The New Sudoku Players' Forum

[Hammerite]

How odd that people should be so very against puzzles requiring "Trial and Error", as it is referred to by everybody here.

"Trial and Error" as the term is used here, appears to refer to the use of proof by contradiction, a.k.a. reductio ad absurdum, to verify an entry in a Su Doku puzzle. I do wonder why puzzles requiring this technique are looked down upon so by many users of this forum. Most feel, it seems, that for a puzzle to require this technique makes it in some way "inelegant". But why is this so? Used correctly, proof by contradiction is as logical as any other method of proof.

To me, the only Su Doku puzzles that are inelegant are the ones with more than one solution - they seem something of a faux pas (though with one of these, the solver might still take pleasure from demonstrating to himself that there exist multiple solutions, and if there are not so many that it tries his patience, identifying all of the solutions). If a Su Doku puzzle has a unique solution then I may take pleasure from proving to myself that it is so, by whatever means.

I now copy a statement that I saw made by another poster on this site (unfortunately I don't remember who made it or where I saw it). Many mathematical theorems can be proven using proof by contradiction, for example the fact that the square root of 2 is not the ratio of two integers. There are lots of others, many of them very elegant proofs (my favourite proof is the proof that the set of real numbers is uncountably infinite, which is a proof by contradiction). Say I have a Su Doku puzzle partially filled in and I see there are two squares within a box, row or column where some symbol could go. If I show that putting it in one box leads to a chain of logical deductions from which a contradiction follows, then the alternative must be correct. How elegant! Of course if one wishes to show that the Su Doku has only one solution, then if one makes the correct choice first time, one must go back and check that the alternative does lead to a contradiction; otherwise, we shuold be unsure of the uniqueness of the solution we have found...

Final point. The choice of "Trial and Error" to describe the technique is a bad one; it is a misnomer. Tral and Error would surely imply that the solver blindly enters numbers in the grid and goes on to see whether or not he finds a solution. This is not what proof by contradiction is about; it's about making a very specific hypothesis and showing that that hypothesis cannot be sustained as taken to its logical conclusion, it produces absurdities. It's clear-cut and above all logical, which actually attempting to solve by guesswork would never be.

[angusj]

...

While I wholeheartedly agree with everything you've said from a mathematical perspective, I do question the value of "proof by contradiction" from a player's perspective.

My main objection with what we non-mathematicians call trial-and-error is that it generally requires a computer to help work through the many possibilities. Consequently, the puzzle becomes a challenge for programmers but not players.

Edited to make what I wrote intelligible.

[zinckiwi]

I can't speak for my father (Wayne), but to me, the context of the word "logical" in the sense that Pappocom uses it is this:

When placing a number, the reason for doing so must a) be provable, and b) eliminate the possibility of any other number appearing in the cell.

[MCC]

How odd that people should be so very against puzzles requiring "Trial and Error", as it is referred to by everybody here.

"Trial and Error" as the term is used here, appears to refer to the use of proof by contradiction, a.k.a. reductio ad absurdum, to verify an entry in a Su Doku puzzle. I do wonder why puzzles requiring this technique are looked down upon so by many users of this forum. Most feel, it seems, that for a puzzle to require this technique makes it in some way "inelegant". But why is this so? Used correctly, proof by contradiction is as logical as any other method of proof.

.

To me Trial and Error is inelegant and I'll use a bit of drama to show you why.

Think of the puzzle as a game of Life and Death.
Starting at the begining, you solve the puzzle using logical steps, one step at a time until you reach the end and finished the puzzle without getting killed.

You solve the puzzle by either placing a big number or eliminating candidates.
Every time you place a big number a gunman appears to protect that number.
You carry on at each step placing a big number or eliminating candidates.
If you don't retrace your steps you'll reach the end, complete the puzzle and live but if you retrace your steps you will meet a gunman and you will die.

Trial and Error is about retracing your steps and having a second go at completing the puzzle. That is what I find inelegant about it.

[Hammerite]

I can't speak for my father (Wayne), but to me, the context of the word "logical" in the sense that Pappocom uses it is this:

When placing a number, the reason for doing so must a) be provable, and b) eliminate the possibility of any other number appearing in the cell.

Proof by contradiction has this property.

[lunababy_moonchild]

Ok, some people like to use T&E and some people don't. The people who like it use it and the people who don't, don't.

Everybody has their reasons and it doesn't mean that anybody is wrong, it's just the way they play the game.

It's a game and it doesn't matter how you play it, it's designed to amuse.

Luna *I don't use T&E myself but don't condemn those who do*

[RFB]

The problem is with non-Pappocom puzzles that may require bifurcation.
You never know for sure whether you have missed an elegant logical deduction that would allow you to solve the puzzle or if you are wasting your time looking for a non existent (non-T&E) technique.

[Hammerite]

Think of the puzzle as a game of Life and Death.
Starting at the begining, you solve the puzzle using logical steps, one step at a time until you reach the end and finished the puzzle without getting killed.

You solve the puzzle by either placing a big number or eliminating candidates.
Every time you place a big number a gunman appears to protect that number.
You carry on at each step placing a big number or eliminating candidates.
If you don't retrace your steps you'll reach the end, complete the puzzle and live but if you retrace your steps you will meet a gunman and you will die.

Trial and Error is about retracing your steps and having a second go at completing the puzzle. That is what I find inelegant about it.

But you don't need to "write in a big number" to use the technique...

