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From: Leroy Quet on 19 Jun 2010 08:47
Leroy Quet wrote:
/ Let r = the sum of the row products, and let c be the sum of the
/ column products.
/ Let q = |c-r|.
/ The solutions where 1 through 9 are all row products (or column
/ products) maximizes q, obviously.
/ Can anyone find a solution that gives the MINIMUM possible q,
whatever
/ that q is?

The minimum q is 1. It is possible to get the row products and column
products to sum to 100 and 101, respectively or irrespectively. But is
the puzzle itself solvable with a sum of row products of 100 and a sum
of column products of 101 (or vice versa)?

Thanks,
Leroy Quet

From: Justin Leck on 9 Jul 2010 11:59
"Carl G." wrote:
> Next Challenge: Is there a solution with the products 1 to 9 all in the
> row-products?

I was surprised to find there are 36 solutions with the row-products in numerical order. Here's an example:
010101010 1
010101011 2
000101010 3
001101010 4
011111010 5
011010001 6
100000001 7
100001101 8
100010111 9

122211111
847060452


"Leroy Quet" wrote:
> Leroy Quet wrote:
> / Let r = the sum of the row products, and let c be the sum of the
> / column products.
> / Let q = |c-r|.
> / The solutions where 1 through 9 are all row products (or column
> / products) maximizes q, obviously.
> / Can anyone find a solution that gives the MINIMUM possible q,
> whatever
> / that q is?
>
> The minimum q is 1. It is possible to get the row products and column
> products to sum to 100 and 101, respectively or irrespectively. But is
> the puzzle itself solvable with a sum of row products of 100 and a sum
> of column products of 101 (or vice versa)?

There are many solutions. Below is an example with the added constraint that if you convert each row and column to a 9 bit binary
value, the sum of the rows equals the sum of the columns.

000000000 9
000000011 14
000000100 12
000111111 18
001111100 20
000001110 15
000010101 4
010000010 5
100010101 3

12 21
876467012

Rows: 9+14+12+18+20+15+4+5+3=100
Cols: 8+7+16+24+6+27+10+1+2=101

Converting binary to decimal:
Rows: 0+3+4+63+124+14+21+130+277=636
Cols: 1+2+16+48+53+56+125+170+165=636


Justin


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