of which the first two shown above are auspicious, and six ways to tile a 6-mat room:
of which again the first two shown above are auspicious.
Japanese rooms traditionally had their floors covered, one layer deep throughout, with tatami mats. The mats were a standard size and shape, about one yarda by two yards, and the size of a room was measured by the number of tatami maps it needed to cover its floor. So a room measuring 3 yards by 4 would be a "six mat room". Half mats were also used as necessary.
A tiling of a room with mats is regarded as "auspicious" (祝儀敷き, shūgijiki) if it has no point at which four mats meet, and as "inauspicious" (不祝儀敷き, fushūgijiki) if it has at least one such point. For example, there are three waysb to tilec a 4½-mat room:
of which the first two shown above are auspicious, and six ways to tile a 6-mat room:
of which again the first two shown above are auspicious.
This document aims to answer "what rectangles can be tiled auspiciously using unlimited 2x1 mats and no more than one 1x1 mat?"
When specifying a rectangle, we always give its height (as portrayed here) before its width. This is because we will build up tilings for rectangles by juxtaposing several smaller rectangles of the same height, as shown in a horizontal line.
For each height h, we show some components of height h, and then show how to append a set of such components to form an auspicious tiling of an h×w rectangle with w≥h. When, for example, it says in row 9 column 3 "27=10+8+8+1", it means that we can auspiciously tile a 27×9 rectangle by appending the structures shown in column 2 of that row and there labelled 10,8,8,1, in that order, but rotated as necessary.
Where this method yields no auspicious tiling, we believe that none exists, and list its width in red.
| height n | Components | Constructions |
|---|---|---|
| 1 | 2:
1:
|
Use as many of 2 as necessary, with one 1 if necessary |
| 2 | 2:
1:
|
Make an alternating sequence of 1s and 2s, and take a subsequence of the required width |
| 3 | 3a:
3b:
1: |
Make an alternating sequence of 3as and 3bs, and add a 1 as necessary |
| 4 | 2:
1:
|
Make an alternating sequence of 1s and 2s, and take a subsequence of the required width |
|
From here on, note that we may rotate any component through 180° (odd-numbered rows), and square components through 90° (even-numbered rows). |
||
| 5 | 4:
6:
1: |
5=4+1, 6=6, 7=6+1, 8=4+4, 9=4+1+4, 10=6+4, 11=6+4+1. 12=6+6, 13=6+6+1. 14=6+4+4, 15=6+4+4+1, etc. |
| 6 | 4:
6:
1:
|
6=6, 7=6+1, 8=1+6+1, 9=4+1+4, 10=6+4, 11=6+4+1. 12=6+6, 13=6+6+1, 14=6+1+6+1, 15=1+6+1+6+1, 16=6+4+6, 17=6+4+6+1, 18=6+6+6, etc. |
| 7 |
6:
8:
1:
5: |
7=6+1, 8=8, 9=8+1, 10, 11=6+5, 12=6+6, 13=6+6+1, 14=8+6, 15=8+6+1, 16=8+8, 17=8+8+1, 18=6+6+6, 19=6+6+6+1, 20=8+6+6, 21=8+6+6+1, etc. |
| 8 |
6:
8:
1:
|
8=8, 9=8+1, 10=1+8+1, 11, 12, 13=8+5, 14=8+6, 15=8+6+1, 16=8+8, 17=8+8+1, 18=8+1+8+1, 19=1+8+1+8+1, 20=6+8+6, 21=6+8+6+1, 22=8+8+6, 23=8+8+6+1, 24=8+8+8, etc. |
| 9 |
10:
1:
7: |
9=8+1, 10=10, 11=10+1, 12, 13, 14, 15=8+7, 16=8+8, 17=8+8+1, 18=10+8, 19=10+8+1, 20=10+10, 21=10+10+1, 22, 23=8+8+7, 24=8+8+8, 25=10+8+7, 26=10+8+8, 27=10+8+8+1, 28=10+10+8, 29=10+10+8+1, 30=10+10+10, 31=10+10+10+10=1, 32=8+8+8+8, etc. |
