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5x5 **: The parity rule for children/beginners
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Posted on: Mon Apr 30, 2012 1:57 pm

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Joined: Fri May 13, 2011 6:51 pm
5x5 **: The parity rule for children/beginners
The parity rule:

If the sum of the numbers inside a cage is even we say that the cage is even; if odd, the cage is odd.

A cage of the type “x-” is even if x is even and odd if x is odd, for all possible combinations (for instance, “0-” or “4-” are always even; “7-” is always odd).

A cage (in the case of 2 cells) of the type “3:”, “5:”, “7:”, ... , is always even (a cage “x:” where x is odd).
A cage of the type ”2:”, “4:”, “6:”, ... , can be even or odd (a cage “x:” where x is even).

Reedited May 01,2012 to correct a mistake in the text (thanks to mparisi).

Last edited by clm on Tue May 01, 2012 8:29 am, edited 1 time in total.

Posted on: Tue May 01, 2012 3:23 am

Posts: 93
Joined: Sat May 14, 2011 1:08 am
Re: 5x5 **: The parity rule for children/beginners
clm wrote:
A cage of the type “3:”, “5:”, “7:”, ... , is always even (a cage “x:” where x is odd).

That's not true for cages with more than two cells. For example a three-cell 3: could be 3,1,1; 6,2,1, (both odd) or even 12,2,2 (even). Admittedly you're not going to find the last two cases in a 5x5 puzzle.

Posted on: Tue May 01, 2012 8:25 am

Posts: 698
Joined: Fri May 13, 2011 6:51 pm
Re: 5x5 **: The parity rule for children/beginners
mparisi wrote:
clm wrote:
A cage of the type “3:”, “5:”, “7:”, ... , is always even (a cage “x:” where x is odd).

That's not true for cages with more than two cells. For example a three-cell 3: could be 3,1,1; 6,2,1, (both odd) or even 12,2,2 (even). Admittedly you're not going to find the last two cases in a 5x5 puzzle.

Absolutely, you are right, thank you very much for your attention and correction, I'm sorry, it's a mistake because I was thinking in two cells. I will reedit the post and modify that in the examples.

Posted on: Tue May 01, 2012 11:55 am

Posts: 246
Location: Lisbon, Portugal
Joined: Sun Sep 18, 2011 5:40 pm
Re: 5x5 **: The parity rule for children/beginners
You are wrong, Clm: This post is not only for beginners, I'm not a beginner ( I think ) but I profited something: I did never perceived that a 2-cage x: with x odd is always even! ( ).

Thank you for this hint.

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Posted on: Tue May 01, 2012 7:32 pm

Posts: 698
Joined: Fri May 13, 2011 6:51 pm
Re: 5x5 **: The parity rule for children/beginners
jomapil wrote:
You are wrong, Clm: This post is not only for beginners, I'm not a beginner ( I think ) but I profited something: I did never perceived that a 2-cage x: with x odd is always even! ( ).

Thank you for this hint.

Last sunday when I was teaching an 8-years old girl on how to do this 5x5 “difficult”, I was explaining her how to avoid the “trial and error” and I observed that when talking about the parity she quickly got the concept, that gave me the idea of preparing the post. If the children, for instance, are trained with this rule, it will become familiar to them in most situations.

Of course the parity rule is useful to everybody in many puzzles, specially in the “subtractions only” puzzles, etc.; in the case of the multiplication cages by reducing the number of combinations to approximately the half, in other words, if that number of combinations is big there is logically a huge probability that half of the combinations be odd and the other half even; for instance, “453600x” has (in a 16x16, 6-cell cage) 45 different combinations (since none of them repeates a number more than 3 times all are valid combinations to spread over many different shapes of a 6-cell cage) and in this case 23 of them are even cages and 22 are odd cages. This big number (45) of combinations would obviously be bigger in a, let’s say, 25x25 puzzle .

In general, applying the parity rule to a wider area that “encircles” the cage under study with the “unknown combination” we can reduce much the possibilities.

By the way, jomapil, a suggestion for the next version of your useful “auxiliary program”: it would be interesting (for the purpose of the parity rule) to include at the end of each combination the sum of the operands.

Large division cages can be even or odd but the example for the 2-cell “x:” cage with x odd is a curiosity since the cages are always even.

Another curious case (which is valid up to the 16x16 size) for a 3-cell “x:” cage with x odd is that now, inversely, all combinations in the case of “5:”, “7:”, “9:”, “11:”, “13:” and “15:” are odd (in-line or L-shape, whichever applies); in the case of “3:” only the [2,2,12] (L-shape) is even and the other five ([1,1,3], [1,2,6], [1,3,9], [1,4,12], [1,5,15] are odd, whichever shape applies); and finally, the cage “1:” has only 5 even cages, the [2,2,4], [2,4,8], [2,6,12], [2,8,16] and [4,4,16] (whichever shape applies) (less than the 20% among the total number of 26 combinations for this result), being the other 21 odd cages: [2,3,6], [2,5,10], [2,7,14], [3,3,9], [3,4,12], [3,5,15] and the 15 of the type [n,n,1] (whichever shape applies).

Posted on: Tue May 01, 2012 8:35 pm

Posts: 246
Location: Lisbon, Portugal
Joined: Sun Sep 18, 2011 5:40 pm
Re: 5x5 **: The parity rule for children/beginners
clm wrote:
By the way, jomapil, a suggestion for the next version of your useful “auxiliary program”: it would be interesting (for the purpose of the parity rule) to include at the end of each combination the sum of the operands.

Good idea, Clm.
Jointly with other suggestions I will include that. Let's see, in the next month of May, if I defeat the inertia and edit the new version.

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