World!Of
Numbers

WON plate
142 |

[ November 6, 2002 ]
Double Cross Palindromes
by 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 3 1 1 5 1 1 9 1

 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 Prime Curios! Prime Puzzle
Wikipedia 142 Le nombre 142
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