Mathmaticians apply within

[zaphod]

booyah!

[neorichieb1971]

[zaphod]

PS: Figs are 10 cents.

[Minzoku]

Here you go

A mother and father have six sons and each son has one sister. How many people are in that family?

r = s
r² = rs [multiply both sides by r]
r²-s² = rs-s² [subtract s² from both sides]
(r+s)(r-s) = s(r-s) [factor]
r + s = s [divide out r-s]
2s = s [substitute r]
2 = 1 [DURP?!]

1
1 1
2 1
1 2 1 1
3 1 1 2
1 3 2 1 1 2
3 1 1 3 2 2
What comes next? [actually not as math-related as you think]

[shiftace]

And now, knowing that all the information is there, we can then take a stab at solving the problem logically. Without assuming all the information is present, we can't get anywhere.

...

This is the tricky part. We know this, because if something else determined cost (such as number of letters on a logarythmic scale of some sort, and or diffrent weighing of individual letters.) we wouldn't have enough information to deduce the answer. But this contradicts the presupposition implied by the question that there is a unique and reachable answer! so we are LOGICALLY correct in our deductions!

Which wonderfully demonstrates that people tend to find exactly what they set out to look for. If you think that the statement of a question generally implies a logical answer, then what are your feelings, on a scale of 1 to 10, about inevitability? Does the set of all sets that do not contain themselves contain itself? Granted, though, in the apples & bananas problem, the person asking it did have a definite answer in mind, and I'm just being pedantic for no good reason.

A mother and father have six sons and each son has one sister. How many people are in that family?

Nine.

r = s
r² = rs [multiply both sides by r]
r²-s² = rs-s² [subtract s² from both sides]
(r+s)(r-s) = s(r-s) [factor]
r + s = s [divide out r-s]
2s = s [substitute r]
2 = 1 [DURP?!]

You can't divide by r-s since it's zero.

I think I saw the sequence problem once... uh, I don't know the pattern. Edit: Looked it up. That's horrible (in a good way).

[zaphod]

Which wonderfully demonstrates that people tend to find exactly what they set out to look for. If you think that the statement of a question generally implies a logical answer, then what are your feelings, on a scale of 1 to 10, about inevitability? Does the set of all sets that do not contain themselves contain itself? Granted, though, in the apples & bananas problem, the person asking it did have a definite answer in mind, and I'm just being pedantic for no good reason.

Actually quite often you must assume the answer exists to solve the problem. it's a classic problem solving technique. It is appropriate in this instance because that is clearly how it was presented. Real life problems do not always have all the information, or always have unique solutions. but the context this contrived problem was presented in made the presumption appropriate.

I'll give you a better example of the principle.

Lets say you have a crossword puzzle. and somewhere in this puzzle are 20 circled letters which you must unscramble.

Now you find a clue that seems to have two answers that fit. HOwever, one of the possibilities leaves two working answers for another clue, but the other possibility only leaves one. Now how do we know which one sis right? if we pick the wrong ones the unscramble puzzle will be unsolvable. SO how do we know which one. We must assume there is on eunique solution to the puzzle. Knowing that there is only one valid solution, we can then deduce that the answer that allows for only one answer for the other word intersecting it, because that one is part of a unique solution, whearas the other one was part of two different solutions, and therefore wrong.

This is done INTENTIONALLY ALL THE TIME in double crosstics or something like that.

[shiftace]

Actually quite often you must assume the answer exists to solve the problem.

I agree that if a problem is not completely specified, and one desires to have a unique solution, then one must supply the missing specifications oneself. Furthermore, I agree that in the apples & bananas problem, assuming that the problem as stated includes enough information to be solved, and that an answer with a concise explanation based on letter frequencies is desirable, is reasonable. And all this, to my knowledge, is just repetition.

I'll give you a better example of the principle.

Lets say you have a crossword puzzle. and somewhere in this puzzle are 20 circled letters which you must unscramble.

