Multisets with 2 Elements
As already mentioned rectangles of size 1 x w can always be constructed by the w=(n+1)*n/2 multisets
of cardinality 2
chosen from n items. The 2x3 rectangle isn't possible to get with the 6 multisets
of size 2 chosen from n=3 items. The same applies for the 2x5 rectangle using the 10 multisets
of size 2 chosen from n=4 items. Otherwise if you take
the 6 combinations of 2 elements from n=4 items or the 10 combinations of 2 elements from n=5 items the above rectangles can be made.
n=5, k=2, 15 multisets
Rectangle 5x3. The solution is very easy to get and it's shown with pieces made from colored stripes.
n=6, k=2, 21 multisets
Rectangle 7x3. The table tennis balls aren't connected to show that the order of the elements doesn't
matter.
n=7, k=2, 28 multisets
Rectangle 14x2
| 5-5 | 5-6 | 6-6 | 3-6 | 3-3 | 3-7 | 7-7 | 2-7 | 2-2 | 2-5 | 1-5 | 1-6 | 4-6 | 4-5 |
| 3-5 | 5-7 | 6-7 | 2-6 | 2-3 | 1-3 | 1-7 | 4-7 | 2-4 | 1-2 | 1-1 | 1-4 | 4-4 | 3-4 |
Rectangle 7x4
| 6-6 | 1-6 | 1-5 | 1-3 | 3-3 | 3-6 | 2-6 |
| 4-6 | 5-6 | 5-5 | 3-5 | 3-7 | 2-3 | 2-2 |
| 4-4 | 4-5 | 2-5 | 5-7 | 7-7 | 2-7 | 1-2 |
| 3-4 | 1-4 | 2-4 | 4-7 | 6-7 | 1-7 | 1-1 |
n=8, k=2, 36 multisets
Rectangle 18x2
| 3-3 | 2-3 | 2-2 | 2-7 | 7-7 | 4-7 | 4-4 | 2-4 | 2-6 | 6-6 | 5-6 | 5-5 | 1-5 | 1-1 | 1-7 | 3-7 | 3-8 | 8-8 |
| 3-4 | 1-3 | 1-2 | 2-8 | 7-8 | 5-7 | 4-5 | 1-4 | 4-6 | 3-6 | 3-5 | 2-5 | 5-8 | 1-8 | 1-6 | 6-7 | 6-8 | 4-8 |
Rectangle 9x4
| 4-4 | 4-5 | 5-5 | 3-5 | 2-3 | 1-3 | 1-1 | 1-6 | 6-6 |
| 3-4 | 4-8 | 5-8 | 2-5 | 2-2 | 1-2 | 1-5 | 5-6 | 3-6 |
| 3-3 | 3-8 | 8-8 | 2-8 | 2-4 | 1-4 | 1-7 | 5-7 | 6-7 |
| 3-7 | 7-8 | 1-8 | 6-8 | 2-6 | 4-6 | 4-7 | 7-7 | 2-7 |
Rectangle 6x6. The beads are connected to ensure that all multisets are used.
n=9, k=2, 45 multisets
Rectangle 9x5
| 1-1 | 1-8 | 8-8 | 6-8 | 6-6 | 5-6 | 4-6 | 3-6 | 3-8 |
| 1-2 | 1-4 | 4-8 | 8-9 | 6-9 | 6-7 | 2-6 | 1-6 | 1-3 |
| 2-2 | 2-4 | 4-4 | 4-9 | 1-9 | 7-9 | 2-7 | 1-7 | 3-7 |
| 2-9 | 2-3 | 3-4 | 4-5 | 1-5 | 5-9 | 2-5 | 5-7 | 7-7 |
| 9-9 | 3-9 | 3-3 | 3-5 | 5-5 | 5-8 | 2-8 | 7-8 | 4-7 |
Rectangle 15x3
| 5-5 | 5-7 | 7-7 | 2-7 | 2-2 | 2-4 | 4-4 | 1-4 | 4-6 | 6-6 | 3-6 | 6-8 | 1-8 | 7-8 | 6-7 |
| 1-5 | 5-9 | 7-9 | 3-7 | 2-3 | 2-8 | 4-8 | 4-9 | 4-5 | 5-6 | 2-6 | 6-9 | 1-6 | 1-7 | 4-7 |
| 1-1 | 1-9 | 9-9 | 3-9 | 3-3 | 3-8 | 8-8 | 8-9 | 5-8 | 3-5 | 2-5 | 2-9 | 1-2 | 1-3 | 3-4 |
n=10, k=2, 55 multisets
Rectangle 11x5
| 4-4 | 4-8 | 8-8 | 8-10 | 10-10 | 7-10 | 1-7 | 1-5 | 5-9 | 9-9 | 9-10 |
| 2-4 | 2-8 | 3-8 | 3-10 | 6-10 | 2-10 | 1-2 | 1-8 | 8-9 | 6-9 | 2-9 |
| 2-2 | 2-5 | 5-8 | 5-10 | 4-10 | 1-10 | 1-1 | 1-6 | 6-8 | 4-6 | 4-9 |
| 2-6 | 2-7 | 7-8 | 5-7 | 4-7 | 1-4 | 1-9 | 1-3 | 3-6 | 5-6 | 4-5 |
| 6-6 | 6-7 | 7-7 | 7-9 | 3-7 | 3-4 | 3-9 | 3-3 | 2-3 | 3-5 | 5-5 |
Home