Or to put it another way, forming a hypothesis and considering the consequences of that hypothesis does not require you to commit yourself to it. If one deduces an absurdity from a hypothesis, then one knows that the hypothesis is false and that its negation is true. If this gives you the remaining information needed to make an entry in the grid, then good. Then, and only then, have you "written in a big number", in your "Life-or-Death situation". Until then, all you are considering is two hypotheses.

It is the mark of an educated mind to be able to entertain a thought without accepting it.

[MCC]

But you don't need to "write in a big number" to use the technique...

Or to put it another way, forming a hypothesis and considering the consequences of that hypothesis does not require you to commit yourself to it. If one deduces an absurdity from a hypothesis, then one knows that the hypothesis is false and that its negation is true. If this gives you the remaining information needed to make an entry in the grid, then good. Then, and only then, have you "written in a big number", in your "Life-or-Death situation". Until then, all you are considering is two hypotheses.

It is the mark of an educated mind to be able to entertain a thought without accepting it.

How to reply to this! How to reply to this!
Woe is me. Am I'm being Hammered.

"But you don't need to "write in a big number" to use the technique..."

I think you do. Whether you write a number down or mentally assign a number to a cell, you've effectively placed a number in that cell.


In the course of solving a puzzle a period of elimination occurs before a number is placed in a cell, i.e. elimination precedes number placement.
Whereas in T&E a number is assigned ( written or mentally) to a cell then elimination occurs, i.e. number placement precedes elimination. Elimination here is used to justify the placement of the number in the first place.

I agree that T&E is a valid logical process, it's just that I find it an inelegant process.

Lewis Carroll

Why, sometimes I've believed as many as six impossible things before breakfast.

In "Through the Looking-Glass"(1872)ch.5.

[Karyobin]

Hammerite, for God's sake please leave off with all the maths jargon! As a maths teacher myself I understand every last ikkle bit and I won't conjecture that anybody else reading your missives understands less than I, but every line seems to miss the point, which I would chose to define as follows:

Trial & error (improvement) may be mathematically sound but it is simply not as satisfying a technique as intuitive logic.

All you need for trial and improvement is patience. X-wings, hidden triplets, swordfish and the like require discovery. Surely anyone would rather discover than trudge.

P.s. I quite liked the death-or-dishonour analogy. T & I does seem to contain the "I'm prepared to go wrong to find out what's right" principle. Well I'm not.

[martinji]

I think the reason many people prefer to avoid "T&E" is it doesn't feel like you truly know WHY the number you're putting in is right. You know it logically has to be that number as others lead to a contradiction, but at some level you don't understand why that's the case; whereas with logical deduction you "know" why that's the right number to be filling in.

Personally however I embrace T&E or hypothesis testing or proof by contradiction or whatever you want to call it. There's still a challenge in reducing the puzzle logically to a stage where a quick T&E can be used.

And tbh I sometimes feel that the "logical deduction" used is mainly the algorithmic implementation of learned techniques, without any true original thought - and can become merely a test of how systematic and observant you are in noticing when a rule can be employed.

[Hammerite]

I guess what I don't like is when people say things like, "There's no way to solve this Su Doku using only logic. You need to use trial and error". (Something that the operator of this site is guilty of!) If a puzzle has a unique solution, then of course it is possible to use logic to arrive at that solution.

[Karyobin]

And tbh I sometimes feel that the "logical deduction" used is mainly the algorithmic implementation of learned techniques, without any true original thought - and can become merely a test of how systematic and observant you are in noticing when a rule can be employed.

Odd criticism. Whereas I agree wholeheartedly with the general cut of what you're saying, what's wrong with "algorithmic implementation of learned teachniques"? That's how we go through life: multiplication tables; driving a car; reading a book, the reason we bother to learn anything is so that we don't have to go back to first principles every time we want to do something. Imagine proving to yourself why 6x7=42 every time you need that particular fact. Down that road lies madness.

[martinji]

And tbh I sometimes feel that the "logical deduction" used is mainly the algorithmic implementation of learned techniques, without any true original thought - and can become merely a test of how systematic and observant you are in noticing when a rule can be employed.

Odd criticism. Whereas I agree wholeheartedly with the general cut of what you're saying, what's wrong with "algorithmic implementation of learned teachniques"? That's how we go through life: multiplication tables; driving a car; reading a book, the reason we bother to learn anything is so that we don't have to go back to first principles every time we want to do something. Imagine proving to yourself why 6x7=42 every time you need that particular fact. Down that road lies madness.

I'm not saying there's anything wrong with the "algorithmic implementation of learned techniques".
But what I'm pointing out is that some people castigate the use of T&E because it's "unthinking", yet I think the implementation of standard techniques can be argued to be equally "unthinking".

[Scozzer]

Hello Hammerite,
I found your explanation of Proof by Condradiction interesting and encouraging. This week the LA Times started a daily Sudoku puzzle and for the last two mornings their puzzles have not been solvable without "Proof by Contradiction". I found that irritating until I read your explabation.
However I must add that I suspect most people will prefer puzzles without "Trial by contradiction" unless they could be informed at some point that they have gone as far as they can with simple process logic alone. This obviously is not possible, so despite your excellent explanation we are all doomed to waste a lot of time figuring out that there is no logical progression other than a simple flip of the metaphorical coin to choose between two or more options. In the meatime we are all going to waste a lot of time trying desperately to find the elusive answer that we still assume exists but actually doesn't.
For that reason I intend to use www.dailysudoku, where all puzzles are pure and virginal, rather than the LA Times, .... unless of course you can persuade me othewise.