Enigma 1562 - Same unused digits
What were the largest and smallest of our
six squares?

Enigma 1562 - Same unused digits
New Scientist magazine, 9 September 2009,
By Richard England.

I chose two four-digit perfect squares (with
no leading zero) that contained eight
different digits. So did Harry, and so did
Tom.

Our six chosen squares were all different,
but the two digits that were unused were the
same for all three of us. If you knew what
those two digits were you would be able to
deduce with certainty what the six squares
that we chose were.

What were the largest and smallest of our
six squares?

Ciao,
Chappy

"Chappy" <petergregorychapman at (no spam) hotmail.com> wrote in message
news:6e0e8905-b565-4b38-b653-a8cbf40c4391 at (no spam) y28g2000prd.googlegroups.com...
Enigma 1562 - Same unused digits
New Scientist magazine, 9 September 2009,
By Richard England.

I chose two four-digit perfect squares (with
no leading zero) that contained eight
different digits. So did Harry, and so did
Tom.

Our six chosen squares were all different,
but the two digits that were unused were the
same for all three of us. If you knew what
those two digits were you would be able to
deduce with certainty what the six squares
that we chose were.

What were the largest and smallest of our
six squares?

Ciao,
Chappy

I tried to do this on paper, honest I did.

But deriving the 36 potential squares took an hour, and I couldnt see an
easy way of imposing the other constraints on them.

So in the end twas prolog to derive the basic lists and then an eyeball of
them to see where six unique squares existed as a list predicated on the
same two digits left over (1,7).

As requested:- 2809 ... 8649

HTH
JJ

"John Jones" <a1jrj at (no spam) hotmail.com> wrote in message
news:hc4pj8$3r6$1 at (no spam) adenine.netfront.net...

"Chappy" <petergregorychapman at (no spam) hotmail.com> wrote in message
news:6e0e8905-b565-4b38-b653-a8cbf40c4391 at (no spam) y28g2000prd.googlegroups.com...
Enigma 1562 - Same unused digits
New Scientist magazine, 9 September 2009,
By Richard England.

I chose two four-digit perfect squares (with
no leading zero) that contained eight
different digits. So did Harry, and so did
Tom.

Our six chosen squares were all different,
but the two digits that were unused were the
same for all three of us. If you knew what
those two digits were you would be able to
deduce with certainty what the six squares
that we chose were.

What were the largest and smallest of our
six squares?

Ciao,
Chappy

I tried to do this on paper, honest I did.

But deriving the 36 potential squares took an hour, and I couldnt see an
easy way of imposing the other constraints on them.

So in the end twas prolog to derive the basic lists and then an eyeball
of them to see where six unique squares existed as a list predicated on
the same two digits left over (1,7).

As requested:- 2809 ... 8649

HTH
JJ



You used the same digits

Enigma 1562 - Same unused digits
New Scientist magazine, 9 September 2009, By Richard England.

I chose two four-digit perfect squares (with no leading zero) that
contained eight different digits. So did Harry, and so did Tom.

Our six chosen squares were all different, but the two digits that were
unused were the same for all three of us. If you knew what those two
digits were you would be able to deduce with certainty what the six
squares that we chose were.

What were the largest and smallest of our six squares?

Ciao,
Chappy

Enigma 1562 - Same unused digits
New Scientist magazine, 9 September 2009,
By Richard England.

I chose two four-digit perfect squares (with
no leading zero) that contained eight
different digits. So did Harry, and so did
Tom.

Our six chosen squares were all different,
but the two digits that were unused were the
same for all three of us. If you knew what
those two digits were you would be able to
deduce with certainty what the six squares
that we chose were.

What were the largest and smallest of our
six squares?

Ciao,
Chappy

"Chappy" <petergregorychapman at (no spam) hotmail.com> wrote in message
news:6e0e8905-b565-4b38-b653-a8cbf40c4391 at (no spam) y28g2000prd.googlegroups.com...
Enigma 1562 - Same unused digits
New Scientist magazine, 9 September 2009,
By Richard England.

I chose two four-digit perfect squares (with
no leading zero) that contained eight
different digits. So did Harry, and so did
Tom.

