The strategy doesn’t seem, however, to prevent a column win for the first player, but rather ensures that any such winning configuration is duplicated for the second player, forcing the draw. If the first player always plays on one of the paired columns, and you copy on the other, then both will make the winning configuration.

Incidentally, I think you probably mean $3m+1$, $3m+2$, instead of $2m+1$, etc. I’m not sure if users can edit comments here, but if you are not able to edit, I think I can change this for you.

]]>Maybe the players can force draw already when it is sufficient to arrange k number of coins in a row, where k is finite? If the board is Z^2 without gravity and the winning configuration is a 2×2 square, there is an easy proof that both players can force a draw (arrange cells in pairs like bricks, if the first player plays X, the second player plays its pair). Similarly, if the winning configuration is a vertical or horizontal line of five. I think this is also the case for a vertical or horizontal or diagonal line for some k, but I do not remember the details. With gravity it could be similar.

]]>https://doi.org/10.1007/s11856-018-1677-1]]>

Accepted, January 2019.]]>

Accepted, December 2018.]]>

Accepted, September 2018.]]>

The book by Hindman&Strauss has a section on filters and compactifications in a later chapter — that might be suitable for 2).

]]>Possibly relevant is my October 2004 sci.math post “Generalized Quantifiers” (URLs below). FYI, the Math Forum version has a lot of strange formatting errors. See also Brian Thomson’s 1985 book “Real Functions”, and see Thomson’s earlier 2-part survey Derivation bases on the real line (which contain examples and side-detours not in his book).

google sci.math URL:

https://groups.google.com/forum/#!msg/sci.math/rhZEhXynVLQ/MI0MJ0ZQIvoJ

Math Forum sci.math URL:

http://mathforum.org/kb/message.jspa?messageID=3556191

Regarding Moving around every few years for postdocs would be exhausting

, I wholeheartedly agree! Every such decision has pros and cons. They are usually not obvious at the outset. Moreover, they change over time. The lesson I learned though my own course is that every decision is right, at least at the time you make it, and there is never any point regretting it later on… Just keep on doing what you do best all the time!

We went on to look at the paradoxical situations that arise when someone gets a positive test result for a rare disease. Should they be worried? If the test is 99% accurate, but the disease occurs in say, 1 in million, then a positive result is not so worrisome: in million people, there will be about 1 true positive result, and about 10,000 false positives, since 1% of a million is 10,000. So the odds that you’ve actually got the disease, given that you tested positive, is 1 in 10,000.

But your situation is completely different! It would be as though we had calculated the odds of getting HHHTTT, and then when I actually flipped the coin on stage, i actually got the same pattern HHHTTT. Totally weird! And very unlikely. But you know, if it wasn’t that, it would have been some other totally unlikely thing, like getting all green lights, or all red lights, or getting the serial number 123456 on your receipt at Starbucks.

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