Combinatorial Game Theory
who wins, and by how much
Series

Heating — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Which of the two operators distributes over a sum. Cooling and heating, each asked whether applying it to a sum is the same as applying it to the parts and adding. The pools are the values born by day two and a set of deliberately hot positions; the counts are of ordered pairs.

    Cooling adds and heating does not

    The two operators are presented as a pair, and they are not one. Cooling a sum is the same as cooling the parts and adding, on every one of the 1,768 pairs tried, at two taxes and on two pools. Heating fails on 263 — and not for the obvious reason: in every failure neither part and not the sum is a number, so the clause exempting numbers never fires at the top. It fires two levels down, where an option of a sum is one part's option plus the whole of the other.

    part 1 · temperature
  2. Where the counterexample should have been. Cooling tested against a disjunctive sum on 107,816 ordered pairs: an 81-position pool of switches and numbers carrying infinitesimals at six taxes from a quarter to two, and a 185-value sample of day three at a half and one. Cooling never fails. On 6,032 pairs the sum freezes while a part does not, and on all of them the cooled parts add to exactly the frozen sum.

    Why cooling has to add

    Cooling came through a sweep of 1,768 sums untouched while heating failed 263 times, and the sweep's own account was cautious: cooling is not additive in general, and a richer pool would break it. Aimed exactly where that account said to look — infinitesimals hidden under low temperatures, at taxes of a quarter and a half — 107,816 more pairs break nothing, because there is nothing to break. Cooling distributes over every sum of short games. Its freezing clause fires on one side of the equation and not the other in 6,032 of those pairs, and costs nothing every time, because what it hands back is the mean, and means add.

    part 2 · temperature
  3. Seven clauses on one recursion. Seven operators built on the same recursion and differing only in the clause that stops it, each tested for f(G + H) = f(G) + f(H) on the 1,768 pairs of the earlier sweep. Failures: cooling 0, cooling with heating's clause 404, heating 263, heating exempting only integers 335, heating with a switch clause 1768, heating with no clause 0, Norton's product by t∗ 0.

    The bonus an integer is owed

    Heating fails to distribute over a sum where cooling never does, and the repair proposed for it was to change what its exception returns: heat a number into a switch instead of leaving it alone. Run on the same 1,768 pairs, that repair fails on every one of them, beginning with nought plus nought. What adds is a heating that changes which numbers it exempts and pays the bonus on the moves an integer holds in reserve — Norton's product, and heating with no exception at all, which at a whole-number bonus is the same operator. Given heating's exception, cooling stops adding too.

    part 3 · temperature
  4. Where exempting an integer starts to cost. Heating that leaves integers alone, tested for additivity at twelve bonuses from an eighth to two on two pools. It fails on no pair below a bonus of one and on dozens from one upward, while heating with no exemption fails at no bonus on these pools.

    An exemption that costs nothing below one

    Heating that leaves integers alone fails to add at a bonus of one and adds on everything at a bonus of a half, and the natural guess was that a half is where it starts to go wrong, since that is where a fraction's moves become worth fighting over. It is not. Across twelve bonuses nothing fails at any value below one, and dozens of pairs fail at one exactly. The reason is the simplicity rule: below one, the recursion sends every integer to itself, so the exemption is not doing anything at all.

    part 4 · temperature

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