# Multisets with 3 Elements

As already mentioned rectangles of size 1 x w can always be constructed by the w=(n+2)C 3 multisets of cardinality 3 chosen from n items. Also a lot of chains with a width of 2 can be made. For n>=6 some rectangles of size a x b with a>=b>=3 are shown.

## n=6, k=3, 56 multisets

The six items to choose from are high and low boxes, high and low octogonal prisms and high and low pyramids. The 56 multisets made from these items were printed on squares with random positions. On the left there are three or two equal items, on the right all items are different.

Rectangle 14x4
 1-1-1 1-1-6 1-6-6 6-6-6 5-6-6 4-6-6 4-4-6 4-4-4 1-4-4 1-4-6 4-5-6 1-4-5 1-2-4 1-2-2 1-1-5 1-1-3 1-3-6 3-6-6 2-6-6 2-4-6 2-4-4 4-4-5 3-4-4 3-4-6 3-5-6 1-5-6 1-2-5 1-2-6 1-1-2 1-2-3 1-3-3 3-3-6 2-3-6 2-2-6 2-2-4 2-4-5 2-3-4 3-4-5 1-3-5 1-5-5 2-5-5 2-5-6 1-1-4 1-3-4 3-3-4 3-3-3 2-3-3 2-2-3 2-2-2 2-2-5 2-3-5 3-3-5 3-5-5 5-5-5 4-5-5 5-5-6

The Rectangle of size 8x7 is shown with the real pieces.

## n=7, k=3, 84 multisets

Rectangle 28x3
 4-4-4 1-4-4 4-4-5 4-5-5 5-5-5 3-5-5 3-3-5 3-3-3 3-3-4 3-4-5 2-3-4 2-2-3 2-2-2 2-2-4 1-2-4 1-1-2 1-1-1 1-1-5 1-1-4 1-4-7 4-5-7 5-6-7 2-6-7 6-6-7 6-6-6 5-6-6 3-5-6 2-3-5 3-4-4 4-4-6 4-5-6 5-5-6 1-5-5 1-3-5 1-3-3 2-3-3 3-3-6 3-4-6 2-3-6 2-2-6 1-2-2 2-2-7 1-2-7 1-1-7 1-1-6 1-1-3 1-3-4 3-4-7 3-5-7 5-7-7 6-7-7 4-6-7 4-6-6 1-6-6 1-5-6 1-2-5 4-4-7 2-4-4 2-4-5 2-5-5 5-5-7 1-5-7 1-3-7 3-3-7 3-6-7 3-6-6 2-6-6 2-5-6 2-2-5 2-5-7 2-7-7 1-7-7 1-6-7 1-3-6 1-2-3 2-3-7 3-7-7 7-7-7 4-7-7 2-4-7 2-4-6 1-2-6 1-4-6 1-4-5

Rectangle 21x4
 2-2-2 2-2-5 2-5-5 5-5-5 3-5-5 3-3-5 3-3-3 1-3-3 2-3-3 2-2-3 1-2-2 1-1-2 1-1-1 1-1-5 1-5-5 1-2-5 1-2-4 1-2-3 2-3-5 3-5-6 4-5-6 2-2-4 2-2-6 2-5-6 5-5-6 5-5-7 3-5-7 3-3-7 3-3-4 3-3-6 2-3-6 1-2-6 1-1-6 1-1-4 1-1-3 1-3-5 1-5-7 1-2-7 2-3-7 2-3-4 3-4-5 4-5-5 2-4-4 2-4-6 2-6-6 5-6-6 5-6-7 5-7-7 3-7-7 3-4-7 3-6-7 3-6-6 1-6-6 1-4-6 1-4-4 1-3-4 1-3-7 1-7-7 2-7-7 2-5-7 2-4-5 4-5-7 4-4-5 4-4-4 4-4-6 4-6-6 6-6-6 6-6-7 6-7-7 7-7-7 4-7-7 4-6-7 3-4-6 1-3-6 1-5-6 1-4-5 1-4-7 1-1-7 1-6-7 2-6-7 2-2-7 2-4-7 4-4-7 3-4-4