| 10 | 8 10 1 built as for rows 6 and 8
No two adjacents 8s |
0=10, 11=10+1, 12=1+10+1, 13, 14, 15, 16, 17=8+1+8, 18=10+8, 19=10+8+1, 20=10+10, 21=10+10+1, 22=10+1+10+1, 23=1+10+1+10+1, 24, 25, 26=8+10+8, 27=8+10+8+1, 28=10+10+8, 29=10+10+8+1, 30=10+10+10, 31=10+10+10+1, 32=10+10+1+10+1, 33=10+1+10+1+10+1, 34=1+10+1+10+1+10+1, 35=8+1+8+1+8+1+8, 36=8+1+8+1+8+1+8+1+8+1, etc. |
| 11 | 10 12 1 9 built as for rows 7 and 9
Only one 1 or 9, at an end |
11=10+1, 12=12, 13=12+1, 14, 15, 16, 17, 18, 19=10+9, 20=10+10, 21=10+10+1, 22=12+10, 23=12+10+1, 24=12+12, 25=12+12+1, 26, 27, 28, 29=10+10+9, 30=10+10+10, 31=10+10+10+1, 32=12+10+10, 33=12+10+10+1, 34=12+12+10, 35=12+12+10+1, 36=12+12+12, 37=12+12+12+1, 38, 39=10+10+10+9, 40=10+10+10+10. 41=10+10+10+10+1, 42=12+10+10+10, 43=12+10+10+10+1, 44=12+12+10+10, 45=12+12+10+10+1, 46=12+12+12+10, 47=12+12+12+10+1, 48+12+12+12+12, 49=12+12+12+12+1, 50=10+10+10+10+10, etc. |
| 12 | 10 12 1 built as for rows 6 and 8
No two adjacents 10s |
12=12, 13=12+1, 14=1+12+1, 15, 16, 17, 18, 19, 20, 21=10+1+10, 22=10+12, 23=10+12+1, 24=12+12, 25=12+12+1, 26=12+1+12+1, 27=1+12+1+12+1, 28, 29, 30, 31, 32=10+1+10+1+10, 33=10+1+10+12=1, 34=10+12+12, 35=10+12+12+1, 36=12+12+12, 37=12+12+12+1, 38=12+12+1+12+1, 39=12+1+12+1+12+1, 40=1+12+1+12+1+12+1, 41, 42, 43=10+1+10+1+10+1+10, 44=10+1+10+1+10+12, 45=10+1+10+12+12, 46=10+12+12+12, 47=10+12+12+12+1,48=12+12+12+12, 49=12+12+12+12+1, 50=12+12+12+1+12+1, 51=12+12+1+12+1+12+1, 52=12+1+12+1+12+1+12+1, 53=1+12+1+12+1+12+1+12+1, 54=10+1+10+1+10+1+10+1+10, etc. |
| 13 | 12 14 1 11 built as for rows 7 and 9
Only one 1 or 11, at an end |
13=10+1, 14=14, 15=14+1, 16, 17, 18, 19, 20, 21, 22, 23=12+11, 24=12+12, 26=14+12, 27=14+12+1, 28=14+14, 29=14+14+1, 30, 31, 32, 33, 34, 35=12+12+11, 36=12+12+12, 37=12+12+12+1, 38=14+12+12, 39=14+12+12+1, 40=14+14+12, 41=14+14+12+1, 42=14+14+14, 43=14+14+14+1, 44, 45, 46, 47=21+12+12+11, 48=12+12+12+12, 49=12+12+12+12+1, 50=12+12+12+14, 51=12+12+12+14+1, 52=12+12+14+14, 53=12+12+14+14+1, 54=12+14+14+14, 55=12+14+14+14+1, 56=14+14+14+14, 57=14+14+14+14+1, 58, 59=12+12+12+12+11, 60=12+12+12+12+12, 61=12+12+12+12+12+1, 62=12+12+12+12+14, 63=12+12+12+12+14+1, 64=12+12+12+14+14, 65=12+12+12+14+14+1, 66=12+12+14+14+14, 67=12+12+14+14+14+14+1, 68=12+14+14+14+14, 69=12+14+14+14+14+1, 70=14+14+14+14+14, 71=14+14+14+14+14+1, 72=12+12+12+12+12+12, etc. |
| 14 | 12 14 1 built as for rows 6 and 8 | All except 17-24 — 32-37 — 47-50 — 62, 63 |
| 15 | 14 16 1 13 built as for rows 7 and 9 | All except 18-26 — 34-40 — 50-54 — 66-68 — 82 |
| 16 | 14 16 1 built as for rows 6 and 8 | All except 19-28 — 36-43 — 53-58 — 70-73 — 87, 88 |
To find if a rectangle with dimensions w×h, w≥h, can be auspiciously tiled:
Calculate r = w modulo h
Calculate q = (w-r)/h
If r ≥ q+2 AND h-r ≥ q+3, we believe there is no auspicious tiling; otherwise there is one, constructible as detailed above.
a. The term "yard" is used for convenience. The exact value varied from province to province.
b. Tilings count as equivalent if they are related by symmetry.
c. The word "tile" is used in its mathematical sense. Japanese sources do not use such a term to describe the arranging of mats.