[ November 6, 2002 ]
Double Cross Palindromes
Enoch Haga (email)
Have a look at Enoch's following double cross squares
which are filled horizontally with different palprimes.
(131, 151 & 191)
(10301, 12421, 15451, 12721 & 18181)
| 1 | 0 | 3 | 0 | 1 |
| 1 | 2 | 8 | 2 | 1 |
| 1 | 5 | 4 | 5 | 1 |
| 3 | 2 | 3 | 2 | 3 |
| 1 | 8 | 1 | 8 | 1 |
The 'double cross' refers to two additional requirements.
First, as indicated in the 3 x 3 example, the middle vertical bar
must be prime, preferably in both directions 
(359 & 953).
Secondly, as shown in the 5 x 5 example, the diagonals
must be palindromic and prime (ie. palprime)
and different from any horizontal one (12421).
Note that the center vertical of this 5 x 5 square
is only unidirectionally prime (38431)
Finally, the next 7 x 7 and 9 x 9 illustrations are the ones to beat !
No doubt there exist larger solutions to be discovered
and who knows with some extra hidden curiosities...
| 1 | 2 | 4 | 2 | 4 | 2 | 1 |
| 9 | 1 | 1 | 0 | 1 | 1 | 9 |
| 1 | 5 | 7 | 9 | 7 | 5 | 1 |
| 7 | 5 | 0 | 7 | 0 | 5 | 7 |
| 9 | 1 | 7 | 4 | 7 | 1 | 9 |
| 9 | 1 | 2 | 7 | 2 | 1 | 9 |
| 1 | 4 | 8 | 9 | 8 | 4 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 1 |
| 1 | 0 | 1 | 4 | 2 | 4 | 1 | 0 | 1 |
| 1 | 0 | 1 | 4 | 1 | 4 | 1 | 0 | 1 |
| 1 | 0 | 2 | 2 | 0 | 2 | 2 | 0 | 1 |
| 1 | 0 | 0 | 8 | 8 | 8 | 0 | 0 | 1 |
| 1 | 4 | 4 | 2 | 0 | 2 | 4 | 4 | 1 |
| 1 | 0 | 1 | 2 | 9 | 2 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 5 | 0 | 0 | 0 | 1 |
| 1 | 0 | 3 | 6 | 9 | 6 | 3 | 0 | 1 |
Of course the ultimate challenge would be to have all verticals
(not just the center) prime and in both directions. I expect that
someone will send us some of these.
[ November 8, 2002 ]
Already Jean Claude Rosa (email) contributes some
modest 3 x 3 solutions :
1 3 1 3 1 3
3 5 3 7 5 7
1 9 1 3 7 3
followed by this 5 x 5 square :
1 3 3 3 1
1 6 5 6 1
9 4 0 4 9
7 6 6 6 7
1 7 9 7 1
Note that all verticals are prime in both directions (a.k.a. emirps)!
(11971 & 17911)
(36467 & 76463)
(35069 & 96053)
[ November 15, 2002 ]
A triple 5 x 5 ALL ODD contribution from Jean Claude Rosa
1 9 9 9 1 7 1 9 1 7 9 3 7 3 9
3 3 5 3 3 9 7 5 7 9 1 7 9 7 1
7 7 3 7 7 7 9 3 9 7 1 5 5 5 1
1 3 9 3 1 7 7 9 7 7 9 7 3 7 9
1 9 3 9 1 7 1 3 1 7 9 3 1 3 9
[ November 22, 2002 ]
J. C. Rosa searched for a 5 x 5 grid embedded in a 7 x 7 grid
( there are no solutions if this 5 x 5 grid is 'all-odd' )
and he found only 3 solutions. Here is one :
3 3 3 1 3 3 3
3 7 7 3 7 7 3
9 9 3 2 3 9 9
7 3 9 2 9 3 7
3 9 3 1 3 9 3
1 7 9 3 9 7 1
3 3 3 7 3 3 3
Alas, all the verticals of the 7 x 7 grid are not emirps :
3397313 prime but 3137933 composite
3793973 palprime
3939373 prime but 3739393 composite
1322137 emirp |
The interior 5 x 5 grid is complete
since all are "palprimes" or "emirps".
[ November 30, 2002 ]
This time J. C. Rosa constructed an 'all-odd' 7 x 7 grid,
complete, and embedded in an 'all-odd' 9 x 9 grid that is alas,
not complete !
9 9 7 7 3 7 7 9 9
1 3 7 9 9 9 7 3 1
1 1 5 7 9 7 5 1 1
3 3 7 9 3 9 7 3 3
1 1 3 1 1 1 3 1 1
9 1 7 9 3 9 7 1 9
7 7 5 1 9 1 5 7 7
1 3 9 1 3 1 9 3 1
9 7 3 3 3 3 3 7 9
911319719 composite but 917913119 prime
931311737 prime but 737113139 composite
775737593 prime but 395737577 composite
797919113 prime but 311919797 composite
399313933 emirp |
A000142 