Rectangle 14x6
 5-5-5 5-5-7 3-5-7 3-3-7 2-3-7 1-3-7 1-2-7 2-6-7 2-6-6 6-6-6 3-6-6 3-5-6 3-4-6 3-4-4 1-5-5 3-5-5 3-3-5 3-3-3 2-3-3 1-2-3 1-2-4 2-4-7 2-4-6 4-6-6 5-6-6 4-5-6 4-4-6 2-4-4 1-1-5 1-3-5 1-3-3 3-3-6 3-3-4 1-3-4 1-4-7 4-4-7 4-6-7 6-6-7 1-6-6 1-4-6 1-4-4 4-4-4 1-1-4 1-1-3 1-3-6 2-3-6 2-3-4 3-4-5 4-5-7 4-7-7 3-4-7 3-6-7 1-6-7 1-5-6 1-4-5 4-4-5 1-1-7 1-1-6 1-2-6 2-2-6 2-2-3 2-3-5 2-5-7 5-7-7 3-7-7 6-7-7 5-6-7 2-5-6 2-4-5 4-5-5 1-1-1 1-1-2 1-2-2 2-2-2 2-2-4 2-2-5 2-2-7 2-7-7 7-7-7 1-7-7 1-5-7 1-2-5 2-5-5 5-5-6

Rectangle 12x7
 5-5-7 5-5-6 4-5-6 3-4-6 3-4-4 4-4-7 4-5-7 4-6-7 2-6-7 2-6-6 6-6-6 1-6-6 5-5-5 4-5-5 3-4-5 3-3-4 3-4-7 4-7-7 5-7-7 5-6-7 6-6-7 3-6-6 5-6-6 4-6-6 1-5-5 3-5-5 3-3-5 3-3-3 3-3-7 3-7-7 7-7-7 6-7-7 3-6-7 3-5-6 2-5-6 2-4-6 1-1-5 1-3-5 1-3-3 3-3-6 2-3-3 2-3-7 2-7-7 1-7-7 1-6-7 1-5-6 1-2-5 2-4-5 1-1-4 1-1-3 1-3-6 2-3-6 2-3-4 2-4-7 1-2-7 1-5-7 1-4-7 1-4-6 1-4-5 4-4-5 1-1-7 1-1-6 1-2-6 2-2-6 2-2-3 2-2-7 2-5-7 3-5-7 1-3-7 1-3-4 1-4-4 4-4-6 1-1-1 1-1-2 1-2-2 2-2-2 2-2-4 2-2-5 2-5-5 2-3-5 1-2-3 1-2-4 2-4-4 4-4-4

## n=8, k=3, 120 multisets

Rectangle 40x3
 4-8-8 5-8-8 3-5-8 3-3-8 3-3-6 3-6-7 3-5-6 5-5-6 4-5-5 2-4-5 1-4-5 1-1-4 1-1-7 1-3-7 1-6-7 1-5-7 1-5-5 3-5-5 2-3-5 2-2-3 2-2-6 2-6-7 2-4-7 3-4-7 3-3-4 1-3-4 3-4-6 4-6-7 4-7-8 3-7-8 1-7-8 1-5-8 1-1-5 1-1-1 1-1-6 1-5-6 1-6-6 1-3-6 1-6-8 6-7-8 1-8-8 2-8-8 3-8-8 3-6-8 3-6-6 6-6-7 5-6-6 4-5-6 4-4-5 2-4-4 1-4-4 1-4-7 1-7-7 3-7-7 6-7-7 5-7-7 5-5-7 5-5-5 2-5-5 2-2-5 1-2-2 2-2-7 2-3-7 3-3-7 1-3-3 1-3-5 3-4-5 4-5-7 4-5-8 3-4-8 1-4-8 1-2-8 1-1-8 1-1-3 1-1-2 1-2-5 1-2-6 2-3-6 2-6-8 2-7-8 8-8-8 7-8-8 6-8-8 6-6-8 6-6-6 2-6-6 4-6-6 4-4-6 4-4-4 4-4-8 3-4-4 4-4-7 4-7-7 7-7-7 2-7-7 7-7-8 5-7-8 5-5-8 2-5-8 2-2-8 2-2-2 2-2-4 2-3-4 2-3-3 3-3-3 3-3-5 3-5-7 5-6-7 5-6-8 4-6-8 2-4-8 2-3-8 1-3-8 1-2-3 1-2-7 1-2-4 1-4-6 2-4-6 2-5-6 2-5-7

