# POLYOMINO TILINGS

## Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

Then click "Show" button.

You may also see list of all polyomino sets for which data is available here.

## Dominoes and I triomino¶

Prime rectangles: 9.

## Smallest rectangle tilings¶

Smallest rectangle (1x5):

Smallest square (3x3):

## Rectangle tilings' solutions count (including symmetric)¶

Blue number (P) - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number (W) - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Red number (C) - composite rectangle (which can be divided into two rectangles tileable by this set).

Gray number - it is unknown whether rectangle is prime or composite.

Question mark (?) - solution count is unknown.

Click on underlined numbers to view picture with one solution.

w \ h1234567N>0
100
20000
300001212P
40022P2727P148148C
522P77C139139C974974C96129612C
6001616P425425C53305330C8142981429C11370421137042C
733P3939C15531553C2912629126C707903707903C1582763915827639C354859954354859954C
833P8989C50675067C159555159555C≥1≥1C≥1≥1C≥1≥1C?
944P193193C1737617376C856560856560C≥1≥1C≥1≥1C≥1≥1C?
1066C417417C5740357403C46130464613046C≥1≥1C≥1≥1C≥1≥1C?
1199P886886C193305193305C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
121010C18591859C642028642028C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
131616C38803880C21473332147333C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
142020C80488048C71486227148622C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
152727C1661816618C2385213523852135C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
163636C3421034210C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
174949C7023770237C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
186363C143914143914C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
198686C294423294423C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
20113113C601585601585C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
21150150C12280051228005C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
22199199C25047872504787C≥1≥1C≥1≥1C≥1≥1C≥1≥1C≥1≥1C?
23265265C51059505105950C???????????
24349349C1040335710403357C???????????
25465465C2118872621188726C???????????
26615615C4314243643142436C???????????
27815815C8782137087821370C???????????
2810801080C178736362178736362C???????????
2914321432C363713823363713823C???????????
3018951895C740038666740038666C???????????
3125132513C?????????????
3233283328C?????????????
3344094409C?????????????
3458415841C?????????????
3577397739C?????????????
361025010250C?????????????
371358113581C?????????????
381799017990C?????????????
392383223832C?????????????
403157131571C?????????????
414182441824C?????????????
425540355403C?????????????
437339673396C?????????????
449722897228C?????????????
45128800128800C?????????????
46170624170624C?????????????
47226030226030C?????????????
48299424299424C?????????????
49396655396655C?????????????
50525455525455C?????????????
N>0allallallallallallall

## Smallest prime reptiles¶

Smallest prime reptiles (2Ix2, 3Ix2):

## Reptile tilings' solutions count (including symmetric)¶

polyomino \ n²
I domino??P?P?C
I triomino??P?P?C

## Smallest common multiples¶

Smallest common multiple (area 6):

area612
solutions4P391C

## Formulas¶

$N(w; h)$ - number of ways to tile $w\times h$ rectangle (including symmetric solutions)

$T(w; h) = \begin{cases} 1, & N(w; h) \geq 1 \\ 0, & \text{else} \end{cases}$ - tileability function, $1$ if tiles rectangle, $0$ otherwise

$A(w; h) = \left(N(w; h)\right)^{\frac{1}{wh}}$ - average number of ways to tile cell in $w\times h$ rectangle (including symmetric solutions)

$G(T; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}T(w; h)x^wy^h$ - bivariate generating function of $T(w; h)$

$G(A; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}A(w; h)x^wy^h$ - bivariate generating function of $A(w; h)$

$N(1; n) = -N(1; n - 1) + N(1; n - 2) + 3\times N(1; n - 3) + 2\times N(1; n - 4) - N(1; n - 5) - 2\times N(1; n - 6) - N(1; n - 7) \tag{1}$

$N(2; n) = 2\times N(2; n - 1) + N(2; n - 2) + N(2; n - 3) - 5\times N(2; n - 4) - 2\times N(2; n - 5) - N(2; n - 6) + 2\times N(2; n - 7) + N(2; n - 8) + N(2; n - 9) + N(2; n - 10) \tag{2}$