Unscheduled Post: A couple of interesting posts
This is a glimpse into my browsing habits. It starts with an interesting post on the Secret Blogging Seminar, on bidding games. The basic idea is to take any game, add a stack of chips and allow the players to bid to make each move.
This gives a new angle to games from Chess to noughts and crosses (tic-tac-toe in the US). These games provide some interesting variants (bid hex is already on Facebook), it also opens up possibilities for mathematical analysis. This excellent paper, does that job very well coming up with delightfully weird results for example in bidding noughts and crosses the centre is an optimal move if and only if the number of starting chips is not 5! (Theorem 6.7 P 26)
It also has an appendix on methods of tie breaking, including chips and and claiming:
Coin Flips, and taking turns nicely. Some tie-breaking methods are not worth discussing.
Coin flips, however are not ruled out entirely. The deepest result of the paper shows that, if bids are allowed to very continuously (allowing any number rather than just whole numbers) then bidding games become equivalent to random turn games (where the next player to move is determined by coin toss).
Random turn games led me to an Ivars Petersons’ blog post, describing random turn Hex. It turns out that this is actually easier to analyse than the same game with alternating moves. Hex is a famous example as it can be shown that there is a winning strategy for the player who moves first (as every game must end and no game ends in a draw), however finding the strategies for large boards is a very difficult problem. Peterson describes a paper that shows that the chance of winning random turn hex (with both players playing optimally) is the same as the chance of winning where moves alternate but are made at random.
Both these papers describe toy examples, however both bidding games and random turn games describe real world conflicts where the right to move can depend on chance, or on how much one is willing to invest. From this point of view bidding games where bids are lost (whether or not you win) might also be interesting.
It was hard for me to believe that I had not already come across Ivars Peterson’s blog, and I look forward to reading the archives. However the latest post really caught my eye. He has dug through the photo archives of LIFE magazine, and unearthed some mathematical gems. These pictures give some glimpses into the history of our mathematical culture. In particular, a couple show geometric models and an incredible collection of tools to teach plane and solid geometry. What has happened to these models and toys? I am reminded of the fact that when Sophus Lie was travelling from Norway his travels where usually funded by the university to collect mathematical models rather than do research. Today, do any universities actively maintain libraries of mathematical models? As I have commented before and will almost certainly do again, these are an important way both to draw people into mathematics and to generate the intuition to do research in many areas.