The other rectangles of size 30x4, 24x5, 20x6, 15x8 and 12x10 are on the 100+ puzzle gallery of my site.

## n=9, k=3, 165 multisets

Rectangle 55x3
 8-8-8 3-8-8 4-8-8 4-4-8 4-4-4 4-4-9 4-4-5 4-5-5 5-5-5 5-5-9 1-5-5 1-1-5 1-1-1 1-1-8 1-1-7 1-7-7 7-7-7 7-7-8 2-7-7 2-2-7 2-2-2 2-2-8 2-2-4 2-4-7 4-7-8 6-7-8 6-6-8 6-6-6 3-6-6 3-3-6 3-3-3 3-3-8 3-3-7 1-3-3 1-1-3 1-1-9 1-5-9 1-4-5 4-5-6 3-4-6 4-6-9 3-6-9 3-6-8 5-6-8 5-8-9 4-8-9 4-7-9 5-7-9 2-5-9 1-2-9 1-2-8 1-7-8 1-4-7 4-6-7 4-6-6 7-8-8 5-8-8 1-8-8 1-4-8 1-4-4 2-4-4 4-4-7 4-5-7 5-5-7 5-5-8 2-5-5 1-2-5 1-1-2 1-1-4 1-1-6 1-6-7 6-7-7 3-7-7 7-7-9 2-7-9 2-2-9 2-2-3 2-2-6 2-6-7 2-7-8 7-8-9 6-8-9 6-6-9 5-6-6 3-5-6 3-3-5 2-3-3 3-3-4 3-3-9 1-3-9 1-9-9 5-9-9 4-5-9 4-5-8 3-4-8 3-4-9 3-8-9 3-5-8 1-5-8 1-8-9 1-4-9 1-7-9 1-5-7 2-5-7 1-2-7 1-2-3 1-3-7 1-3-4 1-4-6 2-4-6 2-8-8 8-8-9 6-8-8 4-6-8 4-4-6 3-4-4 3-4-7 4-7-7 5-7-7 5-7-8 2-5-8 2-2-5 1-2-2 1-2-4 1-2-6 1-6-6 6-6-7 3-6-7 3-7-9 7-9-9 2-9-9 2-3-9 2-6-9 2-6-6 2-6-8 2-8-9 8-9-9 6-9-9 5-6-9 5-5-6 3-5-5 2-3-5 3-4-5 3-5-9 3-9-9 9-9-9 4-9-9 2-4-9 2-4-5 2-4-8 2-3-4 2-3-8 3-7-8 1-3-8 1-6-8 1-6-9 6-7-9 5-6-7 3-5-7 2-3-7 2-3-6 1-3-6 1-3-5 1-5-6 2-5-6

