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.

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\n2345678
1
(Ed Pegg)
????
2
(Joe DeVincentis)

(Mike Reid)
?
(George Sicherman)
?
(George Sicherman)
3?????
4?
(Mike Reid)
?
(George Sicherman)

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

Other Finite Spectra

ShapeSpectrumAuthor
{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
{1,2,3,4,16}Karl Scherer
{1,2,9,14,18}Ed Pegg
{12,48}John Wallace

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

ShapeSpectrumAuthor
NErich 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

ShapeSpectrumAuthor
2NErich Friedman
1 ∪ 2NErich Friedman
2N-{2}Erich Friedman

Spectra with Eventual Period 3

ShapeSpectrumAuthor
3NErich Friedman
{1,2} ∪ 3NErich Friedman

Spectra with Eventual Period 4

ShapeSpectrumAuthor
4NErich Friedman
2k+4NErich Friedman

Spectra with Eventual Period 5

ShapeSpectrumAuthor
3+5NGeorge Sicherman

Spectra with Eventual Period 6

ShapeSpectrumAuthor
{n≠5 (mod 6)} ?Mike Reid

Spectra with Eventual Period 8

ShapeSpectrumAuthor
8NMike Reid
{6,10,14,18, . . . } ∪ {24,32,40, . . . }Mike Reid

Other Spectra

ShapeSpectrumAuthor
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

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


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.