Revisiting the Mutilated Chessboard
solipsys.co.uk
solipsys.co.uk
This would be clearer if the word "closed" were inserted before "tour".
Cheers.
Really minor nit: You could drop one of your nows.
Tangentially, my sons told me that when they start card tricks with an unopened pack of cards and strongly highlight that fact like it proves the cards have not been tampered with, the trick there is that card decks all start in the same order. So an unopened deck will come with a particular card order and they can know the order of the cards and can use that fact to help them with their card trick. It is part of how they fool the audience.
> Really minor nit: You could drop one of your nows.
Good catch - change in the process of being uploaded.
WRT cards in an unopened deck, there are two different orderings that are used. Sometimes there are "kissing kings" and sometimes not. If you always use the same make of cards, probably the order is always the same.
What you say is true, but on its own it's not enough to be convincing.
In this case, if we consider the situation where we take a card from a pile, this leaves 3 cards left in the pile. We can't take another card from this pile this round, but there are only 3 cards left which guarantees that there is still one card available in another pile. This much is correct, but as you say it is not enough to prove the issue.
Once we take a card from one pile, three other cards are not available to draw this round. When we take another card, three more cards are not available. How do we know that there were not 2 of one card in the first pile and 2 of another card in the second pile (or some other annoying combination)? Indeed, this situation will exist. In order to prove the original assertion, you have to prove that there will always be another solution that will allow us to avoid doing that. That's much harder.
But let me begin by saying that it's true that a straight can always be drawn, and showing that someone's false argument of a true statement is indeed false is hard, because you can't give counter-examples. Even so, let's look at what's been said:
> barrkel: For dealing out the cards, it seems that the only thing that would stop you from drawing a straight via one card from each pile would be if any one pile had one card in all 4 suits, along with one of its adjacent cards in the straight; e.g. 4 of hearts, clubs, diamonds and spades, and any 5 or 3. (If you pluck off all the same-suited cards first, then you'll end up with two adjacent cards for the final draw in the same pile). Since the piles are constrained to have no more than 4 cards, this situation can't happen.
> ColinWright: That's very much an argument along the lines of: "I can't imagine anything going wrong, so it must be OK." What you say is true, but on its own it's not enough to be convincing.
So here barrkel is describing the only way it can go wrong. Why is this the only way it can go wrong? What if I choose a card from each pile bar one, and then find that the remaining pile doesn't have any of the final card I'm looking for? Why can't that happen? Well, it can happen, and I need to fix my previous choices. How do I know I can fix my previous choices? Why can't it be the case that my first 12 choices always leave me in this predicament? [X]
In short, the situation described is only one of the gazillion things that might go wrong, and there is no discussion of why they can't happen. That's why it's not convincing.
Then, to go on with your comment:
> if we consider the situation where we take a card from a pile, this leaves 3 cards left in the pile.
Yes.
> We can't take another card from this pile this round, but there are only 3 cards left which guarantees that there is still one card available in another pile.
I don't understand at all what you're trying to say here. What is available in the other piles? (Based on text later in your comment I suspect this doesn't matter.)
Added in edit: Ah - I think you're saying that for any card we want to pick next it must be available somewhere, because there are four of them, and only three can be in the now-forbidden pile. OK.
> This much is correct, but as you say it is not enough to prove the issue.
> Once we take a card from one pile, three other cards are not available to draw this round.
Yes.
> When we take another card, three more cards are not available.
Yes.
> How do we know that there were not 2 of one card in the first pile and 2 of another card in the second pile (or some other annoying combination)? Indeed, this situation will exist. In order to prove the original assertion, you have to prove that there will always be another solution that will allow us to avoid doing that. That's much harder.
Yes, exactly.
So you have outlined one way that it can all go wrong that isn't covered by the original argument put forth by barrkel. I hope you agree that my comment [X] above also provides enough explanation of why barrkel's argument is unconvincing.
Anyway, I enjoyed your blog post and it gave me lots to think about. Thanks for doing it.
But this is a standard problem when teaching proof in math. All too often a proposed "proof" simply doesn't prove the theorem, and if the theorem is actually true it can be extremely difficult to explain why the "proof" is wrong, inadequate, or incomplete. The most common form is "Well, I can't see how anything could go wrong." and that's notoriously hard to get around.
But I'm pleased you posted, and I'm pleased you enjoyed the post. Thank you.
You might be right that it can't go wrong like that, but why can't it go wrong in other ways?
> barrkel: I could flesh it out, but I think there's enough in my comment to get all the way for anyone interested.
I'm letting you know that this turns out not to be the case, for I am interested, and I've read your comment in detail, and for me, there isn't enough there to get all the way.
But you're right that it's not your job to convince me of anything, I'm just trying to let you know that, in my opinion, for what it's worth, you haven't solved the challenge.