Rectangle 33x5
 2-4-4 4-4-5 4-5-7 5-7-7 3-7-7 3-5-7 3-5-5 2-5-5 4-5-5 2-4-5 2-2-4 2-2-7 1-2-2 1-1-2 1-1-3 1-1-5 1-5-7 1-3-7 1-2-7 2-3-7 3-4-7 4-7-8 4-4-8 4-4-9 4-9-9 7-9-9 1-7-9 7-8-9 1-8-9 1-9-9 1-2-9 2-2-9 2-8-9 4-4-4 4-4-7 4-7-7 7-7-7 2-7-7 2-5-7 5-5-7 5-5-5 1-5-5 1-2-5 2-2-5 2-2-2 2-2-3 1-2-3 1-3-3 1-3-5 1-4-5 1-3-4 1-2-4 2-4-7 4-6-7 4-6-8 4-4-6 3-4-4 3-4-9 4-7-9 7-7-9 6-7-9 1-6-9 1-1-9 1-5-9 2-5-9 2-5-8 1-4-4 1-4-7 1-7-7 7-7-8 2-7-8 5-7-8 5-5-8 5-5-6 1-5-6 2-5-6 2-2-6 2-2-8 2-3-8 2-3-3 3-3-5 3-4-5 4-5-8 1-4-8 2-4-8 2-4-6 1-4-6 4-5-6 3-4-6 2-3-4 2-4-9 2-7-9 5-7-9 3-7-9 1-3-9 1-4-9 4-5-9 5-5-9 5-8-9 1-1-4 1-1-7 1-7-8 7-8-8 2-8-8 5-8-8 5-6-8 5-6-6 1-6-6 2-6-6 2-6-7 2-6-8 3-6-8 3-3-6 3-3-4 3-4-8 3-5-8 1-5-8 1-2-8 1-2-6 1-6-7 5-6-7 3-5-6 2-3-5 2-3-9 2-9-9 5-9-9 3-9-9 3-8-9 4-8-9 4-6-9 5-6-9 3-5-9 1-1-1 1-1-8 1-8-8 8-8-8 4-8-8 6-8-8 6-6-8 6-6-6 4-6-6 6-6-7 6-7-7 6-7-8 3-7-8 3-3-7 3-3-3 3-3-8 3-8-8 1-3-8 1-6-8 1-1-6 1-3-6 3-6-7 3-6-6 2-3-6 2-6-9 6-9-9 9-9-9 8-9-9 8-8-9 6-8-9 6-6-9 3-6-9 3-3-9

Rectangle 15x11
 4-4-6 4-4-8 4-7-8 1-7-8 1-5-8 3-5-8 2-5-8 2-4-8 2-4-5 1-2-5 1-5-9 1-9-9 5-9-9 5-5-9 5-5-8 3-4-4 4-4-5 4-5-7 1-5-7 1-1-5 1-3-5 2-3-5 2-3-4 2-4-7 1-2-7 1-7-9 7-9-9 5-7-9 2-5-9 5-8-9 4-4-4 4-4-7 1-4-7 1-1-7 1-1-8 1-3-8 1-2-3 2-3-7 2-7-7 1-7-7 1-6-7 6-7-9 2-7-9 2-3-9 3-5-9 2-4-4 1-4-4 1-1-4 1-1-1 1-1-3 1-3-4 1-3-7 3-4-7 4-7-7 7-7-7 6-7-7 7-7-9 3-7-9 3-3-9 1-3-9 2-2-4 1-2-4 1-1-2 1-1-6 1-3-6 3-4-6 3-6-7 3-5-7 5-7-7 3-7-7 7-7-8 7-8-9 4-7-9 3-4-9 3-6-9 2-2-8 1-2-2 1-2-6 1-6-6 1-4-6 2-4-6 4-6-7 5-6-7 5-7-8 3-7-8 2-7-8 2-8-9 4-8-9 4-9-9 3-9-9 2-2-3 2-2-6 2-6-6 6-6-6 4-6-6 4-6-8 6-7-8 5-6-8 5-8-8 7-8-8 2-8-8 8-8-9 3-8-9 8-9-9 9-9-9 2-2-2 2-2-7 2-6-7 6-6-7 6-6-8 2-6-8 3-6-8 6-8-8 8-8-8 1-8-8 1-2-8 1-8-9 6-8-9 6-9-9 2-9-9 2-2-5 2-5-7 2-5-6 5-6-6 3-6-6 2-3-6 2-3-8 3-8-8 4-8-8 1-4-8 1-6-8 1-6-9 6-6-9 2-6-9 2-2-9 2-5-5 5-5-7 5-5-6 3-5-6 3-3-6 2-3-3 3-3-8 3-4-8 4-5-8 1-4-5 1-5-6 5-6-9 4-6-9 2-4-9 1-2-9 5-5-5 1-5-5 3-5-5 3-3-5 3-3-3 3-3-7 1-3-3 3-3-4 3-4-5 4-5-5 4-5-6 4-5-9 4-4-9 1-4-9 1-1-9

