Pentomino holes
 

What is the largest hole you can make with 2 to 12 [U]different[/U] pentominoes? The pentominoes must join at edges, not just corners. For example, you can enclose a hole of area 2 cells with 2 pentominoes if you use the same pentominoes, but they must be different, so the top score is 1 cell. You can attach your answers as Excel spreadsheets where the pentominoes are highlighted in different colours, or any other format as long as you can distinguish between the pentominoes. I have just answered 2, have a go at the rest!

This puzzle was specially intended to be a non-Pari-crackable puzzle (because these last longer), so I hope it is!

A 2-cell hole has 10 neighboring cells that must be occupied, so no two non-identical pentaminoes suffice.

A 3-cell hole has 12 neighboring cells in either configuration. Three distinct pentaminoes suffice.

A 4-cell square hole also has 12 neighboring cells, likewise solvable by three pentaminoes. The other 4-cell holes have 14 neighboring cells. ... and that's as far as I can go in just my head. (*Where's my chessboard?*)

A strategy that seems to work is to look for the squarest hole that the pentaminos will go round:

two pentaminos have 10 cells which should suffice to go round a 1x2 hole, but as 10metreh said, the best that can be achieved with distinct pentaminos is a single hole.

three pentaminos have 15 cells; enough to go round a 2x3 hole; and as only 14 cells surround the 2x3 hole this is easily achieved.

four pentaminos have 20 cells; enough for a 4x4 hole. The best I can get is two less than a 3x5 hole ie 13 cells.

five pentaminos should surround a 5x5 or 4x6. 5x5 less two is easy, and looks to be best.

six and up may have to wait til tomorrow

Richard

I can get 24 with 5:

[code]1112222
1 2
1 3
5 33
5 3
55 3
544444[/code]

where 1, 2, 3, 4, 5 are the pentominoes.

[QUOTE=10metreh;162681]I can get 24 with 5:

[/QUOTE]

Yes, I missed the "L" in the corner: used "W" instead. Clearly it was too late for me last night. I got the 24 this morning, honest!

I've got 33 with 6 and 43 with 7 so far.

[quote=Richard Cameron;162683]Yes, I missed the "L" in the corner: used "W" instead. Clearly it was too late for me last night. I got the 24 this morning, honest!

I've got 33 with 6 and 43 with 7 so far.[/quote]

I don't know the answers myself, but I did a bit of work on this one a few years ago and I remeber I got 45 for 7. Only I can't remeber how I did it.

[QUOTE=10metreh;162706]I don't know the answers myself, but I did a bit of work on this one a few years ago and I remeber I got 45 for 7. Only I can't remeber how I did it.[/QUOTE]

I've got a 46 for 7 pentominos. Can i attach an excel file or do i have to do something else?

I googled briefly to see if 46 is a maximal solution but couldn't find this problem online. And it doesn't appear to be at OEIS.

[quote=Richard Cameron;163174]I've got a 46 for 7 pentominos. Can i attach an excel file or do i have to do something else?

I googled briefly to see if 46 is a maximal solution but couldn't find this problem online. And it doesn't appear to be at OEIS.[/quote]

Do what I did in post 4. It seems to be easier. I only spent about 10 mins on 7 pentominoes, and 45 was the most I got. What have you got for 6 pentominoes? I have a feeling 34 is possible.

46 in 7
 

[QUOTE=10metreh;163177]What have you got for 6 pentominoes? I have a feeling 34 is possible.[/QUOTE]

I had the same feeling and indeed:

[CODE] FFIIIIIN
FF NN
F N
L N
L V
L V
LLYYYYVVV
Y[/CODE]

Heres the 46 with 7. I got several 45s before I got this
[CODE]
LLLIIIII
L WW
L WW
Y W
Y Z
YY ZZZ
Y Z
L NN
LLLLNNN [/CODE]

I'm now looking at 8. Its getting quite difficult -for me- to see what shape is best without counting squares. The difference between 60 and 61 is not apparent to my eye.

Richard

I'm stuck on 59 with 8 a.t.m.

[quote=Flatlander;163204]I'm stuck on 59 with 8 a.t.m.[/quote]I got 60. Great puzzle b.t.w. :smile: I'm surprised I can find no mention of this on the net.
Of course we'll have to move on to hexominoes soon. lol