mike Reid
	{8,9,10,4R(3R+N)}	John Wallace Dave Barlow Donald Bell George Sicherman Michael Dowle

Joe DeVincentis found a shape whose spectrum consists of at least {4,8,18,24,38,54} ∪ {7,9,11, . . . } ∪ {32,36,40, . . . } and maybe more. George Sicherman found several of these tilings.

Mike Reid found a similar shape whose spectrum is apparently {2,5,6,7,8,11,14,15,16,17,18,19,20,22+}, and again George Sicherman found several of these tilings.

Mike Reid also found this shape whose spectrum is {30,42,46,50,54,55,56,58,61,62,64,70,90,98, ...} , and again George Sicherman found most of these tilings.

George Sicherman found a shape whose spectrum is apparently multiples of 4, together with a quadratically increasing sequence {2, 18, 66, 138, 234, ...}.

George Sicherman says the spectrum of this shape is unknown.

George Sicherman says the spectrum of this shape is also unknown.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 1/30/20.