Our six chosen squares were all different,
but the two digits that were unused were the
same for all three of us. If you knew what
those two digits were you would be able to
deduce with certainty what the six squares
that we chose were.

What were the largest and smallest of our
six squares?

Ciao,
Chappy

I tried to do this on paper, honest I did.

But deriving the 36 potential squares took an hour, and I couldnt see an
easy way of imposing the other constraints on them.

So in the end twas prolog to derive the basic lists and then an eyeball of
them to see where six unique squares existed as a list predicated on the
same two digits left over (1,7).

As requested:- 2809 ... 8649

HTH
JJ

John Jones wrote:
"Chappy" <petergregorychapman at (no spam) hotmail.com> wrote in message
news:6e0e8905-b565-4b38-b653-a8cbf40c4391 at (no spam) y28g2000prd.googlegroups.com...
Enigma 1562 - Same unused digits
New Scientist magazine, 9 September 2009,
By Richard England.

I chose two four-digit perfect squares (with
no leading zero) that contained eight
different digits. So did Harry, and so did
Tom.

Our six chosen squares were all different,
but the two digits that were unused were the
same for all three of us. If you knew what
those two digits were you would be able to
deduce with certainty what the six squares
that we chose were.

What were the largest and smallest of our
six squares?

Ciao,
Chappy

I tried to do this on paper, honest I did.

I admit to resorting to automation as well, but after the event ..

But deriving the 36 potential squares took an hour, and I couldnt see an
easy way of imposing the other constraints on them.

Well, any squares which are permutations of each other can be
eliminated, as that would make the "deduce with certainty" impossible.
That reduces the 36 to 23:

Set Square(s)
0124: 1024 2401
0126: 2601
0145: 5041
0189: 1089 9801
0234: 2304
0235: 3025
0247: 2704
0259: 9025
0289: 2809
0468: 6084
0469: 4096 9604
0567: 7056
1237: 3721
1246: 6241
1269: 1296 2916 9216
1279: 7921
1348: 3481
1369: 1369 1936
1458: 5184
1467: 1764 4761
1489: 1849
2349: 3249
2359: 5329
2467: 6724
3456: 4356
3679: 7396
4567: 5476
4689: 8649
5679: 7569

Keeping the digit sets sorted makes it easier to spot the disjoint ones,
and then you should end up with 17 pairs:

set1 sq1 set2 sq2 excludes
0145: 5041 + 3679: 7396 x 28
0234: 2304 + 5679: 7569 x 18
0235: 3025 + 1489: 1849 x 67
0235: 3025 + 4689: 8649 x 17
0259: 9025 + 1348: 3481 x 67
0289: 2809 + 3456: 4356 x 17
0289: 2809 + 4567: 5476 x 13
0468: 6084 + 1237: 3721 x 59
0468: 6084 + 1279: 7921 x 35
0468: 6084 + 2359: 5329 x 17
0567: 7056 + 1348: 3481 x 29
0567: 7056 + 1489: 1849 x 23
0567: 7056 + 2349: 3249 x 18
1237: 3721 + 4689: 8649 x 05
1279: 7921 + 3456: 4356 x 08
1348: 3481 + 5679: 7569 x 02
1458: 5184 + 3679: 7396 x 02

and only (1,7) occurs three times as the excluded pair.

It's still too much like hard work without a computer, though.

So in the end twas prolog to derive the basic lists and then an
eyeball of
them to see where six unique squares existed as a list predicated on the
same two digits left over (1,7).

As requested:- 2809 ... 8649

HTH
JJ

The following appear to meet these criteria:

Set Number Square Number Square Outstanding
1 53 2809 69 4761 3,5
2 78 6084 89 7921 3,5
3 82 6724 99 9801 3,5

Then the highest square is = 9801
and the smallest square is = 2809

On Tue, 27 Oct 2009 22:57:23 +0800, giovani wrote:

The following appear to meet these criteria:

Set Number Square Number Square Outstanding
1 53 2809 69 4761 3,5
2 78 6084 89 7921 3,5
3 82 6724 99 9801 3,5

Then the highest square is = 9801
and the smallest square is = 2809

33*33 = 1089. So the third person could have selected 1089 and 6724,
meaning that if 3 and 5 were the two digits not used, you wouldn't know
unambiguously which three pairs of squares were selected.