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The addition of all numbers of a row or column
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Posted on: Tue Jun 07, 2011 10:36 pm

Posts: 684
Joined: Fri May 13, 2011 6:51 pm
The addition of all numbers of a row or column
Basic strategies (perhaps trivial but may be useful to someone).
First rule: The addition of all numbers of a row or column or of a group of them.

The addition of all numbers (regardless of its order) in any row or column is a fix number (because all must be different this total equals the addition of all terms of an arithmetic progression of ratio 1). That total is, for the 4x4, 5x5, 6x6, 7x7, 8x8, 9x9, 10x10, 12x12 and 17x17, respectively equal to: 10, 15, 21, 28, 36, 45, 55, 78 and 153. Puzzles “from 0” behave like decreasing the size of the puzzle by one so, for an 8x8 (from 0 to 7), the addition will be 28 (and not 36) since the zero does not modify the total of the other 7 numbers.

Consequently, the addition of a “rectangle” (group of rows or columns), for instance of two adjacent columns will be the double of a line, for instance, 2 x 45 = 90, in a 9x9. This procedure is useful to obtain the value of individual cells that “enter” the “rectangle” or that are an “appendix” of a given “rectangle” (or to obtain the value of the addition of two or more cells or of two or more small subareas, etc), simply by getting the difference between that fix total and the known part. This tool can aid in eliminating some combinations in subareas with the multiplication sign, for instance. It is specially useful in the 5x5’s and 6x6’s.

Posted on: Wed Jun 08, 2011 1:15 am

Posts: 175
Joined: Fri May 13, 2011 2:11 am
Re: The addition of all numbers of a row or column
Though this is a bit basic as a strategy, product of a row is useful as well; That is, a single row in any puzzle multiplies to puzzle size factorial, and for multiple rows you raise the product of 1 row to however many rows you have.

E.g. yesterday's 9x9 became much easier is you did the product of the 3 right most and leftmost columns.

Posted on: Wed Jun 08, 2011 8:08 pm

Posts: 684
Joined: Fri May 13, 2011 6:51 pm
Re: The addition of all numbers of a row or column
starling wrote:
Though this is a bit basic as a strategy, product of a row is useful as well; That is, a single row in any puzzle multiplies to puzzle size factorial, and for multiple rows you raise the product of 1 row to however many rows you have.

E.g. yesterday's 9x9 became much easier is you did the product of the 3 right most and leftmost columns.

Exactly and avery good example indeed for the concept of the multiplication of all the elements of a line. In the Jun 07 9x9, the only possible combinations for 26+ was the 77831 (for a product of 1176) and for 19+ the 75241 (for a product of 280), then inmediately situating all 7’s in the full 9x9. The n! (n factorial) is provided by all scientific calculators and then it’s easy to use (no PC is needed). Another easier example is the singleop (multiplications only) saturday Jun 04 8x8 medium: when multiplying all products of the two first rows and dividing by (8! x 8!) we get 15 for the product of c2 and c7, then c7 must be the 5 and c2 must be the 3 (now, the cage 63x, in the position b2, b3 and c2, has the only posible combination of 733 and not the "791").

Posted on: Wed Jun 08, 2011 11:05 pm

Posts: 36
Joined: Fri May 13, 2011 10:23 am
Re: The addition of all numbers of a row or column
Clever trick, never thought of it.... I will add it to my bag 'o tricks

Posted on: Sun Jun 12, 2011 4:09 pm

Posts: 8
Joined: Fri May 13, 2011 1:15 am
Re: The addition of all numbers of a row or column
I've modified the row/column product for easier use by pen and paper. Perhaps this has already been discovered. Instead of thinking the product of a row/column in a 9x9 as 9! for example, I think of it as 2^7 * 3^4 * 5 * 7. I prime factor each of the multiplication cages and cancel. For example, I see a 144x cage and think (or write) 2^4 * 3^2 and cancel four 2's and two 3's from the overall product.
What can make this more powerful in some cases is to focus on 2's, 3's, 5's, 7's separately. I believe it's common logic to focus on the 5's and 7's (extracting info whether or not a cage is a multiple of 5 or 7), but not on the powers of 2's and 3's. I once got out of a stall by focusing on the power of 3's. There were 3 or 4 multiplication cages contained in two columns in a 9x9. They didn't cover enough squares to make use of the fact that the double column product is twice 9 factorial. Upon prime factoring, I noticed the products of these cages had 7 factors of 3, and since two columns contain 8 factors of 3, I could eliminate 9 as a possibility outside the cages (within those columns).

Posted on: Sun Jun 12, 2011 10:39 pm

Posts: 684
Joined: Fri May 13, 2011 6:51 pm
Re: The addition of all numbers of a row or column
larryb33 wrote:

...What can make this more powerful in some cases is to focus on 2's, 3's, 5's, 7's separately. I believe it's common logic to focus on the 5's and 7's (extracting info whether or not a cage is a multiple of 5 or 7), but not on the powers of 2's and 3's....

The procedure is very good, however if the cages extend over 4 columns, for instance, you would have 2 ^ 28 and 3 ^ 16. I agree with the nuclear idea, and you are right, probably most players use 5’s and 7’s as “keys” for solving 9x9’s (and 11’s for the 12x12’s, etc.). The 5’s and 7’s give much information (only three of them in three columns, etc…, no one in areas outside the involved cages, or when using the “X-ray”, etc.) but, as you say, sometimes we pay less attention to the other prime factors, like 2’s and 3’s, and even the 1. Sometimes the 1’s are the key to advance in the solution of the 9x9 (if there are big products that can not contain the 1). Sometimes all tools must be combined and used together to arrive to the solution.

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