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

Here are some other shapes and their spectra:

Other Finite Spectra

Shape	Spectrum	Author
	{2,12}	Yoshiaki Araki
	{4,16}	Dave Barlow
	{6,7}	Yoshiaki Araki
	{6,9}	Yoshiaki Araki
	{6,12}	Yoshiaki Araki
	{6,24}	Yoshiaki Araki
	{8,9}	Livio Zucca
	{2,18}	George Sicherman
	{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
	{4,5,6}	Yoshiaki Araki
	{6,7,8}	Yoshiaki Araki
	{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
	{1,2,4,10}	Yoshiaki Araki
	{1,2,3,4,6}	George Sicherman
	{1,2,3,6,8}	Teruhisa Sugimoto Yoshiaki Araki
	{1,2,3,4,16}	Karl Scherer
	{1,2,6,7,8}	Yoshiaki Araki
	{1,2,3,4,5,8}	Yoshiaki Araki
	{1,2,3,5,8,12}	Yoshiaki Araki
	{1,2,4,5,6,10}	George Sicherman
	{1,2,4,6,8,10,20}	Yoshiaki Araki
	{1,2,3,4,5,6,8,10,11,12,16,18,20,24,26,48}	Karl Scherer George Sicherman

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–{1,3,5,7,9,15,17,19,23,35,43}?	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
	{1,3,7} ∪ 2N	Kay Young
	2N+2	Erich Friedman
	2N+6	George Sicherman
	{4} ∪ 2N+8	George Sicherman

Spectra with Eventual Period 3

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

Spectra with Eventual Period 4

Shape	Spectrum	Author
	4N	Erich Friedman
	4N+4(k–1), k≥1	Erich Friedman Yoshiaki Araki
	{6} ∪ 4N+4	Yoshiaki Araki
	{1,2,9} ∪ 4N+18	Ed Pegg Gábor Damásdi Yoshiaki Araki

Spectra with Eventual Period 5

Shape	Spectrum	Author
	5N+3	George Sicherman

Spectra with Eventual Period 6

Shape	Spectrum	Author
	{n≠5 (mod 6)} ?	Mike Reid
	{1,2,5,6,20,6N-2}	Yoshiaki Araki

Spectra with Eventual Period 8

Shape	Spectrum	Author
	8N	Mike Reid
	4N+2k ∪ 8N+8k+8	Kay Young

Other Spectra

Shape	Spectrum	Author
	N–{1,9,odd primes}	Mike Reid
	{4r(3r–1), 2r(6r–1)–1, 2(6r²–1), 4r(3r+s), 4r(3r+2), 4r(3r+4)+4, 12r(r+2)+8}	John Wallace Dave Barlow Donald Bell George Sicherman Michael Dowle Yoshiaki Araki

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 these three shapes are unknown:

Yoshiaki Araki found this shape, whose spectrum is 2r(5r–2), 2r(5r–1)–1, 10r²+4r(s–2).

Dividing this tile in half gives spectrum {1, 2, 4r(5r–2), 4r(5r–1)–1, 20r²+8r(s–2)}.

Yoshiaki Araki also found this shape, whose spectrum is {14r²+4r(s–2)}.

Yoshiaki Araki also found this shape, whose spectrum is {(2k+3)r²}.

You can see Yoshiaki Araki's generalizations and results here.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/24/23.