Now you find a clue that seems to have two answers that fit. HOwever, one of the possibilities leaves two working answers for another clue, but the other possibility only leaves one. Now how do we know which one sis right? if we pick the wrong ones the unscramble puzzle will be unsolvable. SO how do we know which one. We must assume there is on eunique solution to the puzzle. Knowing that there is only one valid solution, we can then deduce that the answer that allows for only one answer for the other word intersecting it, because that one is part of a unique solution, whearas the other one was part of two different solutions, and therefore wrong.

Something seems wrong to me in this example. Let me use some notation like so: Let {A,B} be candidate answers for the first clue, {C,D,E} be cross-clues, and F be a valid solution to the unscrambling puzzle. I think you have said something that I can reasonably represent by A -> C and B -> (D or E).

Now, one has to choose between A and B, and you say that A is automatically the correct choice, because B leads to an unresolvable cross-clue. Why couldn't there be another cross-clue G, so that D -> G while C and E don't logically imply anything? Or, why couldn't the puzzle writer make E -> F, while C and D don't imply anything?

More concisely: One knows which of {A,B} to pick by trying both of them and seeing if the unscramble puzzle is solvable. Maybe A is the more appealing one to test first, since there's less potential backtracking, but I don't see why it has to be right. Or have I misread your example?

[zaphod]

With a normal crossword, the solution doesn't have to be unque. it just has to fit every clue. However, that i susually considere d an error in the crossword design.

But we added an extra component to the puzzle, which requires the intended solution be the one used.

For this new puzzle, you have to find the intended answer.

CLue a has two fitting answeers.

clue B has THREE answers. two of them will fit with the second answer for clue A, and the other answer only matches the first answer.

Usually when you se such an event, your first reaction is to go "wttf? multiple answers work! the person who made the puzzle MESSED UP! It's unsolvable!!!"

But by assuming a unique solution does exist, you know which answer is correct for clue A. it's the one that leaves clue b with one unique solution! Because if te other answer was corect for clue A, then there wouldn't be unique solution, and the puzzle would be unsolvable because the scrambled phrase would be wrong.

I repeat puzzle authors do this sorta stuff on purpose.

And the chain of alternate answers coud lpossibly be MUCH longer. It coudl be something like.

okay this clue has two working answers, lets try this one.. and this one fits, and this one fits, and hmm two answers for this one now, this works, wtf? Why can't i solve this clue?

By applying the presumption, we know as soon as we see a second set of dual answers, we know our first answer was slected wrong, and to go back and change it.

[shiftace]

But by assuming a unique solution does exist, you know which answer is correct for clue A. it's the one that leaves clue b with one unique solution! Because if te other answer was corect for clue A, then there wouldn't be unique solution, and the puzzle would be unsolvable because the scrambled phrase would be wrong.

But wouldn't it be reasonable, or at least admissible, for a puzzle author to require the use of clue B and one chain of its consequences to get a phrase that can be unscrambled? There can still be a unique solution; it's just that the unique solution is to the union of the crossword and the scrambling puzzles, rather than to the crossword itself -- the crossword portion has multiple answers, leading to multiple unscrambling puzzles, but only one unscrambling puzzle is valid. Is that considered unsporting or something? I won't pretend to be a crossword hobbyist, it just seems like the uniqueness assumption only works here if the puzzle authors are playing nice.

By applying the presumption, we know as soon as we see a second set of dual answers, we know our first answer was slected wrong, and to go back and change it.

It seems, IMHO, like an artificial way to limit difficulty, not a demonstration of a useful logical principle.

[DarkWolf7]

1
1 1
2 1
1 2 1 1
3 1 1 2
1 3 2 1 1 2
3 1 1 3 2 2
What comes next? [actually not as math-related as you think]

2 3 2 1 2 2
4 2 1 3 1 1
1 4 1 2 3 1 1 3
4 1 1 4 1 2 2 3
2 4 3 1 2 2 1 3
.
.
.

