So we have to look for prime sextuplets of the same pattern. And that page only lists 5 of them (we need 6) and they're into 5 digit numbers. Yeah there's no way we were getting this without some serious help from internet or computer program.

FWIW all of the answers we came up with last night via computer program were less than 1,000.

There are three groups of four prisoners. The rearmost prisoner looks at the three in front of him. If the two immediately in front of him have the same colour hat, he calls out the colour of the hat of the third (frontmost) prisoner. Otherwise, he calls out the opposite colour of the third prisoner.

Call the prisoners ABCD. B knows if A called out D's colour correctly. If It is correct, then B knows he has the same colour as C. If not, he knows he has the opposite colour to C.

C also knows if A called out D's colour correctly, so he knows if the colour called out by B matches his own or is the other colour.

D knows that if B and C call out the same colour, then his hat must be the colour called out by A. If they call different colours, then D has the opposite colour to the one called by A.

So BCD know their colours and will be freed. A has a 50% chance (I assume) of being right about his own hat.

The three groups of four prisoners will each have three freed and one with a 50% chance.

If zero is allowed:
0, 2, 8, 18, 30, 32
11, 29, 71, 149, 431, 569

If positives are required:
4, 6, 16, 30, 34, 40
7, 13, 67, 97, 277, 823

Last guy sees 11 hats, so one color will have an odd number and one will have an even number. He calls out the odd number as his guess for his color hat (the "chosen" color). He has 50/50 chance at being right.

11th guy knows this color should have an odd number of hats - if he sees an odd number in front of him he knows he has the other color. If he sees an even number he knows he has the chosen color.

Each time someone names the chosen color as their hat, the rest of the guys know that the count has changed, from odd to even or from even to odd. If their count matches then they have the other color hat. If their count is different then they have the chosen color.

0, 6, 12, 36, 42, 96 and 5, 17, 31, 47, 61, 71