11. Can be done by an argument similar to the countability of the rationals. Every customer is assigned a pair (a, b), where a is the time they were seated, and b is their table number. Then we serve customer (a, b) at time T(a, b) = (a+b)(a+b+1)/2 + a. Every customer is served, and at every time only one customer is served (this can be shown as follows - if (a, b) and (p,q) are two customers, and if a+b < p+q, then T(a, b) < (a+b+1)(a+b+2)/2 <= T(p, q). If a+b = p+q, then if a < p, then also T(a, b) < T(p, q). All other cases are similar.).
Thank you for pointing me to this article - it is the best exposition of Hilbert's Hotel I have read so far.
One question -- Can one build this Hotel in 3d space such that there is a bound on how much every guest has to walk to their next room? Clearly one can do it for the case where a finite number of guests arrive, but I suspect this is not possible when the bus shows up.
That is a very interesting question! I have to think about it. You mean a finite uniform bound on the distance, but only for the current guests, and not for the bus or train passengers? I think this question is worthy of math.stackexchange.com, if you'd like to ask it there. Just link from here to the question if you do so, and if possible, link from the question to my substack.
It is possible. Consider in the 2D plane, with vertices on the natural number lattice (quarter plane). Let every person in the hotel currently move one step to the right. This frees up the entire first column for the people on the bus. A similar idea works in any higher dimension. So yes, we can limit the current guests to bounded distance movements.
Sounds like a fun class! Is there any course material that is available online?
4. Basically there are two states, on fire and not. From the on fire state, there is always a transition to being not on fire. From the not on fire state, there is only the transition to being on fire.
So at best, que of your moves should be to put the fire out, leaving your opponent no choice but to light it again, alternating states. So if it starts not on fire, you should go second.
Ah, sorry. I had given my students comp access, but otherwise, I am pursuing Substack as an alternative publication model, whereby I am serializing my books here as they are written for about the same (or less) as what one might pay for the book. It is a new way of publishing, which also enables this connection between author and reader. (Paid subscription also gets access to all my other serialized books in the archive.)
11. Can be done by an argument similar to the countability of the rationals. Every customer is assigned a pair (a, b), where a is the time they were seated, and b is their table number. Then we serve customer (a, b) at time T(a, b) = (a+b)(a+b+1)/2 + a. Every customer is served, and at every time only one customer is served (this can be shown as follows - if (a, b) and (p,q) are two customers, and if a+b < p+q, then T(a, b) < (a+b+1)(a+b+2)/2 <= T(p, q). If a+b = p+q, then if a < p, then also T(a, b) < T(p, q). All other cases are similar.).
Yes, very nice. And you used the Cantor polynomial pairing function, whereas many other people would use the back-and-forth zip-zag curve and draw a picture. You can see my discussion and comparison of these pairing functions in the post at https://www.infinitelymore.xyz/i/97164255/the-set-of-pairs-of-natural-numbers-is-countable.
Thank you for pointing me to this article - it is the best exposition of Hilbert's Hotel I have read so far.
One question -- Can one build this Hotel in 3d space such that there is a bound on how much every guest has to walk to their next room? Clearly one can do it for the case where a finite number of guests arrive, but I suspect this is not possible when the bus shows up.
That is a very interesting question! I have to think about it. You mean a finite uniform bound on the distance, but only for the current guests, and not for the bus or train passengers? I think this question is worthy of math.stackexchange.com, if you'd like to ask it there. Just link from here to the question if you do so, and if possible, link from the question to my substack.
It is possible. Consider in the 2D plane, with vertices on the natural number lattice (quarter plane). Let every person in the hotel currently move one step to the right. This frees up the entire first column for the people on the bus. A similar idea works in any higher dimension. So yes, we can limit the current guests to bounded distance movements.
Sounds like a fun class! Is there any course material that is available online?
4. Basically there are two states, on fire and not. From the on fire state, there is always a transition to being not on fire. From the not on fire state, there is only the transition to being on fire.
So at best, que of your moves should be to put the fire out, leaving your opponent no choice but to light it again, alternating states. So if it starts not on fire, you should go second.
11. Sounds like a job for a diagonal argument.
Many of the readings were based on my games posts here on Infinitely More. See the games posts at https://www.infinitelymore.xyz/s/infinite-games.
Thanks. Unfortunately I'm not a paid user.
Ah, sorry. I had given my students comp access, but otherwise, I am pursuing Substack as an alternative publication model, whereby I am serializing my books here as they are written for about the same (or less) as what one might pay for the book. It is a new way of publishing, which also enables this connection between author and reader. (Paid subscription also gets access to all my other serialized books in the archive.)