[Minzoku]

^ I admit I'm curious to see how long that series can go before it becomes meaningless [confusion about what to count] though I think I have better things to do!

[professor ganson]

As for mathematics being a "perfect science," I don't think it's normally called a science, and it certainly isn't perfect (buzzwords: incompleteness, undecidability, Godel, Chaitin).

Nicely put, shitface (sorry to call you that). Back in antiquity geometry (as standardized by Euclid) was commonly thought to be an exemplary science, but today Euclid's postulates or axioms are mostly thought to be inadequate, largely because of the problematic fifth postulate (sometimes called the parallel postulate). But have we really settled on an alternative at this point? Last I heard, the foundations of geometry were still very shaky.

[zaphod]

But wouldn't it be reasonable, or at least admissible, for a puzzle author to require the use of clue B and one chain of its consequences to get a phrase that can be unscrambled? There can still be a unique solution; it's just that the unique solution is to the union of the crossword and the scrambling puzzles, rather than to the crossword itself -- the crossword portion has multiple answers, leading to multiple unscrambling puzzles, but only one unscrambling puzzle is valid. Is that considered unsporting or something? I won't pretend to be a crossword hobbyist, it just seems like the uniqueness assumption only works here if the puzzle authors are playing nice.

Yes, that would be considered an error in a crossword. You aren't supposed to be able to put in any alternate answes and still complete the puzzle.

It is rather common for multiple answers to exist to any one clue, but one of them causes an unsolvable clue later on. This is what you call a difficult crossword. But a clue where answeing one clue wrong also enables you to answer OTHER clues wrong as well before coming to the unsolvable clue, that is an extremely difficult crossword. But most of the time, when such an event happens, you will realise that an alternate solution could exist, and working backwards fom the unsolvable clue can let you retrace your steps.

Double crosstics are an even more annoying case, because you can go all the way through it with a bad answer right to the end, and not know it uintil then. And you won't know where the error is! as i said belie by applying this principle, you can stop earlier. Trial and error is NEVER required to solve one. If so, then it is not a puzzle. Therefore this principle must be followed in it's design.

SOme puzzles ARE trial and error based (the turn out the light puzzle is one of them) but not crosswords oranything based on them.

[shiftace]

Yes, that would be considered an error in a crossword. You aren't supposed to be able to put in any alternate answes and still complete the puzzle.

It is rather common for multiple answers to exist to any one clue, but one of them causes an unsolvable clue later on. [etc.]

But, in this example, I don't think it's right to call the crossword a whole puzzle any more, since there is an additional constraint. The crossword is just part of the puzzle, and it doesn't make sense to arbitrarily require uniqueness before the puzzle is finished. In any event, I can't see how allowing ambiguity in a crossword to be resolved afterwards, possibly requiring backtracking into the crossword, makes the puzzle degenerate or impossible, just annoying.

[About double-crostics:] Trial and error is NEVER required to solve one. If so, then it is not a puzzle.

Then what is it?

Back in antiquity geometry (as standardized by Euclid) was commonly thought to be an exemplary science, but today Euclid's postulates or axioms are mostly thought to be inadequate, largely because of the problematic fifth postulate (sometimes called the parallel postulate). But have we really settled on an alternative at this point? Last I heard, the foundations of geometry were still very shaky.

All I really know is that rejecting or replacing the 5th postulate leads to a bunch of useful geometries (spherical, hyperbolic, projective... I guess this heads off towards topology in a hurry... and I emphasize "guess"), and they mostly act like Euclidean geometry as long as you stay away from infinities and wraparound. Then again, real mathematicians don't care if something's useful or not. I think it's pretty well accepted that these non-Euclidean geometries are interesting, but I don't think physicists have decided which one is most like the universe.

[zaphod]

i must apologise, i had the puzzle type I mentioned wrong.

it's cross sums.

here's an example.
AS in croswords, it is possible for a section of the puzzle to get isolated from the rest of the puzzle,a nd have it's solution not affect the rest of the puzzle.

