User:Peter Luschny/OptimalRulers

Optimal Rulers

All optimal rulers (disregarding their mirror images) which measure all lengths <= L. Definitions are here. Perfect rulers are here.

There are 136495 perfect rulers with length <= 101, but only 67 optimal rulers. Thus below is the Crème de la Crème of all rulers displayed.

For the smaller ones the blueprint looks somewhat arbitrary but for the larger ones a layout seems to emerge which is pretty balanced. A conjecture on the blueprint of these rare (and hard to find) creatures can be found here. For the green numbers see A004137.

Length <= 1

[0, 1]
||

Length <= 3

[0, 1, 3]
||-|

Length <= 6

[0, 1, 4, 6]
||--|-|

Length <= 9

[0, 1, 4, 7, 9]
[0, 1, 2, 6, 9]
||--|--|-|
|||---|--|

Length <= 13

[0, 1, 6, 9, 11, 13]
[0, 1, 4, 5, 11, 13]
[0, 1, 2, 6, 10, 13]
||----|--|-|-|
||--||-----|-|
|||---|---|--|

Length <= 17

[0, 1, 8, 11, 13, 15, 17]
[0, 1, 4, 10, 12, 15, 17]
[0, 1, 2,  8, 12, 15, 17]
[0, 1, 2,  8, 12, 14, 17]
[0, 1, 2,  6, 10, 14, 17]
[0, 1, 2,  3,  8, 13, 17]
||------|--|-|-|-|
||--|-----|-|--|-|
|||-----|---|--|-|
|||-----|---|-|--|
|||---|---|---|--|
||||----|----|---|

Length <= 23

[0, 1, 4, 10, 16, 18, 21, 23]
[0, 1, 2, 11, 15, 18, 21, 23]
||--|-----|-----|-|--|-|
|||--------|---|--|--|-|

Length <= 29

[0, 1, 4, 10, 16, 22, 24, 27, 29]
[0, 1, 3,  6, 13, 20, 24, 28, 29]
[0, 1, 2, 14, 18, 21, 24, 27, 29]
||--|-----|-----|-----|-|--|-|
||-|--|------|------|---|---||
|||-----------|---|--|--|--|-|

Length <= 36

[0, 1, 3, 6, 13, 20, 27, 31, 35, 36]
||-|--|------|------|------|---|---||

Length <= 43

[0, 1, 3, 6, 13, 20, 27, 34, 38, 42, 43]
||-|--|------|------|------|------|---|---||

Length <= 50

[0, 1, 3, 6, 13, 20, 27, 34, 41, 45, 49, 50]
[0, 1, 2, 3, 23, 28, 32, 36, 40, 44, 47, 50]
||-|--|------|------|------|------|------|---|---||
||||-------------------|----|---|---|---|---|--|--|

Length <= 58

[0, 1, 5, 8, 12, 21, 30, 39, 48, 53, 54, 56, 58]
[0, 1, 3, 6, 17, 24, 27, 38, 45, 49, 53, 57, 58]
[0, 1, 3, 6, 17, 20, 27, 35, 45, 49, 53, 57, 58]
[0, 1, 2, 8, 15, 16, 26, 36, 46, 49, 53, 55, 58]
[0, 1, 2, 6,  8, 17, 26, 35, 44, 47, 54, 57, 58]
[0, 1, 2, 3, 27, 32, 36, 40, 44, 48, 52, 55, 58]
||---|--|---|--------|--------|--------|--------|----||-|-|
||-|--|----------|------|--|----------|------|---|---|---||
||-|--|----------|--|------|-------|---------|---|---|---||
|||-----|------||---------|---------|---------|--|---|-|--|
|||---|-|--------|--------|--------|--------|--|------|--||
||||-----------------------|----|---|---|---|---|---|--|--|

Length <= 68

[0, 1, 2, 8, 15, 16, 26, 36, 46, 56, 59, 63, 65, 68]
[0, 1, 2, 5, 10, 15, 26, 37, 48, 54, 60, 66, 67, 68]
|||-----|------||---------|---------|---------|---------|--|---|-|--|
|||--|----|----|----------|----------|----------|-----|-----|-----|||

Length <= 79

[0, 1, 2, 5, 10, 15, 26, 37, 48, 59, 65, 71, 77, 78, 79]
|||--|----|----|----------|----------|----------|----------|-----|-----|-----|||

Length <= 90

[0, 1, 2, 5, 10, 15, 26, 37, 48, 59, 70, 76, 82, 88, 89, 90]
|||--|----|----|----------|----------|----------|----------|----------|-----|-----|-----|||

Length <= 101

[0, 1, 2, 5, 10, 15, 26, 37, 48, 59, 70, 81, 87, 93, 99, 100, 101]
|||--|----|----|----------|----------|----------|----------|----------|----------|-----|-----|-----|||