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}. The spectrum of the shape below (two 30-60-90 triangles joined at the hypotenuse) is {1,2,3,6,9,12, . . . }.

This month's problem is "Which sets of positive integers are the spectrum of some shape?" Is {1,2,3} a spectrum? Is there a spectrum which contains 2 and 3 but not 1?

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. Therefore, a 50 degree sector of a circle has spectrum {1,2,3}. 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.

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.

The following solvers found a shape whose spectrum contains 2 and 3 but not 1: Joseph DeVincentis, Ed Pegg, Mike Reid, and George Sicherman. I liked George's solutions the best:

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.

Joseph Babcock was interested in convex spectra of solid shapes. He noted that most regular polyhedrons have spectra {1}.

No one found a shape with spectrum {1,5}, {1,6}, {2,5}, {3,4}, {3,5}, {3,6}, or {4,5}. Are there such shapes?

Let N={1,2,3,4, . . . }. Here are some other shapes and their spectra:

Finite Spectra

Shape	Spectrum	Author
	{1,4}	Ed Pegg
	{2,3}	Joe DeVincentis
	{2,4}	Mike Reid
	{2,6}	George Sicherman
	{2,2k}	George Sicherman
	{4,6}	Mike Reid
	{4,8}	George Sicherman
	{1,3,4}	Mike Reid
	{1,3,9}	Mike Reid
	{2,3,4}	Erich Friedman
	{2,6,10}	George Sicherman
	{1,2,3,5}	Erich Friedman
	{1,2,4,6}	Erich Friedman
	{1,2,4,10}	Karl Scherer
	{1,2,3,4,16}	Karl Scherer
	{1,2,9,14,18}	Ed Pegg

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

Spectra with Eventual Period 2

Shape	Spectrum	Author
	2N	Erich Friedman
	1 ∪ 2N	Erich Friedman
	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
	{2,18} ∪ 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
	{4,8,18,24} U {7,9,11, . . . } ∪ {32,36,40, . . . } ?	Joe DeVincentis George Sicherman
	{30,42,46,50,54, . . . } ?	Mike Reid George Sicherman
	{2,4,6,8,10,11,12,13,14,16,18,20,21,22, . . . }	Mike Reid
	{2,5,6,7,8,11,14,15,16,17,18,19,21+} ?	Mike Reid George Sicherman
	{8,12,13,16,18,20,23,24,28,32,33, . . . }	Mike Reid George Sicherman

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/16/12.