Every cross sum has a single, uniqu, unambiguous solution.

And yet sometimes you rin into a case while solving a puzzle were it seems more then one solution is possible for an area.

A cross sum works liek this.

each square is filled with either a black, or a digit form 1 thru 9. Digits do not repeat in a single across or down number. you are given only the sum of the digits of each down and each across number.

Now when a single number is the only thing connecting one oart of the puzzle to the rest of it, then that number divides the puzzle into two sections.

Often what will happen is a different number which is correctly known will intersect the connecting number. so now we have a filled in digit partitioning off the puzzle into a small section and a large section.

Now the only part of the small section that affets the large section is the connecting number. if you have the connecting number right, it is suffcient knowledge to deduce the small section. If that were not the case the puzle wwould be unsolvable. So our principle telsl us that knowing the connecting number alows you to solve the smaller section.

Now since we know the sums of the digits, we can write out all possible combinations of digits that can fit in that connecting number. We can then test each one of them, and see if they allow an unambiguous solution of the small area. We know that any arrangement that allows you to plug more than one solution into the small isolated area is automatically wrong, and only numbers that allow a single solution to the area are right. And sometimes this is ALL you have to go on to finish that area, which will give you the missing information you nedot complete the larer area.

If we did not assume the existence of the unambiguous solution, there is no way to logically derive it.

If it is possible to solve an isolated area of a cross sum differently without affecting the rest of the puzzle that is an error in the puzzle.

and, now that i think about it, this applies to crosswords as well. If you can find a clue that appears to have two answers that fit all cross clues, it means you've answered an earlier clue incorrectly, not that there is an error in the puzzle! And again, the knowledge of a single corect solution allows you to make that deduction, even before coming to the unsolvable clue that your incorrect answer caused. ANd, even if the puzzle can still be filled in this way completely, you still know yuo are wrong, and can work at deducing the intended solution despite the apparent error in the design. But te knowledge that it exists is the only thing that allwos you to decude it. Otherwise, any of them could be right.

[shiftace]

Ok, cross sums. Nice, I've never seen these before.

Now the only part of the small section that affets the large section is the connecting number. if you have the connecting number right, it is suffcient knowledge to deduce the small section. If that were not the case the puzle wwould be unsolvable. So our principle telsl us that knowing the connecting number alows you to solve the smaller section.

Ok, I can accept this form. If a solution uses a connector value that leads to ambiguity on one side, that does sound like a deficient puzzle. My issue with this uniqueness business was mostly in how you worded it earlier -- I thought that what you started out with was equivalent to, "if one can look ahead and see the solution branch at least twice, then one is looking at the wrong solution," which is quite broad and absurd.

Now since we know the sums of the digits, we can write out all possible combinations of digits that can fit in that connecting number. We can then test each one of them, and see if they allow an unambiguous solution of the small area.

Trial and error is NEVER required to solve one.

I see.

[zaphod]

My poit is that you dont' have to work it all the way to the point that you get stuck before knowin gthat you made a mistake. the instant you find ambiguity that doesnt' aggect the rest of the puzzle, yo uknow that you have made an error.

However, knowledge that the unambiguous solution exists is required for that logical leap to be made. If this were'nt so you'd just have to keep going until your found the unsolvable part to know that you were incorrect. And it may in fact be possible to fil the whoel puzzle this way. but you'd still be wrong. and the reason you'd be wrong is because your solution is not unambiguous. The existence of an ambiguos solution doesn't prove the non-existence of the unambiguos solution. And knowing that the solution is there allows you to find it, even when alternate unambiguous solutions exist.

Where you get tripped up badly on the realy difficult crosswods is when you don't see the ambiguity, and you then go where did I go wrong?"

WHere i screwed up on my first example is that i forgot to state that the second clue that seems to ahve two answers has only one cross clue. If you've got a clue with multiple answers that fits all cross clues, you've got a cross clue wrong, and it's because you answered another clue somewhere with an incorrect answer.