help with difficult sudoku

Advanced methods and approaches for solving Sudoku puzzles
PaulIQ164 wrote:Well, I've studied it a bit, and I remain unconvinced. Seems to me you're still trying two possible sets of places for the sixes, and rejecting the one that leads to an error.

You don't quite understand it. "Coloring" has a long history in recreational mathematics that pre-dates the Sudoku.

One of the most well known puzzles and simplest is this one:

You are given an 8x8 grid of squares with two opposite corners removed and 31 dominos, each of which will exactly cover two squares. You are challenged to cover the board completely with the dominos. Though there are many ways to solve the problem, including brute force search, the simplest is to color the board like a checkerboard with light and dark squares. The board will have 30 squares of one color, 32 to of the other. Since a domino will always cover exactly one of each, the task is impossible. It doesn't matter which squares are colored dark or light, only that they follow the pattern.

When coloring, either the board or the Sudoku grid, nothing is "assigned" to the light or dark cells, they are merely colored according to a pattern. When the coloring is in place, a direct conclusion can be made. No hypothesis is required. The solution becomes *visable*. We can *see* that there are more squares of one color than the other. We can *see* that two cells in the same row, column or box have the same color or polarity.

And by the way:

there are two 6's with the same sign in box 6 -- they both can't be 6, so they both must NOT be 6.

PaulIQ164 wrote:The two 6s in box 6 have the different signs. The only unit that has two sixes of the same sign that I can see is column 7. Is this what was meant.

Thanks. I fixed it. "New Math" syndrome.:

...
From the three you then use one
To make ten ones...
(And you know why four plus minus one
Plus ten is fourteen minus one?
And so you've got thirteen tens
And you take away seven,
And that leaves five...

Well, six actually...
But the idea is the important thing!
...
tso

Posts: 798
Joined: 22 June 2005

Hm... maybe it's not T&E then. But I'll find some other reason not to like it, mark my words!
PaulIQ164

Posts: 533
Joined: 16 July 2005

From the three you then use one
To make ten ones...
(And you know why four plus minus one
Plus ten is fourteen minus one?
And so you've got thirteen tens
And you take away seven,
And that leaves five...

Well, six actually...

What the hell is all this?!
Karyobin

Posts: 396
Joined: 18 June 2005

Karyobin wrote:What the hell is all this?!

Lyrics from NEW MATH by TOM LEHRER -- a song about a new (in 1959) method of teaching math that stressed method over results. New Math Syndrome is the propensity to do things in a more clever way -- while getting the wrong answer.
tso

Posts: 798
Joined: 22 June 2005

I had a Geometry Teacher in High School who 'proved' that 2 = 1 by starting with 1 = 1 and working both sides of the equal sign with substitutions. He left it on the board all hour and challenged anyone to refute it. None of us could find the mistake and it's always haunted me.
The Central Scrutinizer

Posts: 22
Joined: 10 August 2005

When he divided by (a-b), he divided by zero.

Let a=1
b=1

To wit:

a = b
a^2 = ab
a^2 - b^2 == ab - b^2
(a+b)(a-b) == b(a-b)
(a+b)(a-b)/(a-b) == b(a-b)/(a-b)

...this is the step that is illegal...

(a+b) == b
1 + 1 == 1
2 == 1

QED

EDITED: see below for explanation.
Last edited by Dusty Chalk on Tue Aug 16, 2005 12:02 pm, edited 1 time in total.
Dusty Chalk

Posts: 21
Joined: 15 August 2005

Cheers tso. I knew about 'New Math' of course, I just thought you'd actually cracked up for a moment there.

Yeah, I often put that 'proof' on the board. Dividing by zero is never a good idea.

...this is the step that is illegal...

And I'm sorry to be horribly pedantic, but I'd have written that statement after (a+b)(a-b)/(a-b) == b(a-b)/(a-b), not before. There's nothing illegal about doing that, but great problems arise with the simplification at the next stage, which should of course read 0 == 0.

And there's a lovely illustration by Ian Stewart of how New Math damaged maths teaching in The Science of Discworld Part III. He compares it to building a house, top stuff.
Karyobin

Posts: 396
Joined: 18 June 2005

Karyobin wrote:
...this is the step that is illegal...

And I'm sorry to be horribly pedantic, but I'd have written that statement after (a+b)(a-b)/(a-b) == b(a-b)/(a-b), not before. There's nothing illegal about doing that, but great problems arise with the simplification at the next stage, which should of course read 0 == 0.
No, you're right. I originally didn't have the denominator, and went straight to the simplification.
Dusty Chalk

Posts: 21
Joined: 15 August 2005

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