## n=10, k=3, 220 multisets

Given the 15x11 rectangle with 3 from 9 items you can extend this to a 20x11 rectangle adding all 55 multisets containing at least one item no 10.

 5-5-5 1-5-5 3-5-5 3-3-5 3-3-3 3-3-7 1-3-3 3-3-4 3-4-5 4-5-5 4-5-6 4-5-9 4-4-9 1-4-9 1-1-9 1-1-10 1-10-10 1-8-10 1-6-10 5-6-10 2-5-5 5-5-7 5-5-6 3-5-6 3-3-6 2-3-3 3-3-8 3-4-8 4-5-8 1-4-5 1-5-6 5-6-9 4-6-9 2-4-9 1-2-9 1-2-10 2-10-10 8-10-10 6-8-10 2-6-10 2-2-5 2-5-7 2-5-6 5-6-6 3-6-6 2-3-6 2-3-8 3-8-8 4-8-8 1-4-8 1-6-8 1-6-9 6-6-9 2-6-9 2-2-9 2-2-10 2-8-10 8-8-10 5-8-10 2-5-10 2-2-2 2-2-7 2-6-7 6-6-7 6-6-8 2-6-8 3-6-8 6-8-8 8-8-8 1-8-8 1-2-8 1-8-9 6-8-9 6-9-9 2-9-9 2-9-10 8-9-10 4-8-10 4-5-10 1-5-10 2-2-3 2-2-6 2-6-6 6-6-6 4-6-6 4-6-8 6-7-8 5-6-8 5-8-8 7-8-8 2-8-8 8-8-9 3-8-9 8-9-9 9-9-9 9-9-10 4-9-10 4-4-10 1-4-10 1-9-10 2-2-8 1-2-2 1-2-6 1-6-6 1-4-6 2-4-6 4-6-7 5-6-7 5-7-8 3-7-8 2-7-8 2-8-9 4-8-9 4-9-9 3-9-9 3-9-10 3-4-10 2-4-10 4-7-10 1-7-10 2-2-4 1-2-4 1-1-2 1-1-6 1-3-6 3-4-6 3-6-7 3-5-7 5-7-7 3-7-7 7-7-8 7-8-9 4-7-9 3-4-9 3-6-9 3-6-10 3-8-10 2-3-10 2-7-10 7-8-10 2-4-4 1-4-4 1-1-4 1-1-1 1-1-3 1-3-4 1-3-7 3-4-7 4-7-7 7-7-7 6-7-7 7-7-9 3-7-9 3-3-9 1-3-9 1-3-10 3-3-10 3-7-10 7-7-10 5-7-10 4-4-4 4-4-7 1-4-7 1-1-7 1-1-8 1-3-8 1-2-3 2-3-7 2-7-7 1-7-7 1-6-7 6-7-9 2-7-9 2-3-9 3-5-9 3-5-10 3-10-10 7-10-10 6-7-10 7-9-10 3-4-4 4-4-5 4-5-7 1-5-7 1-1-5 1-3-5 2-3-5 2-3-4 2-4-7 1-2-7 1-7-9 7-9-9 5-7-9 2-5-9 5-8-9 5-9-10 9-10-10 4-10-10 4-6-10 6-9-10 4-4-6 4-4-8 4-7-8 1-7-8 1-5-8 3-5-8 2-5-8 2-4-8 2-4-5 1-2-5 1-5-9 1-9-9 5-9-9 5-5-9 5-5-8 5-5-10 5-10-10 10-10-10 6-10-10 6-6-10

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