The thread: How much is at stake — page 5
What it costs
The cheap fights make the rule cheaper
A conjecture stands that playing the hottest part costs at most the coolest temperature times the number of parts sharing it — proposed on a range where that number never exceeds two. Swept to five-part boards over 10,410 lines it is false, and false the other way round: every line costing more than the coolest part has one or two parts at that temperature, and over the 3,230 lines with three or more, not one does.
Where it stops
The coldest position has the biggest swing
How much a flip is worth is the gap between the coin's two branches, and it is a rival to the temperature — both answer how much is at stake. They disagree at once: the empty position has the lowest temperature there is and a swing of one, twice the hottest thing born on day two. On the small pool the two quantities look like a perfect three-way correspondence, and 255 of day three's values break it.
Out in the world
Four boxes for every chain after the first
Nimstring answers who is forced to open and says nothing about the score. The margin has a formula: the controller nets the total less four boxes for every long chain after the first — two surrendered and two not taken, each time control is kept. Checked against the solver on 175 endgames it is exact on 172, never too generous, and exact wherever it promises the controller anything at all. The three it misses are the three where it promises nothing.