Perimeter 14 Polyominoes

There are 86 perimeter 14 polyominoes covering an area of 641 unit squares.

Unfortunately 641 is prime and so no rectangles can be made but if we use one on the enneominoes twice a variety of rectangles can be made. This gives a total of 9 problems for each rectangle. The possible rectangles are 5x130, 10x65, 13x50 and 25x26 which can all be made from one or more of the following sets of rectangles..

Alternatively rectangles can be made with a dekomino used twice - the only possible are 7x93 and 21x31 both of which can be made from the set of rectangles below.

Similarly if we used an octomino twice we are able to form the 11x59 rectangle above. Also if we omit the 3x4 rectangle from the set we can form a 17x37 rectangle.

Probably the hardest problem for this set is to form simultaneous squares. These solutions were produced by computer using Gerard's Universal Polyomino Solver. The first took under 30 minutes, the second about 20 and the third about 15 although each had to be given different piece orders to achieve these times.

The above shows solutions with 9, 10, 11 and 12 squares. The question naturally arises - what is the maximum number of squares which can be made with this set? There are over 2000 ways of expressing 641 as the sum of squares even when we reject any with a 1, 2 or 3 and only allow a maximum of four 4s which is all the set can produce.

Patrick Hamlyn has found 15 squares.

If, however, we look for squares which are all different then we find there are just 19 possible sets of squares.

We could, alternatively, look for squares with a single hole. One solution is shown here.

The one-sided polyominoes of perimeter 14 can form a 12 x 97 rectangle. This solution was found using Gerard's Universal Polyomino Solver which got the solution in about 20 seconds after a little careful planning. The second solution which consists of two 6x97 rectangles was found using Peter Esser's solver which can be downloaded from his site.

Other symmetrical constructions are also possible as well as sets of twelve congruent pieces.

There is a also a large number of sets of squares which could be made with this set. Examples with 5, 13 and 17 squares are show here.

The maximum number of different squares which could be made is ten with two possible sets - 4,5,6,7,8,9,12,13,16,18 and 4,5,6,8,9,11,12,14,15,16.

We can also look at sets of pieces with the same perimeter and area. The perimeter 14 rectangles with areas 6, 7, 8 and 9 can for a number of rectangles.

The one sided sets can also produce a number of rectangles.

The 21 heptominoes of perimeter 14 can form a 7x21 rectangle and various other symmetric figures. Brendan Owen has shown that three 7x7 squares cannot be made with this set a result which has been confirmed by Patrick Hamlyn. Both Brendan and Patrick used a computer to search for solution but Mike Reid has produced a deductive proof.